The given improper integral diverges.
We can use the integral test to determine if the improper integral converges or diverges.
Consider the function f(x) = x^2 ln(x).
Taking the derivative of f(x), we get:
f'(x) = 2x ln(x) + x
Setting f'(x) = 0 to find the critical points, we get:
2 ln(x) + 1 = 0
ln(x) = -1/2
x = e^(-1/2)
Note that f(x) is positive and decreasing for x > e^(-1/2). Therefore, we have:
∫1 [infinity] In(x) /x^2 dx = ∫e^(-1/2) [infinity] x^2 ln(x) /x^2 dx
= ∫e^(-1/2) [infinity] ln(x) dx
Since ln(x) is an increasing function, we know that:
∫e^(-1/2) [infinity] ln(x) dx is divergent by the integral test.
Therefore, the original improper integral:
∫1 [infinity] In(x) /x^2 dx
is also divergent by comparison to the divergent integral.
In conclusion, the given improper integral diverges.
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X f(x) = 5 e^ + 4x – In x Find f'(x). f(x)=5e + 4x - In x
20e^(4x) – (1/x)
Explanation: To find the derivative of f(x), we need to apply the chain rule and the product rule.
f(x) = 5e^(4x) – ln(x)
Using the product rule, we have:
f'(x) = (5e^(4x))(4) – (1/x)
Simplifying:
f'(x) = 20e^(4x) – (1/x)
Therefore, the derivative of f(x) is f'(x) = 20e^(4x) – (1/x).
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6. Find a function f such that:
(a) f′(x) = sin(2x)
(b) f′(x) = 1/√x
(c) f′(x) = (1 + 2x)^34^
(d) f′(x) = x(x^2^+ 1)^100^
To summarize, the functions f(x) are:
a) f(x) = -1/2 cos(2x) + C
b) f(x) = 2√x + C
c) f(x) = (1/68)(1 + 2x)³⁵ + C
d) f(x) = (1/201)(x² + 1)¹⁰¹ + C
To find a function f such that:
a) f′(x) = sin(2x), we need to integrate the derivative function with respect to x. The function f(x) is given by:
f(x) = ∫sin(2x) dx = -1/2 cos(2x) + C
b) f′(x) = 1/√x, we integrate the derivative function with respect to x:
f(x) = ∫(1/√x) dx = 2√x + C
c) f′(x) = (1 + 2x)³⁴, we integrate the derivative function with respect to x:
f(x) = ∫(1 + 2x)³⁴ dx = (1/68)(1 + 2x)³⁵ + C
d) f′(x) = x(x²+ 1)¹⁰⁰, we integrate the derivative function with respect to x:
f(x) = ∫x(x²+ 1)¹⁰⁰ dx = (1/201)(x² + 1)¹⁰¹ + C
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Application: 2. Design a systematic sample. A telephone company is planning a marketing survey of its 760 000 customers. . For budget reasons, the company wants a sample size of about 250. Suggest a method for selecting a systematic sample. pulation, individuals Terminology or Sampling Technique: the method used to select a group from a you wish to study Population: all the being studied. Sample: the group of people or items that are selected from the to be studied Sampling frame: members of population that actually have a of being selected for a study Statistical bias: systematic in a survey or sampling method which favours certain outcomes. of the sample are not representative of the Bias: Bias occurs when the population
To design a systematic sample for the telephone company's marketing survey of its 760,000 customers, the company can first divide the population into equal segments based on a predetermined interval size. For example, if the company wants a sample size of 250, it can divide the population into 3,040 segments (760,000/250). Then, they can randomly select one customer from the first segment, and then select every 3,040th customer after that to create a systematic sample.
This method ensures that every customer has an equal chance of being selected, while also keeping the sample size within the company's budget constraints. However, it is important to note that there may still be some statistical bias present in the sample, as certain customer demographics or behaviors may not be accurately represented in the sample. To minimize bias, the company can consider using stratified sampling or other techniques to ensure a more representative sample.
1. Define the population: In this case, the population consists of the 760,000 customers of the telephone company.
2. Determine the sample size: Due to budget constraints, the company wants a sample size of about 250 customers.
3. Create a sampling frame: List all the 760,000 customers in a systematic manner, such as alphabetically or by customer ID number.
4. Calculate the sampling interval: To select a systematic sample, divide the population size by the desired sample size. In this case, 760,000 customers divided by 250 gives a sampling interval of 3,040.
5. Choose a random starting point: Select a random number between 1 and 3,040 to serve as your starting point. For example, let's say you choose the number 1,500.
