Assume that a surface S has the property that |k₁| ≤ 1,|k₂| ≤ 1 everywhere. Is it true that the curvature k of a curve on S also satisfies |k| ≤ 1?

The flux can be calculated as follows Flux = ∫₀⁹ ∫₀²π ∫₀ᴨ (y + z + x) ρ^2 sin(φ) dρ dθ dφ. This triple integral will give us the flux of F out of the sphere.

To find the flux of the vector field F = xy i + yz j + zx k out of a sphere of radius 9 centered at the origin, we need to evaluate the surface integral of the vector field over the sphere.

The flux of F across a closed surface S is given by the surface integral ∬S F · dS, where F is the vector field, dS is the outward-pointing vector normal to the surface element, and ∬S represents the double integral over the surface S.

In this case, the surface S is the sphere of radius 9 centered at the origin. We can represent this sphere using the equation x^2 + y^2 + z^2 = 9^2.

To evaluate the flux, we can use the divergence theorem, which states that the flux of a vector field across a closed surface is equal to the triple integral of the divergence of the vector field over the volume enclosed by the surface.

The divergence of F is given by ∇ · F, which can be computed as follows:

∇ · F = (∂(xy)/∂x) + (∂(yz)/∂y) + (∂(zx)/∂z)

= y + z + x

Now, we can apply the divergence theorem to calculate the flux:

Flux = ∭V (∇ · F) dV

Since we are interested in the flux out of the sphere, we can convert the triple integral into a spherical coordinate system. The volume element in spherical coordinates is given by dV = ρ^2 sin(φ) dρ dθ dφ.

The limits of integration for ρ, θ, and φ will be as follows:

ρ: 0 to 9 (radius of the sphere)

θ: 0 to 2π (full revolution around the sphere)

φ: 0 to π (hemisphere)

Thus, the flux can be calculated as follows:

Flux = ∫₀⁹ ∫₀²π ∫₀ᴨ (y + z + x) ρ^2 sin(φ) dρ dθ dφ

Evaluating this triple integral will give us the flux of F out of the sphere.

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Flowcharts are used to summarize large amounts of statistical data and indicate trends over time. The correct options for the flow charts are to summarize large amounts of statistical data and indicate trends over time

Flowcharts are a powerful tool for both technical and non-technical people, providing a visual representation of complex information. They are used to map out workflows, decision-making processes, and other systems.

By breaking down a process into simple steps and depicting them visually, flowcharts allow users to understand the flow of information and actions, making it easier to identify bottlenecks, inefficiencies, and opportunities for improvement. Flowcharts can also help to standardize processes, ensuring that all stakeholders are aligned on the correct procedures.

Overall, flowcharts are a valuable tool for any organization seeking to streamline processes, improve efficiency, and communicate complex ideas in a simple and easy-to-understand format.

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The derivative of a function in calculus is a measure of how quickly the function alters in relation to its independent variable. It calculates the function's slope or rate of change at every given point.

The limit of the difference quotient as the interval approaches 0 is known as the derivative of a function f(x), denoted as f'(x) or dy/dx:

Using the notion of a derivative, we can show that f'(x) = -sin(x) for the function f(x) = cos(x):

lim(h0) = f'(x) [f(x + h) − f(x)] / h

First, let's calculate f(x + h) and f(x):

cos(x + h) = f(x + h).

x = cos(f(x))

We now change these values in the derivative definition to read:

lim(h0) = f'(x) [cos(h + x) - cos(x)] / h

The trigonometric formula cos(a + b) = cos(a)cos(b) - sin(a)sin(b) is then used:

lim(h0) = f'(x) [sin(x)sin(h) − cos(x)cos(h)] / h

Making the numerator simpler:

lim(h0) = f'(x) Sin(x)sin(h) = [cos(x)(cos(h) - 1)] / h

Using the formula cos(0) = 1, say:

lim(h0) = f'(x) Sin(x)sin(h) = [cos(x)(cos(h) - 1)] / h

Next, we divide the numerator's two terms by h:

lim(h0) = f'(x) Sin(x)sin(h) = [cos(x)(cos(h) - 1) / h - h]

As h gets closer to 0, we now take the bounds of each term:

lim(h)[cos(h) - 1][h 0] By applying L'Hôpital's rule and the limit definition of cos(h), / h = 0

According to the limit definition of sin(h), lim(h0) sin(h) / h = 1.

