lim (1 - a^(1-a)) / (ln(a)) as x -> a This is the value of the given limit.
To find the value of the given limit, which can be represented as lim (a^x - x^a) / (x^x - a^a) as x approaches 'a', you can apply L'Hôpital's Rule, which states that if the limit of the ratio of the derivatives of two functions exists, then the limit of the original functions is equal to that limit.
First, differentiate the numerator and denominator with respect to x:
Numerator: d(a^x - x^a) / dx = a^x * ln(a) - a * x^(a-1)
Denominator: d(x^x - a^a) / dx = x^x * ln(x)
Now, we can find the limit of the ratio of the derivatives as x approaches 'a':
lim (a^x * ln(a) - a * x^(a-1)) / (x^x * ln(x)) as x -> a
After substituting 'a' for 'x' in the limit:
lim (a^a * ln(a) - a * a^(a-1)) / (a^a * ln(a)) as x -> a
Now, cancel out the common term a^a * ln(a):
lim (1 - a^(1-a)) / (ln(a)) as x -> a
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secθ in simplest radical form.
The value of secθ in simplest radical form is:
[tex]sec\theta = -\frac{\sqrt{61} }{5}[/tex]
If the point is given on the terminal side of an angle, then:
Calculate the distance between the point given and the origin:
[tex]r = \sqrt{x^{2} +y^2}[/tex]
Here, x = -5 and y = -6
The secant can be found by the following trigonometric relation:
[tex]sec\theta = \frac{1}{cos\theta}[/tex]
[tex]sec\theta = \frac{1}{\frac{x}{r} }\\ \\sec\theta = \frac{r}{x}\\ \\sec\theta = \frac{\sqrt{x^{2} +y^2} }{x}\\ \\sec\theta =\frac{\sqrt{(-5)^2+(-6)^2} }{-5}\\ \\sec\theta = -\frac{\sqrt{61} }{5}[/tex]
The secant function ‘or’ Sec Theta is one of the trigonometric functions apart from sine, cosine, tangent, cosecant, and cotangent. In right-angled trigonometry, the secant function is defined as the ratio of the hypotenuse and adjacent side.
Now , the secθ functions:
[tex]sec\theta = -\frac{\sqrt{61} }{5}[/tex]
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The given question is incomplete, complete question is:
If θ is an angle in standard position and its terminal side passes through the point (-5,-6), find the exact value of secθ in simplest radical form.
Identify which type of sampling is used: random, stratified, cluster, systematic, or convenience.
1. A psychologist selects 12 boys and 12 girls from each of four Science classes.
2. When he made an important announcement, he based his conclusion on 10 000 responses, from
100 000 questionnaires distributed to students.
3. A biologist surveys all students from each of 15 randomly selected classes.
4. The game show organizer writes the name of each contestant on a separate card, shuffles the cards, and
draws five names.
5. Family Planning polls 1 000 men and 1 000 women about their views concerning the use of contraceptives.
6. A hospital researcher interviews all diabetic patients in each of ten randomly selected hospitals.
1. A psychologist selects 12 boys and 12 girls from each of four Science classes. = Stratified sampling
2. When he made an important announcement, he based his conclusion on 10 000 responses, from 100 000 questionnaires distributed to students= Convenience sampling
3. A biologist surveys all students from each of 15 randomly selected classes = Cluster sampling
4. The game show organizer writes the name of each contestant on a separate card, shuffles the cards, and draws five names= Random sampling
5. Family Planning polls 1 000 men and 1 000 women about their views concerning the use of contraceptives= Stratified sampling
6. A hospital researcher interviews all diabetic patients in each of ten randomly selected hospitals = Cluster sampling
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A tile maker makes triangular tiles for a mosaic. Two triangular tiles form a square. what is the area of one of the triangular tiles
Answer: [tex]\frac{x^2}{2}[/tex]
Step-by-step explanation:
If two of the tiles form a square together, then one of them must be half of a square. this means that the square is split diagonally.