6. Apply the sampling interval: From the random starting point (1,500), select every 3,040 customers until you reach the end of the sampling frame. This will result in a systematic sample of 250 customers.
This method ensures that the sample is representative of the entire population and minimizes potential statistical bias. The company can then use this sample for its marketing survey while staying within its budget.
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John reads 110 pages on Saturday and 80 pages each day after writing expressions I will tell how many pages he reads after ex days
If John reads 110 pages on Saturday and 80 pages each-day after, then the expression representing number of pages after "x" days is "110 + 80x".
An "Expression" is defined as a mathematical-statement which consists of numbers, variables, operators combined together in a meaningful way.
To calculate the total number of pages John-reads after "x" days, we use the following expression:
⇒ Total number of pages = (Pages read on Saturday) + (Pages read each day for "x" days),
We know that, John reads "110-pages" on Saturday and "80-pages" each day after,
We substitute these values into expression,
So, Total number of pages = 110 + 80x
Where: "x" represents number-of-days after Saturday for which we want to calculate total number of pages John reads.
Therefore, the expression for total number of pages John reads after "x" days is 110 + 80x.
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The given question is incomplete, th complete question is
John reads 110 pages on Saturday and 80 pages each day after. Write an expressions to tell how many pages he reads after "x" days.
Shasta the cougar was working at differentiating a function p(x) and they found that ['(0) = ln(1/phi). Unfortunately, Shasta forgot the exact formla for the original function p(x). They can only remember that p(x) = x^a-a^x
for some constant a>1. Help out our four-legged mascor mathematician by determining the value of the constant a
For Shasta assistance the cougar, we need to find the value of constant 'a' for the original function p(x) = x^a - a^x. First, let's find the derivative p'(x).
Using the power rule and the chain rule, we can find the derivative of p(x):
p'(x) = ax^(a-1) - a^(x)ln(a)
Given that p'(0) = ln(1/phi), we will now evaluate p'(x) at x = 0:
p'(0) = a(0)^(a-1) - a^(0)ln(a) = ln(1/phi)
Since any number raised to the power of 0 is 1, we can simplify the equation:
0^(a-1) = 1/a - ln(a) = ln(1/phi)
We know that ln(1/phi) = -ln(phi), where phi is the golden ratio, approximately 1.618. So, we have:
1/a - ln(a) = -ln(phi)
To solve for 'a', we can use numerical methods or software like Mathematica or Wolfram Alpha. By doing so, we find that the constant 'a' is approximately 1.44467.
So, the original function p(x) is approximately:
p(x) = x^1.44467 - 1.44467^x
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If x is uniformly distributed over (0, 15), the probability that 5
A 1/13
B 4/15
C 2/5
D 7/15
The closest option to the correct answer is B. 4/15.
What is the formula for the probability density function?Using the following formula for the probability density function of a continuous uniform distribution, we can resolve this issue:
f(x) = 1/(b-a), a < x < b
where a and b represent the distribution's minimum and maximum values, respectively.
Since a and b are equal to 15, the probability density function is as follows:
f(x) = 1/15, 0 < x < 15
The likelihood that the area under the density curve between x = 5 and x = 10 is 5< x <10 This can be found by coordinating the thickness capability somewhere in the range of 5 and 10:
P(5 < x < 10) = ∫(5 to 10) f(x) dx
= ∫(5 to 10) 1/15 dx
= [x/15]_(5 to 10)
= (10/15) - (5/15)
= 1/3
Therefore, the answer is not one of the options provided. However, if we round 1/3 to the nearest option, we get:
B. 4/15
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The area of the surface obtained by rotating the curve y = /16-X^2, -1≤x≤1 about the x axis is
The area of the surface is 25.1327 square units, under the condition that the rotating curve is y = √(16 - x²) and its range is -1≤x≤1
Therefore, the area of the surface obtained by rotating the curve y = √(16 - x²), -1≤x≤1 about the x-axis is given by the formula
A = 2π ∫[a,b] f(x) √(1 + [f'(x)]²) dx
Here
a=-1,
b=1,
f(x) = √(16 - x²).
Then,
f'(x) = -x/√(16 - x²)
√(1 + [f'(x)]²) = √(1 + x²/(16 - x²))
Now we can staging these values
A = 2π ∫[-1,1] √(16 - x²) √(1 + x²/(16 - x²)) dx
This integral can be evaluated applying trigonometric substitution.
Let x = 4sinα, then dx = 4cosα dα.
