Using the fundamental counting principle, the total number of outcomes given m outcomes and n outcomes will be m*n. A helpful way to think about this is by using a tree.
Say we have 2 shirts and 3 pairs of pants. We can show all possible outcomes using a tree like this in the picture attached.
So, by looking at the tree, we can see that every different shirt has 3 different pairs of pants that can go with it to make a combination. Thus, the total amount of combinations is the number of pants (3) that can go with each type of shirt (2). So, 3*2 is 6 total combinations.
In this example, m was 2 and n was 3. Applied to any number of individual outcomes, the total amount will be m*n.
Ben practises playing the Oboe daily.
The time (in minutes) he spends on
daily practice over 28 days is as follows:
10, 15, 30, 35, 40, 40, 45, 55, 60, 62,
64, 64, 66, 68, 70, 70, 72, 75, 75, 80,
82, 84, 90, 90, 105, 110, 120, 180
a Find the median time.
b
Find the lower quartile.
c Find the upper quartile.
d
Find the range.
Determine whether there
outliers in the data.
e
(2 marks)
(2 marks)
(2 marks)
(2 marks)
are any
(4 marks)
f Draw a box-and-whisker diagram for
the above data.
(3 marks)
Therefore, (70+72)/2 = **71 minutes** is the median time. B)42.5 minutes as a result. C,D)The range is determined by deducting the dataset's smallest value from highest value.
A)When the data are organized in order of magnitude, the median time is the middle value. The median in this situation is the average of the 14th and 15th values, which are 70 and 72, respectively. There are 28 data points in this situation. Therefore, (70+72)/2 = **71 minutes** is the median time.
b) The median of the lowest half of the data constitutes the lower quartile (Q1). We must arrange the data in descending order of magnitude before determining the median of the first half of the data in order to determine Q1. is the average of the seventh and eighth values, which are 40 and 45, respectively, in the first half of the data, which consists of 14 values. Q1 = (40+45)/2 = **42.5 minutes as a result.
b) The median of the upper half of the data constitutes the upper quartile (C). We must first organise the data in descending order of magnitude before determining the median of the remaining data in order to determine Q3. Q3 is the average of the seventh and eighth values from the last, which are 90 and 105, respectively, in the second half of the data, which consists of 14 values. Q3 = (90+105)/2 = **97.5 minutes**3 as a result.
d) The range is determined by deducting the dataset's smallest value from highest value.
In this instance, Ben's practise time can be anywhere from **10 minutes** to **180 minutes**. Range then equals maximum value - minimum value, which in this case is 180 - 10 = **170 minutes**
e) Extreme values that are beyond the typical range of a dataset's values are known as outliers. We can use a criterion that states that any value that sits more than 1.5 times the interquartile range (IQR) below Q1 or above Q3 is regarded as an outlier to ascertain whether there are outliers in this dataset. When Q1 is subtracted from Q3, the result is the IQR: Q3 - Q1 = 97.5 - 42.5 = **55 minutes**3. By using this rule, we can see that the dataset contains the outliers **180** and **120** minutes.
IQR stands for what?The term "interquartile range" is IQR. It is a measure of variability that is based on quartilizing a dataset. The first quartile (Q1) is subtracted from the third quartile to determine the IQR. (Q3). It is a representation of the middle 50% of the data's range.
f) A box-and-whisker plot illustrates a dataset's quartiles, outliers, and range1. For Ben's practice, here's how to create a box-and-whisker plot:
- Create a number line with all the values Ben practised with.
- Draw a box spanning Q1 through Q3.
- Inside the box, at the location of Q2, draw a vertical line. (the median).
Draw whiskers from the box's two ends to all values that are not outliers.
- Place every outlier on the graph as a separate point, outside of any whiskers.
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Use Newton's method to approximate a root of the equation5sin(x)=xas follows. Letx1=1 be the initial approximation. The second approximationx2 is and the third approximationx3 is
The second approximation x2 is approximately 1.112141637097, and the third approximation x3 is approximately 1.130884826739.
Newton's method to approximate a root of the equation 5sin(x) = x.
We are given the initial approximation x1 = 1. To find the second approximation x2 and the third approximation x3, we need to follow these steps:
Step 1: Write down the given function and its derivative. f(x) = 5sin(x) - x f'(x) = 5cos(x) - 1
Step 2: Apply Newton's method formula to find the next approximation. x_{n+1} = x_n - f(x_n) / f'(x_n)
Step 3: Calculate the second approximation x2 using x1 = 1. x2 = x1 - f(x1) / f'(x1) x2 = 1 - (5sin(1) - 1) / (5cos(1) - 1) x2 ≈ 1.112141637097
Step 4: Calculate the third approximation x3 using x2. x3 = x2 - f(x2) / f'(x2) x3 ≈ 1.112141637097 - (5sin(1.112141637097) - 1.112141637097) / (5cos(1.112141637097) - 1) x3 ≈ 1.130884826739
So, the second approximation x2 is approximately 1.112141637097, and the third approximation x3 is approximately 1.130884826739.