If you've ever done trigonometry, you'll know this is a 45-45-90 special right triangle. The side lengths are in the ratio of x, x, and xsqrt(2).
so we know the area of one of these tiles will be [tex]\frac{x^2}{2}[/tex], where x is the side length of the square formed.
Peter owns a currency conversion shop.
Last Monday, Peter changed a total of £20,160 into a number of different currencies.
He changed
3/10
of the £20,160 into euros.
He changed the rest of the pounds into dollars, rupees and francs in the ratios 9:5:2
Peter changed more pounds into dollars than he changed into francs.
Work out how many more.
If Peter changed more pounds into dollars than he changed into francs then Peter changed £6,168 more into dollars than into francs.
First, we need to find out how much money Peter changed into euros:
(3/10) × £20,160 = £6,048
Next, we need to find out how much money Peter changed into dollars, rupees, and francs combined:
£20,160 − £6,048 = £14,112
We can use the ratios to find out how much of this total amount goes to each currency:
- Dollars: (9/16) × £14,112 = £7,932
- Rupees: (5/16) × £14,112 = £4,420
- Francs: (2/16) × £14,112 = £1,764
We can see that Peter changed more pounds into dollars than into francs. To find out how many more, we can subtract the amount changed into francs from the amount changed into dollars:
£7,932 − £1,764 = £6,168
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Last question find measure of arc Su
Check the picture below.
[tex]52x=\cfrac{154-50x}{2}\implies 104x=154-50x\implies 154x=154\implies x=\cfrac{154}{154} \\\\\\ x=1\hspace{9em}\stackrel{ 50(1) }{\widehat{SU}=50^o}[/tex]
Use the given acceleration function and initial conditions to find the velocity vector v(t), and position vector r(t). Then find the position at time t = 9. a(t) = −cos ti − sin tj v(0) = j + k, r(0) = i v(t) = r(t) = r(9) =
find the position at time t = 9. a(t) = −cos ti − sin tj v(0) = j + k, r(0) = i v(t) = r(t) = r(9) = This gives you the position vector r(9) as a function of sin(9) and cos(9).
To find the velocity vector v(t) and position vector r(t), we need to integrate the given acceleration function a(t) and apply the initial conditions. Here's a step-by-step explanation:
1. Given acceleration function: a(t) = -cos(t)i - sin(t)j
2. Integrate a(t) with respect to t to find v(t):
v(t) = ∫(-cos(t)i - sin(t)j) dt = (sin(t)i + cos(t)j) + C, where C is a constant vector.
3. Apply initial condition v(0) = j + k:
v(0) = sin(0)i + cos(0)j + C = j + k
C = -i + j + k
4. The velocity function is: v(t) = sin(t)i + cos(t)j - i + j + k
Now let's find the position vector r(t):
5. Integrate v(t) with respect to t to find r(t):
r(t) = ∫(sin(t)i + cos(t)j - i + j + k) dt = (-cos(t)i + sin(t)j + t(k) + D, where D is another constant vector.
6. Apply initial condition r(0) = i:
r(0) = -cos(0)i + sin(0)j + 0(k) + D = i
D = i
7. The position function is: r(t) = -cos(t)i + sin(t)j + tk + i
Finally, let's find the position at time t = 9:
8. r(9) = -cos(9)i + sin(9)j + 9k + i
This gives you the position vector r(9) as a function of sin(9) and cos(9).
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11
Differentiate the function and find the slope of the tangent line at the given value of the independent variable s=8-41², 1=-3 s'(t)=0 The slope of the tangent line is at t= -3.
The slope of the tangent line to the function [tex]s(t) = 8 - 41t^2[/tex] at t = -3 is 246.
Process of finding slope:1. Differentiate the function s(t) with respect to the independent variable t: [tex]s(t) = 8 - 41t^2[/tex].
2. Calculate the derivative s'(t).
3. Evaluate the derivative at the given value of t.
Step 1: Differentiate the function [tex]s(t) = 8 - 41t^2[/tex].
To differentiate this function, we apply the power rule for differentiation.