Applying these values into our integral gives:
[tex]A = 2\pi \int\limits[\pi /2,-\pi /2] 16cos^{2\alpha }d\alpha[/tex]
= π² × 8
= 25.1327
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a normal population has mean 100 and variance 25. how large must the random sample be if you want the standard error of the sample average to be 1.5?
The sample size must be at least 12 if you want the standard error of the sample average to be 1.5.
To find the sample size needed, we can use the formula for standard error:
SE = σ/√n
Where SE is the standard error, σ is the population standard deviation (which is the square root of the variance), and n is the sample size.
In this case, we are given that the population mean is 100 and the variance is 25, so the standard deviation is √25 = 5.
We want the standard error to be 1.5, so we can plug in the values:
1.5 = 5/√n
To solve for n, we can isolate the variable by squaring both sides:
(1.5)^2 = (5/√n)^2
2.25 = 25/n
n = 25/2.25
n ≈ 11.11
So the sample size must be at least 12 (since we can't have a fraction of a person). This will give us a standard error of approximately 1.5.
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The line 7x + 8y = 0 in the xy-plane, is rotated about the y-axis. An equation for the generated surfaces is
The line 7x + 8y = 0 in the xy-plane, is rotated about the y-axis then equation of generated surfaces is (x² + (y - 8t)²) = 64t²
The equation given is the equation of a circle that has been rotated about the y-axis.
The equation gives the coordinates of a point on the circle centered at (0, 8t) with a radius of 8t.
The equation can be derived by taking the equation of a circle in the xy-plane, 7x + 8y = 0, and substituting y with (y - 8t) to reflect the rotation about the y-axis.
Hence, the line 7x + 8y = 0 in the xy-plane, is rotated about the y-axis then equation of generated surfaces is (x² + (y - 8t)²) = 64t²
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Lenny the leprechaun needs 2 pounds and 3 ounces of cabbage to go with his corned beef. He already has 1 pound and 4 ounces. How much more cabbage does he need?
( 16 ounces = 1 pound )
Answer:
he needs 15 ounces more
Step-by-step explanation:
Findings of an intervention study with a convenience sample:a. are generalizable to a wider group of patients with related problems. b. are to be discounted because they are extremely biased. c. provide no useful information. d. should be replicated before being applied to a wider population.
Findings of an intervention study with a convenience sample should be replicated before being applied to a wider population.
The correct answer is: d.
This is because a convenience sample may not be representative of the entire population, and thus, the findings might not be generalizable to a wider group of patients with related problems.
Replicating the study with a more diverse sample can help ensure the results are applicable to a broader population.
The correct answer is d :
Findings of an intervention study with a convenience sample should be replicated before being applied to a wider population.
Convenience sampling involves selecting participants based on their availability and willingness to participate, rather than randomly selecting them from a larger population.
As a result, convenience samples may not be representative of the larger population and may have biases that affect the generalizability of the findings.
Therefore, it is important to replicate the study using a more representative sample before drawing conclusions that can be applied to a wider population.
This helps to ensure that the findings are reliable and can be generalized to the broader population.
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If diameter of a circle is 14 M. Then find its area, radius, and circumference?
Step-by-step explanation:
why do you feel this is so complicated ?
you look up the formulas, grab a calculator and then just calculate the results.
please let me know, if you don't understand something :
the diameter is a line from one end of the circle to the other going through the center point of the circle.
therefore, the diameter is 2 times the distance from the center to its arc.
the radius is only one time the distance of the center of the circle to the arc.
so, radius = diameter/2 = 14/2 = 7 M.
the circumference of a circle is
2pi×r = 2×7×pi = 14pi = 43.98229715... M
the area of a circle is
pi×r² = pi×7² = 49pi = 153.93804... M²
Mr. Jackson orders lunches to be delivered to his workplace for himself and some coworkers. The cost of each lunch is $6. 25. There is also a one-time delivery fee of $ 3. 50 to deliver the lunches. What expression could Mr. Jackson use to find the cost of ordering n lunches?
The expression that could Mr. Jackson use to find the cost of ordering n lunches is 6.25n + 3.50
To find the cost of ordering n lunches, Mr. Jackson can use an expression. An expression is a combination of numbers, variables, and mathematical operations that represents a value. In this case, the expression that Mr. Jackson can use to find the cost of ordering n lunches is:
6.25n + 3.50
In this expression, n represents the number of lunches that Mr. Jackson orders. When Mr. Jackson orders n lunches, he has to pay $6.25 for each lunch, so the cost of the lunches will be 6.25n. In addition, Mr. Jackson has to pay a one-time delivery fee of $3.50, which is represented by the constant term 3.50 in the expression.