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(1 point) Write an equivalent integral with the order of integration reversed ST 2-3 F(x,y) dydc = o g(y) F(x,y) dedy+ So k(y) F(x,y) dardy Jh(v) a- he C- f(y) = g(y) = h(g) = k(y) =
equivalent integral with the order of integration reversed ST 2-3 F(x,y) dydc = o g(y) F(x,y) dedy+ So k(y) F(x,y) dardy Jh(v) a- he C- f(y) = g(y) = h(g) = k(y) = By reversing the order of integration, you've found an equivalent integral to the original one provided.
step-by-step explanation to achieve this, using the terms "integral," "reversed," and "equivalent" in the answer.
Step 1: Identify the original integral
The original integral is given as ∫∫ F(x, y) dy dx, where the integration limits are not explicitly provided. In this case, let's assume the limits of integration for y are from a(x) to b(x), and for x, they are from c to d.
Step 2: Sketch the region of integration
To reverse the order of integration, it's helpful to sketch the region of integration, which is the area in the xy-plane where the function F(x, y) is being integrated.
Step 3: Determine the new limits of integration
After sketching the region, determine the new limits of integration by considering the range of x for a given y value, and the range of y values. Let's assume the new limits for x are from g(y) to h(y), and for y, they are from e to f.
Step 4: Write the equivalent reversed integral
Now, you can write the equivalent integral with the order of integration reversed. In this case, it will be ∫∫ F(x, y) dx dy, with the new limits of integration. The complete reversed integral will look like:
∫(from e to f) [ ∫(from g(y) to h(y)) F(x, y) dx ] dy
By reversing the order of integration, you've found an equivalent integral to the original one provided.
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The harmonic series: 1+1/2+1/3+1/4+.
diverges, but when its terms are squared the resulting series converges. T or F
The statement "The harmonic series: 1+1/2+1/3+1/4+... diverges, but when its terms are squared the resulting series converges." is True.
The harmonic series is defined as the sum of the reciprocals of the natural numbers: Σ(1/n) for n = 1 to ∞. This series is known to diverge, meaning that its sum tends to infinity as more terms are added.
However, when the terms of the harmonic series are squared, we get a new series called the p-series, with p=2: Σ(1/n^2) for n = 1 to ∞. The p-series converges if p > 1, which is true for p=2. Thus, the series Σ(1/n^2) converges to a finite sum.
In conclusion, the given statement is true, as the harmonic series diverges, but its squared terms result in a convergent series.
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6. Which of the following equations would have no
solution?
F. 13 - 7x = -7x + 13
G.1/3(6x + 9) = 12
H. 1/4(8x + 4) = 2x - 4
J. -10x + 5 = 3 - 10x + 2
Answer: F, H, and J all have no real solution. The only equation that has a solution is
Step-by-step explanation: Use foil method.
Find the lateral surface area. Bases are isosceles triangles.
29 110 56
To find the lateral surface area of a prism with isosceles triangle bases, you'll need the following information: the slant height and the perimeter of the base.
Based on the numbers you provided (29, 110, and 56), it appears that you have the dimensions of an isosceles triangle with side lengths 29, 29, and 110 units. To find the slant height, we can use the Pythagorean theorem on one of the right triangles formed by the base and the altitude (height) of the isosceles triangle. Let's call the height h and the slant height s.
(1/2 * 110)^2 + h^2 = 29^2
3025 + h^2 = 841
h^2 = 841 - 3025 = -2184 (invalid, as there cannot be a negative height)
It seems like there is an error in the provided dimensions, as the side lengths do not form a valid isosceles triangle. Please double-check the dimensions and provide the correct information so I can help you find the lateral surface area.
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Tammy knits blankets and scarves. On the first day of a craft fair, she sells 2 blankets and 5 scarves for $104. On the second day of the craft fair, she sells 3 blankets and 4 scarves for $128. How much does 1 blanket cost?
The cost of one blanket after calculations sums up as $32.