The derivative of a constant (8) is 0, and the derivative of 41t^2 is -82t
(since we multiply the exponent 2 by the coefficient 41 and then subtract 1 from the exponent).
Step 2: Calculate the derivative s'(t).
s'(t) = 0 - 82t
Step 3: Evaluate the derivative at the given value of t (t = -3).
s'(-3) = -82(-3) = 246
The slope of the tangent line to the function [tex]s(t) = 8 - 41t^2[/tex] at t = -3 is 246.
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On a number line, point A is located at -3 and point B is located at 19. Find coordinate of a point between A and B such that the distance from A to point B is 3/11 of distance A to B
The coordinate of a point between A and B, such that the distance from A to point B is 3/11 of distance A to B, is 1.
Let's denote the unknown point between A and B as P, and let the distance from A to P be x. Then the distance from P to B is (11/3)x. Since the distance from A to B is 19 - (-3) = 22, we have the equation x + (11/3)x = 22(3/11), which simplifies to (14/3)x = 6, or x = 9/7. Therefore, the coordinate of point P is -3 + (9/7)(19 - (-3)) = 1.
To check our answer, we can verify that the distance from A to P is (10/7)(22) and the distance from P to B is (1/7)(22)(11), and that (10/7)(22) = (3/11)(22), which is indeed true.
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The school store sells spiral notebooks in four colors and three different sizes. The table shows the sales
by size and color for 386 notebooks What is the experimental probability that the next customer buys a
red notebook with 150 pages? Enter your answer as a simplified fraction.
Red
53
100 Pages
150 Pages
200 Pages
Green
31
47
16
Blue
21
57
22
Yellow
12
27
12
63
25
The experimental probability is
The experimental probability that the next customer buys a red notebook with 150 pages is 53/386 or 13.73%.
To find the experimental probability of the next customer buying a red notebook with 150 pages, we need to first identify the total number of red 150-page notebooks sold and then divide that by the total number of notebooks sold.
From the table, we can see that 53 red 150-page notebooks were sold. The total number of notebooks sold is 386.
The experimental probability is therefore the ratio of red 150-page notebooks sold to the total number of notebooks sold:
Probability = (Number of red 150-page notebooks) / (Total number of notebooks)
Probability = 53 / 386
The simplified fraction for the experimental probability is 53/386 or 13.73%.
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A non-government that supports palay production in the philippines conducted the research that answer the question: is the proportion of palay harvested different from 0. 50 of all the farm crops are harvested? in a sample of 200 one-hectare farm lands, 96 harvested palay.
For the sample size of 200 and sample data 96 there is no sufficient evidence to conclude that proportion of palay harvested is different from 0.50.
Sample size 'n' = 200
The proportion of palay harvested is different from 0.50
Use a hypothesis test.
Let us assume the null hypothesis H₀ is that the proportion of palay harvested is equal to 0.50,
and the alternative hypothesis Hₐ is that the proportion of palay harvested is different from 0.50.
H₀: p = 0.50 proportion of palay harvested is equal to 50%
Hₐ: p ≠ 0.50 proportion of palay harvested is not equal to 50%
where p is the population proportion of palay harvested.
To test this hypothesis, use the sample data of 96 out of 200 one-hectare farm lands that harvested palay.
The sample proportion of palay harvested is,
p₁= 96/200
= 0.48
To determine if this sample proportion is significantly different from the hypothesized proportion of 0.50,
Use a two-tailed z-test with a significance level of α = 0.05.
The test statistic is calculated as,
z = (p₁ - p) / √(p(1-p)/n)
where n is the sample size.
Substituting the values, we get,
z = (0.48 - 0.50) / √(0.50(1-0.50)/200)
⇒ z = -0.5658
Using a z-table,
The probability of getting a z-value of -0.5658 or lower in the left tail of the distribution is approximately 0.7123.
Since this is a two-tailed test,
Probability of getting a z-value of 0.5658 or higher in the right tail of the distribution is also approximately 0.7123
p-value for this test is 0.7123+ 0.7123 = 1.4246
Since the p-value is greater than the significance level of α = 0.05,
Fail to reject the null hypothesis.