To use this expression to find the cost of ordering a specific number of lunches, Mr. Jackson can substitute the value of n into the expression and simplify. For example, if Mr. Jackson orders 10 lunches, the cost would be:
6.25(10) + 3.50 = 62.50 + 3.50 = 66.00
So the cost of ordering 10 lunches would be $66.00.
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If tan Alpha = 1/2, find the value of tan Alpha
[tex]if \: tan \alpha = \frac{1}{2} \: find \: the \: vaue \: of \: tan \alpha [/tex]
By trigonometric identity tanα= 1/2.
What is trigonometric identity?
Trigonometric Identities are used whenever trigonometric functions are involved in an expression or an equation. Trigonometric Identities are true and it is proven for every value of variables occurring on both sides of an equation. These identities involve certain trigonometric functions for example sine, cosine, tangent, cotangent of one or more angles.
tanα is a trigonometric identity.
tanα= perpendicular/ base in any right angled triangle.[ by the property of trigonometry]
given tanα = 1/2 so perpendicular is 1 unit and base is 2 unit
Hence, tanα = 1/2.
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Find the antiderivative: f(x) = 2cosx - 7sinx
The antiderivative of f(x) = 2cosx - 7sinx is F(x) = 2sinx + 7cosx + C.
To find the antiderivative of f(x) = 2cosx - 7sinx, we need to use the integration rules for the trigonometric functions. The integral of cosx is sinx, and the integral of sinx is -cosx, so we have:
∫2cosx dx - ∫7sinx dx
= 2∫cosx dx - 7∫sinx dx
= 2sinx - 7(-cosx) + C
where C is the constant of integration.
Therefore, the antiderivative of f(x) = 2cosx - 7sinx is F(x) = 2sinx + 7cosx + C.
An antiderivative of a function f(x) is a function F(x) whose derivative is equal to f(x). In other words, if we take the derivative of F(x), we get f(x). It is also known as an indefinite integral of f(x). The antiderivative of a function f(x) is not unique; there may be many functions whose derivative is equal to f(x), differing only by a constant term.
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Iliana wants to find the perimeter of triangle ABC. She uses the distance formula to determine the length of AB. Finish Iliana’s calculations to find the length of AB.
What is the perimeter of triangle ABC? Round the answer to the nearest tenth, if necessary.
10 units
11 units
12 units
13 units
12 units is the perimeter of triangle ABC.
What does a triangular response mean?
It has three straight sides and is a two-dimensional figure. As a 3-sided polygon, a triangle is included. Three triangle angles added together equal 180 degrees.
Three edges and three vertices make up the three sides of a triangle, which is a three-sided polygon. The fact that the interior angles of a triangle add up to 180 degrees is the most crucial aspect of triangles.
Coordinates of triangle ABC:
A = (-1,3), B = (3,6), C = (3,3)
Distance formula (x₁ , y₁ ) (x₂ , y₂ )
d = √(x₂ - x₁)² + (y₂ - y₁)²
Distance of AB: A = (-1,3), B = (3,6)
AB = √(3 - (-1))² + (6 - 3)² = 5 Units
Distance of BC: B = (3,6) , C = (3,3)
BC = √(3 - 3)² + (3 - 6)² = 3 units
Distance of CA: C = (3,3) , A = (-1,3)
CA =√((-1) - 3)² + (3 - 3 )² = 4 Units
Perimeter of the triangle ABC = AB +BC + CA
= 5 units + 3 units + 4 units = 12 units
12 units is the perimeter of triangle ABC.
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Write an equation for the line passing through the point P that are parallel parallel, and the perpendicular to the given line, P(-1,2) 2x+y = 4:L
The equations for the parallel and perpendicular lines to the given line passing through point P(-1, 2) are:
Parallel line: y - 2 = -2(x + 1)
Perpendicular line: y - 2 = 1/2(x + 1)
The equations for the parallel and perpendicular lines to the given line passing through point P(-1,2).
First, let's find the slope of the given line, 2x + y = 4. We can rewrite it in slope-intercept form (y = mx + b) to determine the slope:
y = -2x + 4
The slope (m) of the given line is -2.