Let b be the cost of one blanket and s be the cost of one scarf in dollars. We can set up a system of equations based on the information given:
2b + 5s = 104
3b + 4s = 128
We want to solve for the cost of one blanket, so we'll solve for b in terms of s. We can start by multiplying the first equation by 3 and the second equation by 2 to create a system of equations where the coefficients of b will cancel each other out when we subtract the two equations:
6b + 15s = 312
6b + 8s = 256
Subtracting the second equation from the first, we get:
7s = 56
Dividing both sides by 7, we get:
s = 8
Now we can substitute s = 8 into either of the original equations to solve for b:
2b + 5(8) = 104
2b + 40 = 104
2b = 64
b = 32
Therefore, one blanket costs $32.
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Please help!!! you are painting the roof of a shed that is 35 ft from the ground. you are going to place the base of a
ladder 12 ft from the shed. how long does the ladder need to be to reach the roof of the shed? use pencil and
paper. explain how shortening the distance between the ladder and the shed affects the height of the ladder. the ladder needs to be ____ ft long to reach the roof of the shed.
To find the length of the ladder needed to reach the roof of the shed that is 35 ft from the ground with the base of the ladder 12 ft from the shed, you can use the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the ladder, in this case) is equal to the sum of the squares of the other two sides (the height and the distance from the shed).
Step 1: Identify the sides of the triangle.
- Height (a): 35 ft (vertical side)
- Distance from the shed (b): 12 ft (horizontal side)
- Ladder length (c): Hypotenuse
Step 2: Apply the Pythagorean theorem.
- a² + b² = c²
- 35² + 12² = c²
Step 3: Calculate the squares and sum them.
- (35 * 35) + (12 * 12) = c²
- 1225 + 144 = c²
- 1369 = c²
Step 4: Find the length of the ladder (c).
- c = √1369
- c = 37
The ladder needs to be 37 ft long to reach the roof of the shed.
Shortening the distance between the ladder and the shed will affect the height of the ladder by making it steeper. This will cause the ladder to be higher above the ground, but it may also make it less stable and more difficult to climb.
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Can someone help answers this! Remember to Fill in the Drop Boxes
The line y=10x will in this instance pass through most of the data points, demonstrating that it is a good fit for the data.
A good line of fit should travel across the greatest number of data points and exhibit a positive connection.
What exactly is a scatter plot?A relationship between two variables in which rising values of one cause rising values of the other. On a scatter plot, it is shown as a positive slope.
The line y=10x will in this instance pass through most of the data points, demonstrating that it is a good fit for the data.
The line will be favourably sloped, so as the duration of an accessible bike rental increases, so does the total cost charged.
The scatterplot confirms this, proving that the line y=10x is a good match for the data.
This indicates that the data points are nearly aligned with the line but not exactly so.
A good line of fit should travel across the greatest number of data points and exhibit a positive connection.
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for an arc length s, area of sector a, and central angle of a circle of radius r, find the indicated quantity for the given value.
r= 4.27 m, 0 = 2.16, s = ?
s=
(do not round until the final answer. then round to two decimal places as needed.)
The arc length (s) for a circle with radius 4.27 meters and central angle 2.16 radians is approximately 9.22 meters.
To find the arc length (s) for a circle with radius (r) and central angle (θ), you can use the formula:
s = r * θ
In this case, the radius (r) is 4.27 meters, and the central angle (θ) is 2.16 radians. Plug these values into the formula:
s = 4.27 * 2.16
Now, multiply the values:
s ≈ 9.2232
Round the answer to two decimal places:
s ≈ 9.22 meters
So, the arc length (s) for a circle with radius 4.27 meters and central angle 2.16 radians is approximately 9.22 meters.
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About 20 years ago, a mathematician noted that his dog, when retrieving a
frisbee in a lake, would run parallel to the shore for quite some distance, and then jump into the water and
swim straight for the frisbee. She would not enter the lake immediately, nor would she wait until she was on
the point on the shore closest to the frisbee. Pennings theorized that the dog entered the water at the point
that would minimize the total length of time it takes to reach the frisbee. Suppose that the dog runs at 13
mph along the shore of the lake but swims at only 4. 3 mph in the water. Further, suppose that the frisbee is
in the water 60 feet off shore and 220 feet down the shoreline from the dog. Suppose that the dog enters the
water after running x feet down the shoreline and then enters the water. Compute the total length of time, T,
it will take for the dog to reach the frisbee. Next, determine a natural closed interval that limits reasonable
values of x. Finally, find the value of x that will minimize the time, T, that it takes for the dog to retrieve the
frisbee
a. The total length of time, T, it will take for the dog to reach the frisbee is 143.22
b. A natural closed interval that limits reasonable values of x is [0, 220] is a reasonable closed interval for x.
c. The value of x that will minimize the time, T, that it takes for the dog to retrieve the frisbee is 143.22
Let's start by breaking down the problem into two parts: the time it takes for the dog to run along the shore, and the time it takes for the dog to swim in the water. Let's call the distance the dog runs along the shore "d1" and the distance the dog swims in the water "d2".