Replacing these restrictions in the derivative expression:

cos(x)(0) = f'(x) - sin(x)(1)

F'(x) = sin(x).

By applying the notion of a derivative, we have demonstrated that if f(x) = cos(x), then f'(x) = -sin(x).

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7.54 and 0.46 are the solutions to the given quadratic equations

Given the quadratic equation below:

x^2-18x+8=0

We need to determine the solutions to the given quadratic expression. Using the general formula below:

x = -b±√b²-4ac/2a

From the equation

a = 1

b = -18

c = 8

Substitute

x = 18±√18²-4(1)(8)/2(1)
x= 18±√324-32/2

x =18± 17.08/2

x = 35.08/2 and 0.92/2

x = 17.54 and 0.46

Hence the two solutions to the given quadratic equation are 17.54 and 0.46

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The probability that the sample mean of 46 racing cars would be greater than 116.9 gallons is 0.0043, or 0.43%.

To solve this problem, we can use the central limit theorem, which states that the sampling distribution of the sample means approaches a normal distribution as the sample size increases.
First, we need to calculate the standard error of the mean, which is the standard deviation of the population divided by the square root of the sample size:
standard error = 7 / sqrt(46) = 1.032
Next, we can standardize the sample mean using the formula:
z = (sample mean - population mean) / standard error
In this case, the population mean is 114 and the sample mean we're interested in is 116.9. So:
z = (116.9 - 114) / 1.032 = 2.662
Finally, we can use a standard normal distribution table or calculator to find the probability that a z-score is greater than 2.662. This probability is approximately 0.0043, rounded to four decimal places.
Therefore, the probability that the sample mean of 46 racing cars would be greater than 116.9 gallons is 0.0043, or 0.43%.

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hello

the answer to the question is B)

explanation:

a point shown on the g(x) graph is (3,3)

if x = 3 and y = 3, therefore:

─ answer A) is incorrect

─ answer B) is the answer since:

(1/3)(x²) = (1/3)(9) = 3

─ answer C) is incorrect since:

((1/3)(x))² = ((1/3)(9))² = 9

─ answer D) is incorrect since:

3x² = 3 × 3² = 27

The sampling distribution is : E. Approximately normal with mean 10 and standard deviation 0.5

The mean of the sampling distribution of the sample means (x-bar) is equal to the population mean (μ). And the standard deviation of this distribution, known as the standard error (SE), is equal to the standard deviation of the population (σ) divided by the square root of the sample size (n).

Given: μ = 10, σ = 10, n = 400

The mean of the sampling distribution (μ_x-bar) is equal to the population mean (μ): μ_x-bar = μ = 10

The standard error (SE) is σ/√n = 10/√400 = 10/20 = 0.5

Therefore, the correct answer is:

E. Approximately normal with mean 10 and standard deviation 0.5

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3. Let X be a random variable that has a skewed distribution with mean = 10 and the standard deviation s =10. Based on random samples of size 400, the sampling distribution of x is  

A. highly skewed with mean 10 and standard deviation 10

B. highly skewed with mean 10 and standard deviation 5

C. highly skewed with mean 10 and standard deviation 5

D. approximately normal with mean 10 and standard deviation 10

E. approximately normal with mean 10 and standard deviation .5

There can be 6,760,000 different license plates manufactured for this state.

To calculate the number of different license plates that can be manufactured for this state, we need to consider the number of options for each character position.

For the two letters, there are 26 options for each letter (A-Z), so the total number of combinations is 26 × 26 = 676.

For the four numbers, there are 10 options for each number (0-9), so the total number of combinations is 10 × 10 × 10 × 10 = 10,000.

To find the total number of different license plates, we multiply the number of combinations for the letters by the number of combinations for the numbers:

676 × 10,000 = 6,760,000.

Therefore, there can be 6,760,000 different license plates manufactured for this state.

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Based on the linear best-fit model, when Mitchell is 62 inches tall, So, approximately he will weigh 312 pounds.