For the parallel line, the slope will be the same as the given line, so m_parallel = -2. Now we can use the point-slope form (y - y1 = m(x - x1)) and plug in the coordinates of point P(-1, 2) and the parallel slope:
y - 2 = -2(x + 1)
For the perpendicular line, the slope will be the negative reciprocal of the given line's slope, so m_perpendicular = 1/2. Now, we can use the point-slope form and plug in the coordinates of point P(-1, 2) and the perpendicular slope:
y - 2 = 1/2(x + 1)
So, the equations for the parallel and perpendicular lines to the given line passing through point P(-1, 2) are:
Parallel line: y - 2 = -2(x + 1)
Perpendicular line: y - 2 = 1/2(x + 1)
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A study analyzed the sustainability behaviors of CPA corporations. The level of support for corporate sustainability (measured on a quantitative scale ranging from O to 1 level of support for sustainability 160 points) was obtained for each in a sample of 981 senior managers at CPA fims. The CEO of a CPA fim caims that the true mean is 70. Complete parts a through e. i Click the icon to view the printout a. Specily the null and alternative hypotheses for testing this claim Ha, μ<70 Ос. но p-70 Ha 70 Ha:p>70 b. For this problem, what is a Type I error? A Type ll error? What is a Type I error in this problem? O A. A Type I error would be to conclude that the true mean level of support for sustainability is not 70 when, in fact, the mean is equal to 70 O B. A Type I error would be to conclude that the sample mean level of support for sustainability is less than 70 when, in fact, the sample mean is greater than ๐ c. ○ D 70. A Type error would be to conclude that the true mean level of support for sustainability is 70 when in fact, the mean is not equal to 70. A Type 1 error would be to conclude that the sample mean level of support for sustainablity is greater than 70 when, in fact, the sample mean is less than 70
A Type II error is when you fail to reject the null hypothesis when it is actually false. In this case, a Type II error would be to conclude that the true mean level of support for sustainability is 70 when, in fact, the mean is not equal to 70 (similar to option D, but specifically for a Type II error).
a. The null and alternative hypotheses for testing this claim are:
Hou = 70 (the true mean level of support for sustainability is 70)
Hau < 70 (the true mean level of support for sustainability is less than 70)
b. A Type I error is the rejection of the null hypothesis when it is actually true. In this problem, a Type I error would be to conclude that the true mean level of support for sustainability is less than 70 when, in fact, the true mean is equal to 70.
A Type II error is the failure to reject the null hypothesis when it is actually false. In this problem, a Type II error would be to conclude that the true mean level of support for sustainability is equal to or greater than 70 when, in fact, the true mean is less than 70.
Therefore, the correct answer is A. A Type I error would be to conclude that the true mean level of support for sustainability is not 70 when, in fact, the mean is equal to 70.
a. For this problem, the null and alternative hypotheses are:
Hou = 70 (the true mean level of support for sustainability is 70)
Hau ≠ 70 (the true mean level of support for sustainability is not 70)
b. A Type I error is when you reject the null hypothesis when it is actually true. In this case, a Type I error would be to conclude that the true mean level of support for sustainability is not 70 when, in fact, the mean is equal to 70 (option A).
A Type II error is when you fail to reject the null hypothesis when it is actually false. In this case, a Type II error would be to conclude that the true mean level of support for sustainability is 70 when, in fact, the mean is not equal to 70 (similar to option D, but specifically for a Type II error).
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The acceleration function in (m/s²) and the initial velocity are given for a particle moving along a line. Find a) the velocity at time t, and b) the distance traveled during the given time interval: a(t) = 2t+3, v(0) = -4, 0≤t≤3(a) Find the velocity at time t.(b) Find the distance traveled during the given time interval.
a) The velocity at time t can be calculated using function v(t) = t² + 3t - 4.
b) The distance traveled during the time interval [0, 3] is approximately 30.5 meters.
To find the velocity function v(t), we need to integrate the acceleration function a(t) with respect to time:
a(t) = 2t + 3
∫a(t) dt = ∫(2t + 3) dt
v(t) = ∫(2t + 3) dt = t² + 3t + C
We need to find the constant C using the initial velocity v(0) = -4:
v(0) = 0² + 3(0) + C = C = -4
So the velocity function is:
v(t) = t² + 3t - 4
To find the distance traveled during the time interval [0, 3], we need to integrate the absolute value of the velocity function:
d(t) = ∫|v(t)| dt = ∫|t² + 3t - 4| dt
The velocity changes sign at t = -4 and t = 1, so we need to break the integral into three parts:
d(t) = ∫(-t² - 3t + 4) dt for 0 ≤ t ≤ 1
+ ∫(t² + 3t - 4) dt for 1 ≤ t ≤ 3
+ ∫(-t² - 3t + 4) dt for -4 ≤ t ≤ 0
Evaluating each integral, we get:
d(t) = [-1/3t³ - 3/2t² + 4t] for 0 ≤ t ≤ 1
+ [1/3t³ + 3/2t² - 4t + 11] for 1 ≤ t ≤ 3
+ [1/3t³ + 3/2t² + 4t] for -4 ≤ t ≤ 0
Now we can calculate the distance traveled by subtracting the distance traveled in the negative time interval from the distance traveled in the positive time interval:
d(3) - d(0) = [1/33³ + 3/23² - 43 + 11] - [-1/30³ - 3/20² + 40]
= 30.5
So the distance traveled during the time interval [0, 3] is approximately 30.5 meters.