Therefore, do not have sufficient evidence to conclude that the proportion of palay harvested is different from 0.50.
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What is the local rate of change on this parabola at the point , (-6,8)?
To find the local rate of change on a curve at a specific point, we need to find the slope of the tangent line at that point. The tangent line represents the instantaneous rate of change or the rate of change at that particular point.
To find the slope of the tangent line at (-6,8) on the parabola, we need to take the derivative of the function at that point.
Assuming that the parabola is defined by the equation [tex]y = ax^2 + bx + c,[/tex]
where a, b, and c are constants, we can find the derivative of the function as follows:
[tex]dy/dx = 2ax + b[/tex]
Substituting [tex]x = -6,[/tex] we get:
[tex]dy/dx = 2a(-6) + b[/tex]
To find the values of a and b, we need more information about the parabola.
If we have the equation of the parabola or another point on the curve, we can use it to find the values of a and b.
Once we have the values of a and b, we can substitute them into the derivative equation and evaluate it at [tex]x = -6[/tex] to find the slope of the tangent line at (-[tex]6,8[/tex]), which is the local rate of change at that point.
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Jose rented a truck for one day. there was a base fee of $17.99, and there was an additional charge of 83 cents for each mile driven. jose had to pay $194.78 when he returned the truck. for how many miles did he drive the truck?
Jose drove the truck for approximately 213 miles.
Let's assume that Jose drove the truck for m miles.
We know that there was a base fee of $17.99, so the remaining amount after that base fee went towards the additional charge of 83 cents per mile.
So, the additional charge for the miles driven can be represented as 0.83m.
The total cost that Jose had to pay was $194.78. Therefore, we can write the equation:
17.99 + 0.83m = 194.78
Solving for m:
0.83m = 194.78 - 17.99
0.83m = 176.79
m = 213.072
So, Jose drove the truck for approximately 213 miles.
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The human gestation times have a mean of about 266 days, with a standard deviation of about 10 days. Suppose we took the average
gestation times for a sample of 100 women.
days
Where would the center of the histogram be?
What would the standard deviation of that histogram?
My sample shows a mean of 264. 8 days. What is my z-score?
days (Round to the thousandth place)
My sample shows a mean of 264. 8 days. What is my z-score?
(Round to the tenth place)
The z-score is -1.2, rounded to the tenths place.
The center of the histogram would be around the population mean of 266 days.
The standard deviation of the histogram would be the standard error of the mean, which is the standard deviation of the population divided by the square root of the sample size. Thus, the standard deviation of the histogram would be 10 / sqrt(100) = 1 day.
To calculate the z-score for a sample mean of 264.8 days, we can use the formula:
z = (sample mean - population mean) / (standard deviation / sqrt(sample size))
Substituting the given values, we get:
z = (264.8 - 266) / (10 / sqrt(100)) = -1.2
Therefore, the z-score is -1.2, rounded to the tenths place.
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By using integration by parts, find the integral 2∫⁷ in x dx b) Hence, find 2∫⁷ in √x dx
The integral is:
[tex](4/3)x^(3/2) ln(x) - (2/3)∫x^(1/2) dx = (4/3)x^(3/2) ln(x) - (4/5)x^(5/2) + C[/tex]
Solve the integrals using integration by parts.