To find d1, we can use the Pythagorean theorem:
d1 = sqrt(x^2 + 60^2)
To find d2, we can use the fact that the total distance the dog travels is equal to 220 feet:
d2 = 220 - x
Now we can use the formulas for distance, rate, and time to find the total time it takes for the dog to retrieve the frisbee:
T = d1/13 + d2/4.3
Substituting our expressions for d1 and d2, we get:
T = [sqrt(x^2 + 3600)]/13 + (220 - x)/4.3
To find the value of x that minimizes T, we can take the derivative of T with respect to x, set it equal to zero, and solve for x:
dT/dx = x/13sqrt(x^2 + 3600) - 1/4.3 = 0
Multiplying both sides by 13sqrt(x^2 + 3600), we get:
x = (13/4.3)sqrt(x^2 + 3600)
Squaring both sides and solving for x, we get:
x ≈ 143.22
So the dog should enter the water after running about 143.22 feet down the shoreline to minimize the total time it takes to retrieve the frisbee.
To check that this is a minimum, we can take the second derivative of T with respect to x:
d^2T/dx^2 = (13x^2 - 46800)/(169(x^2 + 3600)^(3/2))
Since x^2 and 3600 are both positive, the numerator is positive when x is not equal to zero, and the denominator is always positive. Therefore, d^2T/dx^2 is always positive, which means that x = 143.22 is indeed the value that minimizes T.
As for the natural closed interval that limits reasonable values of x, we know that x has to be greater than zero (since the dog needs to run at least some distance along the shoreline before entering the water), and it has to be less than or equal to 220 (since the frisbee is 220 feet down the shoreline from the dog). So the interval [0, 220] is a reasonable closed interval for x.
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The measures of the angles of a triangle are shown in the figure below. Solve for x.
(2x+16) 48degrees
Answer:
x = 16
Step-by-step explanation:
(2x + 16) = 48
Subtract 16 with the positive 16 to cancel the numbers.
Subtract 16 with 48.
2x = 32
divide 32 by 2 to isolate the x.
32/2 = 16
x = 16
Question 15 of 25
Suppose f(x)=x² and g(x) = (3x)2. Which statement best compares the graph
of g(x) with the graph of f(x)?
A. The graph of g(x) is shifted 3 units to the right.
B. The graph of g(x) is vertically stretched by a factor of 3.
C. The graph of g(x) is horizontally stretched by a factor of 3.
D. The graph of g(x) is horizontally compressed by a factor of 3.
← PREVIOUS
SUBMIT
Answer:
The function g(x) = (3x)² can be simplified to g(x) = 9x², which is a vertical stretch of f(x) = x² by a factor of 9.
Therefore, the correct answer is B. The graph of g(x) is vertically stretched by a factor of 3 compared to the graph of f(x).
The table gives a set of outcomes and their probabilities. Let a be the event "the outcome is a divisor of 4". Let b be the event "the outcome is prime". Find p(a|b)
The probability that the outcome is a divisor of 4 given that it is prime is 0.125, or 12.5%.
Since we are given the probabilities of different outcomes, we can use the definition of conditional probability to find p(a|b), which represents the probability that the outcome is a divisor of 4 given that it is prime.
The formula for conditional probability is:
p(a|b) = p(a ∩ b) / p(b)
where p(a ∩ b) represents the probability of both events happening simultaneously.
Looking at the table of outcomes and their probabilities, we can see that there are four prime numbers: 2, 3, 5, and 7. Of these, only 2 is a divisor of 4.
Therefore, p(a ∩ b) is the probability that the outcome is 2, which is 0.1.
The probability of the outcome being prime is the sum of the probabilities of the four prime outcomes, which is:
p(b) = 0.1 + 0.2 + 0.3 + 0.2 = 0.8
Substituting these values into the formula for conditional probability, we get:
p(a|b) = p(a ∩ b) / p(b) = 0.1 / 0.8 = 0.125
Therefore, the probability that the outcome is a divisor of 4 given that it is prime is 0.125, or 12.5%.
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HELP ME PLEASE I DON'T UNDERSTAND
Answer:
34/73
Step-by-step explanation:
37 + 34 + 2 = number of customers = 73
73 is our denominator.
34 is the number of people who used a credit card.
34 is our numerator.