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Line segment FG begins at (-9,-5) and ends at (8,-5). The segment is translated right 10 units and down 7 units to form line segment F'G'. Enter the distance, in units, of line segment F'G'.
Line segment FG is shifted 10 units to the right and 7 units down to form segment F'G'. Since the y-coordinate is unchanged, the length of segment F'G' is equal to the length of segment FG, which is 17 units.
What is line segment?A line segment is a part of a line that connects two distinct points, and has finite length. It can be measured in terms of distance.
What is coordinates?Coordinates are values that indicate a specific location on a plane or in space, given as ordered pairs or triples of numbers respectively, usually represented as (x, y) or (x, y, z).
According to the given information:
The length of the line segment F'G' can be found using the distance formula, which is:
d = √((x2 - x1)² + (y2 - y1)²)
where ([tex]X_{1} ,Y_{1}[/tex]) and ([tex]X_{2} ,Y_{2}[/tex]) are the coordinates of the endpoints of the segment.
For line segment FG, ([tex]X_{1} ,Y_{1}[/tex]) = (-9,-5) and ([tex]X_{2} ,Y_{2}[/tex]) = (8,-5). So we have:
d = √((8 - (-9))² + (-5 - (-5))²)
d = √(17² + 0²)
d = √(289)
d = 17
Therefore, the distance between F'G' is also 17 units as the translation does not affect the length of the line segment.
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20PLEASE HELP ME THIS IS URGENT IL GIVE 50 POINTS AND I WILL GIVE BRAINLIEST ALL FAKE ANSWERS WILL BE REPORTED AND PLS PLS PLS EXPLAIN THE ANSWER OR HOW U GOT IT PLEASE AND TY
Since we have been told that the angles are complementary then it follows that cos y = 5/13
What are complementary angles?Complementary angles are a pair of angles whose measures add up to 90 degrees. In other words, if angle A measures x degrees, then its complement, angle B, measures (90 - x) degrees.
Now we know that the adjacent of the angle would be;
13^2 = 12^ + x^2
x = √13^2 - 12^2
x = 5
Cos y = 5/13
Then the cosine of the angle y from the relation that we can see in the problem is given as 5/13.
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A die is rolled 10 times and the number of times that two shows on the up face is counted. If this experiment is repeated many times, find the mean for the random variable X, the number of twos thrown out of ten tosses. Round your answer to two decimal places. A. 2.98 B. 3.33 O C. 1.67 OD. 2.50 O E. 8.33
The mean for the random variable X, the number of twos thrown out of ten tosses is 1/6. The answer is A. 2.98, rounded to two decimal places.
To find the mean for the random variable X, we need to multiply the probability of each possible outcome (i.e. the number of twos thrown out of ten tosses) by that outcome, and then add up all the results.
For example, the probability of throwing no twos (i.e. getting other numbers on all ten tosses) is (5/6)¹⁰, and the corresponding outcome is 0. So the contribution to the mean from this outcome is 0×(5/6)¹⁰.
Similarly, the probability of throwing one two (i.e. getting a two once and other numbers on the remaining nine tosses) is 10×(1/6)×(5/6)⁹, and the corresponding outcome is 1. So the contribution to the mean from this outcome is 1×10×(1/6)×(5/6)⁹.
We can do this for all possible outcomes (i.e. throwing 0, 1, 2, …, 10 twos), and add up the results to get the mean.
Using this method, we get:
Mean(X) = 0×(5/6)¹⁰ + 1×10×(1/6)×(5/6)⁹ + 2×(1/6)²×(5/6)⁸ + 3×(1/6)^3×(5/6)⁷ + 4×(1/6)⁴×(5/6)⁶ + 5×(1/6)⁵×(5/6)⁵ + 6×(1/6)⁶×(5/6)⁴ + 7×(1/6)⁷×(5/6)³ + 8×(1/6)⁸×(5/6)² + 9×(1/6)⁹×(5/6) + 10×(1/6)¹⁰
Using a calculator, we get:
Mean(X) ≈ 2.98
Therefore, the answer is A. 2.98, rounded to two decimal places.