a) To find [tex]2∫x⁷ln(x) dx[/tex], we'll use integration by parts with the formula: [tex]∫u dv = uv - ∫v du. Let's choose:u = ln(x) = > du = (1/x) dxdv = x⁷ dx = > v = (1/8)x⁸[/tex]
Now, apply the integration by parts formula:
[tex]2∫x⁷ln(x) dx = 2[uv - ∫v du] = 2[((1/8)x⁸ ln(x) - ∫(1/8)x⁸(1/x) dx)]= (1/4)x⁸ ln(x) - (1/4)∫x⁷ dx = (1/4)x⁸ ln(x) - (1/32)x⁸ + C[/tex]
b) To find 2∫√x ln(x) dx, we'll use a similar approach. Let's choose:
[tex]u = ln(x) = > du = (1/x) dxdv = √x dx = > v = (2/3)x^(3/2)[/tex]
Now, apply the integration by parts formula:
[tex]2∫√x ln(x) dx = 2[uv - ∫v du] = 2[((2/3)x^(3/2) ln(x) - ∫(2/3)x^(3/2)(1/x) dx)]= (4/3)x^(3/2) ln(x) - (2/3)∫x^(1/2) dx = (4/3)x^(3/2) ln(x) - (4/5)x^(5/2) + C[/tex]
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A cylindrical swimming pool has a diameter of 12 feet and a height of 4 feet. How many gallons of water can the pool contain? Round your answer to the nearest whole number. (1 ft3 ≈ 7. 5 gal)
The number of gallons of water the pool can contain is approximately 3393 gallons.
To find the amount of water in gallons the pool can contain, we must find the volume of the cylindrical swimming pool, you can use the formula:
Volume = π * r² * h
Where r is the radius (half the diameter), and h is the height.
In this case, r = 12 feet / 2 = 6 feet, and h = 4 feet.
Volume = π * (6 ft)² * 4 ft ≈ 452.39 ft³
To convert cubic feet to gallons, use the given conversion factor (1 ft³ ≈ 7.5 gal).
Volume ≈ 452.39 ft³ * 7.5 gal/ft³ ≈ 3392.93 gal
Rounding to the nearest whole number, the pool can contain approximately 3393 gallons of water.
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A flare is launched from a boat and travels in a parabolic path until reaching the water. Write a quadratic function that
models the path of the flare with a maximum height of 300 meters, represented by a vertex of (59, 300), landing in the water at the point
(119, 0).
f(x) =
Answer:
We can start by using the vertex form of a quadratic function:
f(x) = a(x - h)^2 + k
where (h, k) is the vertex of the parabola.
We know that the vertex is (59, 300), so we can plug in these values:
f(x) = a(x - 59)^2 + 300
To determine the value of "a", we can use the fact that the parabola passes through the point (119, 0). So we substitute these values for x and y and solve for "a":
0 = a(119 - 59)^2 + 300
-300 = 3600a
a = -1/12
Substituting this value of "a" back into the equation for f(x), we get:
f(x) = (-1/12)(x - 59)^2 + 300
This quadratic function models the path of the flare, with a maximum height of 300 meters at the vertex (59, 300), and landing in the water at the point (119, 0).
Chris buys 19 raffle tickets. A total of 250 tickets were sold. Find the probability that Chris does not win the prize
Answer:For 19 to 250 odds against winning;
Probability of:
Winning = (0.9294) or 92.9368%
Losing = (0.0706) or 7.0632%
"Odds for" winning: 250:19
"Odds against" winning: 19:250
Step-by-step explanation:
Find the radius of the cylinder. Round to the nearest whole centimeter.
The cylinder has a height of 6 centimeters and a radius of r1. The volume of the cylinder is 302 cubic centimeters.
___ centimeters
Answer:
To find the radius of the cylinder, we can use the formula for the volume of a cylinder, which is pi*(r1^2)*h, where r1 is the radius and h is the height. Given that the cylinder has a height of 6 centimeters and a volume of 302 cubic centimeters, we can solve for r1 by dividing the volume by pi times the height, and then taking the square root of the result. After rounding to the nearest whole centimeter, the radius of the cylinder is approximately 5 centimeters.
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A square has sides of length s. A rectangle is 6 inches shorter than the square and 1 inch longer. Which of the following expressions represents the perimeter of the rectangle?
The perimeter of the rectangle is represented by the expression 4s - 10.
How to calculate perimeter of a rectangle?
To calculate the perimeter of a rectangle, you need to add up the lengths of all four sides.
In the problem given, we know that the rectangle is 6 inches shorter than the square and 1 inch longer.
Let's call the length of the rectangle l and the width w.