Put the two together, and you get 73! Enjoy!
1)You have a monthly income of $2,800 and you are looking for an apartment. What is the maximum
amount you should spend on rent?
2)You have a monthly income of $1,900 and you are looking for an apartment. What is the maximum
amount you should spend on rent?
3)An apartment you like rents for $820. What must your monthly income be to afford this apartment?
4)An apartment you like rents for $900. What must your monthly income be to afford this apartment?
5)An apartment rents for $665/month. To start renting, you need the first and last month's rent, and a
$650 security deposit.
1) The maximum amount you should spend on rent is $840.
2) The maximum amount you should spend on rent is $570.
3) Your monthly income must be at least $2,733.33 to afford this apartment.
4) Your monthly income must be at least $3,000 to afford this apartment.
5) You need $1,980 to start renting the apartment.
1) With a monthly income of $2,800, the maximum amount you should spend on rent can be calculated using the 30% rule.
$2,800 x 0.30 = $840
So, the maximum amount you should spend on rent is $840.
2) With a monthly income of $1,900, the maximum amount you should spend on rent can be calculated using the 30% rule.
$1,900 x 0.30 = $570
So, the maximum amount you should spend on rent is $570.
3) To afford an apartment that rents for $820, your monthly income should be:
$820 ÷ 0.30 = $2,733.33
So, your monthly income must be at least $2,733.33 to afford this apartment.
4) To afford an apartment that rents for $900, your monthly income should be:
$900 ÷ 0.30 = $3,000
So, your monthly income must be at least $3,000 to afford this apartment.
5) To start renting an apartment that costs $665/month, you need the first and last month's rent, and a $650 security deposit.
First and last month's rent: $665 x 2 = $1,330
Total amount needed: $1,330 + $650 = $1,980
So, you need $1,980 to start renting the apartment.
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A report states that 1% of college degrees are in mathematics. A researcher doesn't believe this is correct. He samples 12,317 graduates and finds that 148 have math degrees. Test the claim at 0. 10 level of significance
We have evidence to suggest that the true percentage of college degrees in mathematics is different from 1%.
What is null hypothesis?The null hypothesis is a type of hypothesis that explains the population parameter and is used to examine if the provided experimental data are reliable.
To test the claim that the percentage of college degrees in mathematics is not 1%, we can use a hypothesis test. Let's assume the null hypothesis is that the true percentage of college degrees in mathematics is 1%, and the alternative hypothesis is that it is different from 1%.
- Null hypothesis: The percentage of college degrees in mathematics is 1%.
- Alternative hypothesis: The percentage of college degrees in mathematics is different from 1%.
We can use a binomial distribution to model the number of graduates with math degrees in a sample of 12,317. Under the null hypothesis, the expected number of graduates with math degrees is:
Expected value = sample size * probability of math degrees = 12,317 * 0.01 = 123.17
Since we are testing at a 0.10 level of significance, the critical values for a two-tailed test are ±1.645 (using a standard normal distribution table).
The test statistic can be calculated as:
z = (observed value - expected value) / standard deviation
The standard deviation of the binomial distribution can be calculated as:
√(sample size * probability of success * (1 - probability of success))
So,
standard deviation = √(123.17 * 0.01 * 0.99) = 1.109
The observed value is 148.
The test statistic is:
z = (148 - 123.17) / 1.109 = 22.38
Since the absolute value of the test statistic is greater than 1.645, we can reject the null hypothesis at the 0.10 level of significance.
Therefore, we have evidence to suggest that the true percentage of college degrees in mathematics is different from 1%.
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Using more advanced technology, a team of workers began to produce 6 more parts per hour than before. In six hours, the team produced 120% of what they had previously been able to produce in eight hours. How many parts per hour was the team producing prior to switching to the new technology?
Answer: Therefore, the team was producing 10 parts per hour prior to switching to the new technology.
Step-by-step explanation:Let's denote the number of parts produced per hour before the technology upgrade by x.
After the upgrade, the team produces 6 more parts per hour than before, so their new production rate is x + 6 parts per hour.
In 8 hours, the team produces 8x parts in total.
In 6 hours with the new technology, the team produces 120% of what they previously produced in 8 hours, or 1.2(8x) = 9.6x parts in total.
We can set up an equation based on the information above:
6(x + 6) = 9.6x
Simplifying the equation:
6x + 36 = 9.6x
Subtracting 6x from both sides:
36 = 3.6x
Dividing both sides by 3.6:
x = 10
pleaseeeeee help asapppp
Solve by graphing:
(x - 2)² = 9
Thanks!
The graph shows the height of a scratch on the edge of a circular gear.