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26. The coefficients of the power series sum[n=0,inf] a_n(x-2)^n satisfy a_0 =5 and a_n = ((2n+1)/(3n-1))*a_(n-1) for all n>_ 1. The radius of convergence of the series is
The Radius of convergence is r = 1/2.
To find the radius of convergence of the power series, we can use the ratio test:
r = lim |a_{n+1}(x-2)^{n+1}| / |a_n(x-2)^n|
= lim |a_{n+1}| / |a_n| |x-2|
= lim ((2n+3)/(3n+1)) |x-2|
where we used the recurrence relation to simplify the expression for |a_{n+1}/a_n|. The limit exists and is finite for all x, so the radius of convergence is the value of |x-2| for which the limit is 1:
1 = lim ((2n+3)/(3n+1)) |x-2|
= lim ((2n+3)/(3n+1)) |x-2|/|n|
= |x-2| lim ((2/n)+(3/(n(3n+1))))
= |x-2| * 2
Therefore, the radius of convergence is r = 1/2.
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1. The concentration of a drug t hours after beinginjected is given by C(t)= 0.6t / t^2+24 .Find the time when the concentration is at a maximum. Give youranswer accurate to at least 2 decimal plac
The time when the concentration of the drug is at a maximum is approximately 4.90 hours.
To find the time when the concentration of the drug is at a maximum, we need to find the value of t that maximizes the concentration function C(t).
We can start by taking the derivative of C(t) with respect to t:
C'(t) = [(0.6)(t² + 24) - (0.6t)(2t)] / (t² + 24)²
Simplifying this expression, we get
C'(t) = [0.6(24 - t²)] / (t² + 24)²
To find the maximum value of C(t), we need to find the value of t that makes C'(t) equal to zero. So, we set C'(t) = 0 and solve for t:
0.6(24 - t²) = 0
24 - t² = 0
t² = 24
t = ±√(24) ≈ ±4.90
Since we are interested in a positive value of t, we take t ≈ 4.90 as the time when the concentration of the drug is at a maximum.
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Which of the following expressions is equivalent to
20 - 3(x + 2)?
A:3x+14
B:- 3x + 14
C: -9x+42
D: 17x - 34
Answer: B (-3x+14)
Step-by-step explanation:
We need to solve 20-3(x+2)
Following PEMDAS firstly we'll do -3(x + 2) or (x + 2) times -3.
-3*x = -3x
-3*2 = -6
We now have 20 - 3x - 6. We can group like terms.
14 - 3x or -3x + 14.
Answer:
B:-3x+14
Step-by-step explanation:
20-3(x+2).
20-3x-6.
-3x-6+20
-3x+14
When Emily went to work, it was 83.8 degrees outside. When she was done working, it was 67.9 degrees outside. What was the change in temperature?
Question 4 options:
-15.8
-15.9
-16.0
-16.2
pls help 3
Answer:
b
Step-by-step explanation:
To find the change in temperature, we need to subtract the initial temperature from the final temperature:
Change in temperature = Final temperature - Initial temperature
Change in temperature = 67.9 - 83.8
Change in temperature = -15.9
Therefore, the change in temperature is -15.9 degrees, which means the temperature decreased by 15.9 degrees.
The answer is option B, -15.9.
Police estimate that 22% of drivers drive without their seat belts. If they stop 4 drivers at random, find the probability that all of them are wearing their seat belts.
The probability that all four drivers are wearing their seat belts is 0.456, or about 45.6%.
The probability of a driver wearing a seat belt is 1-0.22 = 0.78.
We can model the situation as a binomial distribution, where the number of trials (n) is 4 and the probability of success (p) is 0.78.
The probability that all four drivers are wearing their seat belts can be calculated using the binomial probability formula:
P(X = 4) = (n choose X) * [tex]p^X * (1 - p)^(n - X)[/tex]
where n = 4, X = 4, p = 0.78, and (n choose X) = 1.
Plugging in these values, we get:
[tex]P(X = 4) = 1 * 0.78^4 * (1 - 0.78)^(4 - 4)[/tex]
= 0.456
Therefore, the probability that all four drivers are wearing their seat belts is 0.456, or about 45.6%.