We know that the length of the square is equal to its width (since it's a square), so the length of the rectangle must be l = s - 6, and the width must be w = s + 1.
To find the perimeter, we add up all four sides: P = 2l + 2w = 2(s-6) + 2(s+1) = 4s - 10.
Therefore, the expression that represents the perimeter of the rectangle is 4s - 10.
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A neighborhood watch association surveyed 40 neighbors about their feelings of safety in the neighborhood. They will survey an additional 80 neighbors. Based on the information, predict how many of the 80 neighbors will feel safe?
We can predict that around 50 of the additional 80 neighbors will feel safe in the neighborhood.
To make a prediction about the number of neighbors who will feel safe, we need to know the proportion of the initial 40 neighbors who felt safe. Let's say that 25 of the 40 neighbors surveyed felt safe.
Then, we can estimate the proportion of the larger group of 120 neighbors (the initial 40 plus the additional 80) who will feel safe as follows:
proportion feeling safe = number feeling safe / total number surveyed
proportion feeling safe = 25 / 40
proportion feeling safe = 0.625
We can use this proportion to estimate the number of the 80 additional neighbors who will feel safe:
number feeling safe = proportion feeling safe x total number surveyed
number feeling safe = 0.625 x 80
number feeling safe ≈ 50
So based on the information given, we can predict that around 50 of the additional 80 neighbors will feel safe in the neighborhood.
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Portfolio for Unit 5
Part 1: Car Wheel Project
For the Portfolio for Unit 5 Part 1, the Car Wheel Project, you'll want to include several key components.
First, be sure to include a detailed description of the project itself, including any goals or objectives you had in mind when you started. This could include things like improving your engineering or design skills, learning more about the materials used in car wheels, or simply creating a visually impressive final product.
Next, include some documentation of your process as you worked on the project. This might include sketches, diagrams, or photos of different stages of the process. Be sure to highlight any challenges or roadblocks you encountered along the way, and how you overcame them.
Finally, be sure to include a final showcase of your completed car wheel project. This might include photos of the finished product from different angles, a video demonstrating how it works or how it was made, or even a physical prototype that you can bring in to show off.
Overall, the key to a successful portfolio for the Car Wheel Project is to demonstrate your creativity, your technical skills, and your ability to work through challenges and solve problems.
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Unit 5 Portfolio Car Wheel Project Answer the following questions: 1. How do you find the area of a circle?
2. How do you find the circumference of a circle?
3. What information do you need in order to find the area or circumference of a circle
4. To determine the number of full rotations your tires complete before you need to switch thefront and back tires, will you be working with area or circumference?
Answer:
How do you find an area of a crilce?
-Use the formula A=[tex]\pi r^2[/tex]
How do you find the cirumference of a cirlce?
-Use the formula c=2[tex]\pi r[/tex]
What information do you need in order to find the area or circumference of a cirlce?
-I believe it's the diameter.
To determine the number of full rotations your tires complete before you switch the front and back tires will you be working with area or circumference?
-Circumference. Because ifts the distance around something, like a circle, or a wheel.
(My three examples of the circumference of tires)
Bike:
C=2(3.14)r
C=2 (3.14) 8
C=50.24
Car:
C=2(3.14)r
C=2 (3.14) 20
C=125.6
Scooter:
C=2(3.14)r
C=2 (3.14) 3
C=18.24
(My tire rotations how far you'll go in 10,000 miles)
car-
20x3.1416=62.83
10,000/62.83= 159.2
scooter-
3x3.1416=9.4248
10,000/9.4248=1,061
bike-
8x3.1416=25.13
10,000/25.13=397.9
Using one of your vehicles how far can you get in a week?
-We're driving the car with twenty inch wheels and we're going seventy miles an hour. So in a week we would have driven 490 miles.
When will you need to change your tires?
-When we reach 225 miles we'd change the tires. Because these tires are heavier than the average tire they're wear out faster.
I hope this helps!!! Let me know if you need some help with anything else.
The hypotenuse of a right triangle measures 29 cm. One leg is 1 cm shorter than the other. What are the lengths of the legs?