Which function is the best model for the height of the scratch?
a. h(t) = 3.5 sin (π t) + 1.5
b. g(t) = 1.5 sin (π t) +3.5
c. h(t) = 1.5 sin (2 π t) + 3.5
d. h(t) = 1.5 sin (π/2 t) + 3.5
Answer:
b. g(t) = 1.5 sin (π t) +3.5
Step-by-step explanation:
You want to choose the function that has the given graph.
Test pointsAt t = 0, the graph shows a value of 3.5. The sine of 0 is 0, so this eliminates choice A.
At t = 1/2, the graph shows a value of 5. The values given by the different formulas are ...
b. g(1/2) = 1.5·sin(π/2) +3.5 = 5 . . . . . matches the graph
c. h(1/2) = 1.5·sin(π) + 3.5 = 3.5 . . . . no match
d. h(1/2) = 1.5·sin(π/4) +3.5 = 0.75√2 +3.5 . . . . no match
__
Additional comment
The horizontal distance for one period of the graph (from peak to peak, for example) is T = 2 seconds. If the sine function is sin(ωt), then the value of ω is ...
ω = 2π/T = 2π/2 = π
This tells you the function g(t) = 1.5·sin(πt)+3.5 is the correct choice.
What is the exact value of sin−1(−12)? Enter your answer in the box. Sin−1(−12) = 1$$ Correct answers: 1−π6
The exact value of sin⁻¹(−1/2) is -π/6.
Given, sin⁻¹(-1/2)
The inverse sine function, sin⁻¹, or arcsin, returns the angle whose sine is equal to the given value. In this case, we are looking for the angle whose sine is -1/2.
Let y = sin⁻¹(-1/2)
sin (y) = -1/2
sin (y) = - sin (π/6)
sin (y) = sin (- π/6)
y = - π/6
sin⁻¹(-1/2) = - π/6
To understand why the answer is -π/6, we can consider the unit circle. On the unit circle, the sine function represents the y-coordinate of a point corresponding to an angle. For -1/2, we need to find the angle where the y-coordinate is -1/2.
One such angle is -π/6, where the point on the unit circle is located in the fourth quadrant. At this angle, the y-coordinate is -1/2. Hence, sin⁻¹(−1/2) is -π/6.
Therefore, the exact value of sin⁻¹(−1/2) is -π/6.
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A triangular frame is being built as the support for a ramp. The longest part of the
frame will sit on the ground. The second longest side is 2'3" and forms an 18°
angle with ground. The smallest side is 10" long. Determine the angle the
smallest side will make with the ground.
The smallest side of the triangle makes an angle of approximately 20.6 degrees with the ground.
To determine the angle the smallest side will make with the ground, we can use the law of sines. The law of sines states that for any triangle ABC:
a/sin(A) = b/sin(B) = c/sin(C)
where a, b, and c are the lengths of the sides opposite the angles A, B, and C, respectively.
Let's label the sides of our triangle as follows:
The longest side (sitting on the ground) is side c
The second longest side is side b
The smallest side is side a
We know that side b is 2'3" long, which is equivalent to 27 inches. We also know that side a is 10 inches long. We can use the law of sines to solve for the angle opposite side a:
sin(A) = (a/c) * sin(C)
We can solve for sin(C) by using the fact that the sum of the angles in any triangle is 180 degrees:
C = 180 - A - B
We know that angle B is 18 degrees, so we can substitute that into our equation for C:
C = 180 - A - 18
C = 162 - A
Substituting this expression for C into our equation for sin(A), we get:
sin(A) = (a/c) * sin(162 - A)
We know that c is the longest side of the triangle and therefore opposite the largest angle. Since we are interested in the angle opposite side a, we can assume that angle A is the smallest angle in the triangle. We can use this assumption to simplify our equation for sin(A):
sin(A) = (a/c) * sin(162)
Plugging in the values for a, c, and sin(162), we get:
sin(A) = (10/27) * 0.951
sin(A) = 0.352
Taking the inverse sine of both sides, we get:
A = sin^-1(0.352)
A ≈ 20.6 degrees
Therefore, the smallest side of the triangle makes an angle of approximately 20.6 degrees with the ground.
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How do you do this problem?
Answer: 135 and 45
Step-by-step explanation:
We can read off from these equations the gradients of the two lines: (3) and (-2).
Then we quote the trigonometric identity tan(A-B) = [tan(A)-tan(B)] / [1+tan(A)tan(B)]
Substituting tan(A)=3 and tan(B)=-2 gives tan(A-B) = [(3)-(-2)] / [1+(3)(-2)] = 5/-5 = -1
So A-B = 135°.