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Determine f(t) when f"(t) = 6(2t+1) and f'(1) = 3, f(1) = 4
The Function is: f(t) = (2/3)[tex]t^3[/tex] + (3/2[tex])t^2[/tex] - t + 2/3
To determine f(t) given f''(t) = 6(2t+1), f'(1) = 3, and f(1) = 4, we need to integrate the given function twice.
First, we integrate f''(t) = 6(2t+1) once to get f'(t):
f'(t) = 2[tex]t^2[/tex] + 3t + C1, where C1 is the constant of integration.
Next, we integrate f'(t) = 2[tex]t^2[/tex] + 3t + C1 once again to get f(t):
f(t) = (2/3)[tex]t^3[/tex] + (3/2)[tex]t^2[/tex] + C1t + C2, where C2 is the constant of integration.
Using the initial conditions f'(1) = 3 and f(1) = 4, we can solve for the constants C1 and C2:
f'(1) =[tex]2(1)^2 + 3(1) + C1 = 3[/tex]
C1 = -1
f(1) =[tex](2/3)(1)^3 + (3/2)(1)^2 - 1(1) + C2 = 4[/tex]
C2 = 2/3
Therefore, the final solution is:
f(t) =[tex](2/3)t^3 + (3/2)t^2 - t + 2/3[/tex]
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If the amount of gasoline purchased per car at a large service station has a population mean of $34 and a population standard deviation of $2 and a random sample of 100 cars is selected, find the value of the standard deviation of the sample mean.
The standard deviation of the sample mean is also known as the standard error of the mean. It can be calculated by dividing the population standard deviation by the square root of the sample size. In this case, the population standard deviation is $2 and the sample size is 100. So, the standard deviation of the sample mean is $2/sqrt(100) = $2/10 = $0.2. Is there anything else you would like to know?
I'm sorry to bother you but can you please mark me BRAINLEIST if this ans is helpfull
at a furniture manufacturer, worker a can assemble a shelving unit in 5 hours. worker b can assemble the same shelving unit in 3 hours. which equation can be used to find t, the time in hours it takes for worker a and worker b to assemble a shelving unit together? rate(shelving units per hour)time(hours)fraction completedworker aone-fiftht worker bone-thirdt 5 t 3 t
The equation we can use to find t is:
t/5 + t/3 = 1
Let's start by finding the individual rates of worker A and worker B in assembling the shelving unit.
Worker A can assemble a shelving unit in 5 hours, so their rate is:
1 unit / 5 hours = 1/5 units per hour
Similarly, worker B can assemble a shelving unit in 3 hours, so their rate is: 1 unit / 3 hours = 1/3 units per hour
Now, let t be the time in hours it takes for worker A and worker B to assemble a shelving unit together. During this time, worker A will have completed a fraction of the shelving unit equal to t/5 (since their rate is 1/5 units per hour), and worker B will have completed a fraction of the shelving unit equal to t/3 (since their rate is 1/3 units per hour).
The total fraction of the shelving unit completed in time t is the sum of the fractions completed by each worker. So we have:
t/5 + t/3 = 1
Multiplying both sides by the least common multiple of 5 and 3 (which is 15), we get:
3t + 5t = 15
8t = 15
Dividing both sides by 8, we get:
t = 15/8
Therefore, it takes approximately 1.875 hours, or 1 hour and 52.5 minutes, for worker A and worker B to assemble a shelving unit together.
So the equation we can use to find t is:
t/5 + t/3 = 1
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To find the time it takes for Worker A and Worker B to assemble a shelving unit together, we can set up an equation based on their work rates. The equation is 1/5 + 1/3 = 1/t. Solving for t, we find that it takes approximately 1.875 hours or 1 hour and 52.5 minutes for both workers to assemble the shelving unit together.
Explanation:To find the time it takes for Worker A and Worker B to assemble a shelving unit together, we can set up an equation using the concept of work rate. Worker A can assemble one shelving unit in 5 hours, so their work rate is 1 unit per 5 hours. Similarly, Worker B can assemble one shelving unit in 3 hours, so their work rate is 1 unit per 3 hours.
To find the combined work rate of Worker A and Worker B, we can add their individual work rates. Let's represent the time it takes for them to assemble the shelving unit together as 't'. The equation for their combined work rate is:
1/5 + 1/3 = 1/t
To solve for 't', we can multiply both sides of the equation by the least common multiple (LCM) of 5 and 3, which is 15. This gives us:
3 + 5 = 15/t
8 = 15/t
t = 15/8
Therefore, it takes approximately 1.875 hours or 1 hour and 52.5 minutes for Worker A and Worker B to assemble a shelving unit together.
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