The length of the legs are 20 cm and 21 cm
What is the length of the legs?The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
We know that from the Pythagoras theorem;
[tex]c^2 = a^2 + b^2[/tex]
Let the hypotenuse be c and the other two sides be a and b
We have that;
[tex]29^2 = x^2 + (x -1)^2\\841 = x^2 + x^2 - 2x + 1\\841 = 2x^2 - 2x + 1\\2x^2 - 2x + 1 - 841 = 0\\2x^2 - 2x - 840 = 0\\x = -20 or 21[/tex]
Since length can not be negative, x = 21 cm
Thus the other leg is 20 cm
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PLEASEEEE HELPPP ASAP 20 PTS
Use long division to determine the quotient of the following expression.
Write the quotient in standard form with the term of largest degree on the left. (10x^(2)+3x-77)-:(2x+7)
The quotient of the division 10x² + 3x - 77 ÷ 2x + 7 is 5x - 16
Evaluating the long division expressionsThe quotient expression is given as
10x² + 3x - 77 ÷ 2x + 7
The long division expression is represented as
2x + 7 | 10x² + 3x - 77
So, we have the following division process
5x - 16
2x + 7 | 10x² + 3x - 77
10x² + 35x
--------------------------------
-32x - 77
-32x - 112
-------------------------------------
35
Hence, the quotient of the long division is 5x - 16
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find the missing no 3,4,13,?,8,168
Answer:
I believe it is 38
Step-by-step explanation:
To the nearest hundredth, what is the value of x?
Use a trigonometric ratio to compute a distance
Therefore, to the nearest hundredth, the value of x is 42.31 units.
What is triangle?A triangle is a three-sided polygon, which is a closed shape made up of straight lines. It is one of the simplest geometric shapes and is used extensively in mathematics, science, and engineering. In a triangle, each side connects two vertices or corners, and each vertex is where two sides intersect. The three angles of a triangle always add up to 180 degrees, and the sum of the lengths of any two sides is always greater than the length of the third side. Triangles can be classified by the lengths of their sides and the sizes of their angles, which gives rise to different types such as equilateral, isosceles, scalene, acute, right, and obtuse triangles. Triangles have many applications, such as in geometry, trigonometry, physics, and engineering, and they are fundamental to understanding the properties of other shapes and mathematical concepts.
Here,
In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. So, for this triangle, we have:
sin(53°) = opposite / hypotenuse
sin(53°) = x / 53
To solve for x, we can rearrange the equation as follows:
x = 53 * sin(53°)
Using a calculator to evaluate sin(53°), we get:
sin(53°) = 0.7986 (rounded to four decimal places)
Substituting this value into the equation, we get:
x = 53 * 0.7986
x = 42.308 (rounded to two decimal places)
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Marcus estimated the mass of a grain of sugar as 6 x 10-4 gram. Based on that
estimate, about how many grains of sugar are there in a small bag of sugar
that weighs 0. 24 kilogram?
There are 400,000 grains of sugar in a small bag of sugar that weighs 0.24 kilograms.
To find out how many grains of sugar are there in a small bag of sugar that weighs 0.24 kilograms, based on Marcus' estimate, follow these steps:
1. Convert the mass of the bag of sugar from kilograms to grams: 0.24 kg * 1000 g/kg = 240 g.
2. Use Marcus' estimate of the mass of a grain of sugar: 6 x 10^-4 g.
3. Divide the total mass of the bag of sugar by the mass of a single grain of sugar: 240 g / (6 x 10^-4 g/grain).
Now, let's perform the calculation:
240 g / (6 x 10^-4 g/grain) = 240 g / 0.0006 g/grain = 400,000 grains.
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Which function is graphed to the right?
A. f(x) = ¹₂ +3 x-2
B. f(x)=¹+2
C. f(x) =3x+2
The rational function graphed in this problem is given as follows:
B. f(x) = 1/(x - 3) + 2.
How to obtain the rational function?From the graph, the asymptotes of the rational function are given as follows:
Vertical asymptote at x = 3 -> the function is not defined at x = 3.Horizontal asymptote at y = 2 -> as x goes to infinity, f(x) approaches y = 2.Considering that the function has a vertical asymptote at x = 3, we have that:
f(x) = 1/(x - 3).