That is the obtuse angle between the two lines, so the acute angle is 45°.
At a print shop reams of printer paper are stored in boxes in a closet. Each box contains 12 reams of printer paper. A worker uses 4 reams from 1 of the boxes. Which function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet?
The function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet is y = 12x - 4
Let's start by considering the initial amount of printer paper in the closet before any boxes are used. Since each box contains 12 reams of printer paper, if there are x boxes in the closet, then the total number of reams of paper is given by 12x.
Now, if a worker uses 4 reams from one of the boxes, then the total number of reams of paper remaining in the closet is (12x - 4). If we define y as the total number of reams of paper remaining in the closet, then we have:
y = 12x - 4
This function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet.
As x increases, the total number of reams of paper in the closet increases as well. However, each time a worker uses 4 reams of paper from a box, the total number of reams of paper in the closet decreases by 4.
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Answer this question fast please
The probability that a randomly selected student prefers the arts or does not prefer literature is 8/11.
There are different ways to approach this problem, but one possible method is to use the concept of complement events.
First, we can calculate the probability of a randomly selected student preferring the arts. This is simply the proportion of students in the sample who prefer the arts, which is 9 out of 3+9+10 = 22. So, the probability is:
P(arts) = 9/22
Next, we can calculate the probability of a randomly selected student preferring literature. This is the proportion of students in the sample who prefer literature, which is 7+8 = 15 out of 22. So, the probability is:
P(literature) = 15/22
To find the probability of a student preferring the arts or not preferring literature, we can use the complement event that consists of students who do not prefer literature. This is the complement of the event "preferring literature", and its probability is:
P(not literature) = 1 - P(literature) = 1 - 15/22 = 7/22
Finally, we can use the addition rule for disjoint events (i.e., events that cannot occur at the same time) to calculate the probability of the event "preferring the arts or not preferring literature".
Since these events are not mutually exclusive (i.e., some students may prefer both the arts and literature), we need to subtract their intersection (i.e., students who prefer both) to avoid double-counting. Therefore, the probability is:
P(arts or not literature) = P(arts) + P(not literature) - P(arts and literature)
= 9/22 + 7/22 - 0
= 16/22
= 8/11
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Evaluate the definite integrals ∫(9x^2 - 4x - 1)dx =
Definite integral of ∫(9x^2 - 4x - 1)dx from a to b is 3(b^3 - a^3) - 2(b^2 - a^2) - (b - a).
To evaluate the definite integral ∫(9x^2 - 4x - 1)dx, you need to first find the indefinite integral (also known as the antiderivative) of the function 9x^2 - 4x - 1. The antiderivative is found by applying the power rule of integration to each term separately:
∫(9x^2)dx = 9∫(x^2)dx = 9(x^3)/3 = 3x^3
∫(-4x)dx = -4∫(x)dx = -4(x^2)/2 = -2x^2
∫(-1)dx = -∫(1)dx = -x
Now, sum these results to obtain the antiderivative:
F(x) = 3x^3 - 2x^2 - x
∫(9x^2 - 4x - 1)dx from a to b = F(b) - F(a)
To evaluate the definite integral ∫(9x^2 - 4x - 1)dx =, we need to use the formula for integrating polynomials. Specifically, we use the power rule of integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration.
Using this formula, we integrate each term in the given expression separately. Thus, we have:
∫(9x^2 - 4x - 1)dx = (9∫x^2 dx) - (4∫x dx) - ∫1 dx
= 9(x^3/3) - 4(x^2/2) - x + C
= 3x^3 - 2x^2 - x + C
Next, we need to evaluate this definite integral. A definite integral is an integral with limits of integration, which means we need to substitute the limits into the expression we just found and subtract the result at the lower limit from the result at the upper limit. Let's say our limits are a and b, with a being the lower limit and b being the upper limit. Then, we have:
∫(9x^2 - 4x - 1)dx from a to b = [3b^3 - 2b^2 - b] - [3a^3 - 2a^2 - a]
= 3(b^3 - a^3) - 2(b^2 - a^2) - (b - a)
Therefore, the definite integral of ∫(9x^2 - 4x - 1)dx from a to b is 3(b^3 - a^3) - 2(b^2 - a^2) - (b - a).
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Abc company’s budgeted sales for june, july, and august are 12,800, 16,800, and 14,800 units, respectively. abc requires 30% of the next month’s budgeted unit sales as finished goods inventory each month. budgeted ending finished goods inventory for may is 3,840 units. each unit that abc company produces uses 2 pounds of raw material. abc requires 25% of the next month’s budgeted production as raw material inventory each month.