Considering the horizontal asymptote at y = 2, we have that:
f(x) = 1/(x - 3) + 2.
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CDE is a tangent to the circle below.
Calculate the size of angle θ.
Fully Justify your answer.
Applying the inscribed angle theorem, the measure of the size of angle ∅ = 85 degrees.
How to Apply the Inscribed Angle Theorem?If an inscribed angle in a circle is subtended by an arc, the inscribed angle theorem states that the measure of the intercepted arc would be twice the measure of the inscribed angle.
Therefore, we have:
measure of arc DF = 2(31) = 62 degrees [inscribed angle theorem]
measure of arc BD = 2(54) = 108 degrees.[inscribed angle theorem]
∅ = 1/2(measure of arc BDF) [inscribed angle theorem]
∅ = 1/2(m(DF) + m(BD))
Substitute:
∅ = 1/2(62 + 108)
∅ = 1/2(170)
∅ = 85 degrees.
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On the same coordinate plane, mark all points (x,y) such that (A) y=x-2, (B) y=-x-2, (C) y=|x|-2
The points that satisfy equations (A), (B), and (C) are (-2,-4), (4,2), and (-4,2).
we can plot the graphs of each of these equations on the same coordinate plane and then identify the points where they intersect.
To mark all the points that satisfy the equations (A) [tex]y=x-2[/tex], (B) y=x-2[tex]y=x-2[/tex] and (C) [tex]y=|x|-2[/tex],
For equation (A), we can see that the slope is 1 (the coefficient of x) and the y-intercept is -2 (the constant term). This means that the graph of equation (A) is a straight line that passes through the point (0,-2) and has a slope of 1.
We can plot this line on the coordinate plane by marking the point (0,-2) and then drawing a line with slope 1 that passes through this point.
For equation (B), we can see that the slope is -1 (the coefficient of x) and the y-intercept is -2 (the constant term).
This means that the graph of equation (B) is a straight line that passes through the point (0,-2) and has a slope of -1. We can plot this line on the coordinate plane by marking the point (0,-2) and then drawing a line with slope -1 that passes through this point.
For equation (C), we can see that the y-intercept is -2 and that the graph of the equation is symmetric with respect to the y-axis.
This means that we only need to plot the part of the graph that lies in the first quadrant, and then we can use symmetry to find the part that lies in the other quadrants.
To plot the graph of equation (C) in the first quadrant, we can start by marking the point (2,0) (since y=|x|-2 when x=2) and then draw a V-shape with the vertex at this point and the arms of the V going up and to the right.
To find the points where these three graphs intersect, we can look for the points where any two of the graphs intersect. For example, we can see that the graphs of equations (A) and (B) intersect at the point (-2,-4).
Similarly, we can see that the graphs of equations (A) and (C) intersect at the point (4,2), and the graphs of equations (B) and (C) intersect at the point (-4,2).
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how to find the polynmial closest to another polynomial in an inner product space
To find the polynomial closest to another polynomial in an inner product space, you can follow these steps:
Choose an inner product on the space of polynomials. One common inner product on this space is the L2 inner product, which is defined as:
<f,g> = ∫a^b f(x)g(x) dx,
where a and b are the endpoints of the interval on which the polynomials are defined.
Let P be the space of polynomials of degree at most n, where n is the degree of the polynomial you want to approximate. Let f be the polynomial you want to approximate, and let g be an arbitrary polynomial in P.
Define the error between f and g as e = f - g.
Compute the inner product of e with itself:
<e,e> = ∫[tex]a^b (f(x) - g(x))^2 dx.[/tex]
Minimize this inner product with respect to g. This can be done by setting the derivative of <e,e> with respect to g equal to zero and solving for g.
The polynomial that minimizes the error is the polynomial closest to f in the L2 sense.
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