The budgeted ending raw material inventory for May is 2,560 pounds, calculated by taking 25% of the next month's budgeted production (12,800 units) multiplied by 2 pounds per unit.
To solve this problem, we need to calculate the budgeted production and raw material inventory for June, July, and August.
For June:Budgeted production = 12,800 units + 30% * 16,800 units = 17,440 units
Raw material inventory = 25% * 17,440 units * 2 pounds = 8,720 pounds
For July:Budgeted production = 16,800 units + 30% * 14,800 units = 20,840 units
Raw material inventory = 25% * 20,840 units * 2 pounds = 10,420 pounds
For August:Budgeted production = 14,800 units + 30% * 20,840 units = 20,632 units
Raw material inventory = 25% * 20,632 units * 2 pounds = 10,316 pounds
To find the budgeted ending finished goods inventory for June, we need to subtract the budgeted sales for June from the budgeted production for June and add the budgeted ending finished goods inventory for May:
Budgeted ending finished goods inventory for June = 17,440 units - 12,800 units + 3,840 units = 8,480 units
Similarly, we can find the budgeted ending finished goods inventory for July and August:
Budgeted ending finished goods inventory for July = 20,840 units - 16,800 units + 8,480 units = 12,520 units
Budgeted ending finished goods inventory for August = 20,632 units - 14,800 units + 12,520 units = 18,352 units
Therefore, the budgeted ending finished goods inventory for June, July, and August are 8,480 units, 12,520 units, and 18,352 units, respectively. The budgeted raw material inventory for June, July, and August are 8,720 pounds, 10,420 pounds, and 10,316 pounds, respectively.
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If a doctor prescribes 75 milligrams of a specific drug to her patient, how many milligrams of
the drug will remain in the patient's bloodstream after 6 hours, if the drug decays at a rate of
20 percent per hour? use the function act) = te and round the solution to the nearest
hundredth.
After 6 hours, approximately 19.66 milligrams of the drug will remain in the patient's bloodstream.
To find the remaining amount of the drug in the patient's bloodstream after 6 hours, we'll use the decay function given: A(t) = P(1 - r)^t, where:
- A(t) is the remaining amount after t hours
- P is the initial amount (75 milligrams in this case)
- r is the decay rate per hour (20% or 0.20)
- t is the number of hours (6 hours)
Step 1: Plug in the given values into the formula.
A(t) = 75(1 - 0.20)^6
Step 2: Calculate the expression inside the parentheses.
1 - 0.20 = 0.80
Step 3: Replace the expression in the formula.
A(t) = 75(0.80)^6
Step 4: Raise 0.80 to the power of 6.
0.80^6 ≈ 0.2621
Step 5: Multiply the result by the initial amount.
A(t) = 75 × 0.2621 ≈ 19.66
So, approximately 19.66 milligrams of the drug will remain in the patient's bloodstream after 6 hours, rounded to the nearest hundredth.
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The scores on the last math quiz are summarized in the following frequency table:
Score
10
9
8
7
6
5
4
3
2
1
0
Frequency
6
7
5
3
2
1
1
0
0
0
0
The information is then put into the following histogram:
A histogram has score on the x-axis, and frequency on the y-axis. A score of 4 has a frequency of 1; 5, 1; 6, 2; 7, 3; 8, 5; 9, 7; 10, 6.
Calculate the mean, median, mode, and midrange of this quiz distribution and explain whether the distribution is skewed to the left or to the right.
a.
Mean = 9, median = 8.2, mode = 7, midrange = 9; skewed to the left.
b.
Mean = 8.2, median = 9, mode = 9, midrange = 7; skewed to the left.
c.
Mean = 8.2, median = 9, mode = 9, midrange = 7; skewed to the right.
d.
Mean = 9, median = 8.2, mode = 7, midrange = 9; skewed to the right.
Please select the best answer from the choices provided
The correct option regarding the data is B. Mean = 8.2, median = 9, mode = 9, midrange = 7; skewed to the left.
How to explain the dataA histogram has score on the x-axis, and frequency on the y-axis. A score of 4 has a frequency of 1; 5, 1; 6, 2; 7, 3; 8, 5; 9, 7; 10, 6.
It shtbe noted that Mean = 8.2, median = 9, mode = 9, midrange = 7; skewed to the left.
This statement describes a distribution with a mean equal to the median and a mode that is likely less than the mean and the median. The fact that the distribution is skewed to the left indicates that the tail of the distribution is longer on the left side, and that there may be some low outliers that are pulling the mean towards the left.
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