To estimate Mitchell's weight when he is 62 inches tall using a linear best-fit model, we need to determine the equation of the line that best represents the relationship between height and weight based on the given data.

We can use the least squares method to find the equation of the line. By fitting a line to the data points, we can determine the slope (m) and y-intercept (b) of the line.

Using statistical software or calculations, the equation of the best-fit line for the given data is estimated to be:

Weight = 4.96 * Height + 4.48

To find Mitchell's estimated weight when he is 62 inches tall, we substitute 62 for Height in the equation:

Weight = 4.96 * 62 + 4.48

Weight = 307.52 + 4.48

Weight = 312 pounds

Therefore, based on the linear best-fit model, Mitchell is estimated to weigh approximately 312 pounds when he is 62 inches tall.

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The Fourier series of the function f(x) on the interval −8 < x < 8 is given by the following expression: f(x) = A0 + Σ(Akcos(kπx/8) + Bksin(kπx/8)). The series consists of a constant term A0 and an infinite sum of cosine and sine terms, where k represents the harmonic frequencies.

To find the Fourier series of f(x), we need to decompose the function into a sum of harmonically related sinusoidal functions. The interval given is divided into two parts: −8 < x < 0 and 0 ≤ x < 8. In the first interval, −8 < x < 0, f(x) is a constant function with a value of 1. The constant term A0 in the Fourier series represents the average value of the function and is given by A0 = 1/2.

In the second interval, 0 ≤ x < 8, f(x) is a linear function with a slope of 1. This part of the function can be expressed as f(x) = x. The coefficients Ak and Bk in the Fourier series represent the amplitudes of the cosine and sine terms, respectively. Ak is given by 1/(kπ), and Bk is given by (2/π)*sin(kπ/2).

By combining the constant term A0 with the cosine and sine terms, we obtain the Fourier series representation of f(x) on the interval −8 < x < 8: f(x) = A0 + Σ(Akcos(kπx/8) + Bksin(kπx/8)). This series represents the function f(x) as an infinite sum of harmonics, which can be used to approximate the original function over the given interval.

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The two tow trucks are exerting forces of 2,850 N and 2,655 N on a stuck truck via 12 m long towing straps attached to its hitch. The angle between the two trucks is 42. We have to determine the magnitude of the resultant force.

The formula to find the magnitude of the resultant force is given below:[tex]F = √(F₁² + F₂² + 2F₁F₂cosθ) where, F₁ = 2,850 NF₂ = 2,655 Nθ = 42 degrees F = √(2,850² + 2,655² + 2(2,850)(2,655)cos(42))F = 4,325 N (rounded off to th[/tex]e nearest whole number) Hence, the magnitude of the resultant force is 4,325 N.

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The point using Mean Value Theorem for function f(x) = x⁻¹ and interval [1, 7] is,

c = √7.

Mean Value Theorem states that if f(x) is continuous on [a, b] and is differentiable on (a, b) so there is at least one point a < c < b such that

f'(c) = (f(b) - f(a))/(b - a)

Given the function is,

f(x) = x⁻¹ and the interval is = [1, 7]

f(1) = 1⁻¹ = 1 and f(7) = 7⁻¹ = 1/7

Differentiating the function with respect to 'x' we get,

f'(x) = -1 x⁻¹⁻¹ = - x⁻²

Clearly the function f(x) is continuous and differentiable on [1, 7] and (1, 7) respectively since it is polynomial and exists for all points of [1, 7].

So by Mean Value Theorem there exist 1 < c < 7 such that

f'(c) = (f(7) - f(1))/(7 - 1)

- c⁻² = (1/7 - 1)/6 = (-6/7)/6 = - 1/7

- 1/c² = - 1/7

c² = 7

c = ± √7

Since 1 < c < 7 so, c = √7.

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Answer:

4x^3 and -12x^3

Step-by-step explanation:

4x^3 and -12x^3 because they both have x^3 in the expressions which means you can add or take them away from each other to simplify it

In this scenario, no lurking variable is mentioned. The study found a strong correlation between SAT scores and college grades, indicating a direct relationship between the two variables.

1. Identify the variables: The variables mentioned in the scenario are SAT scores and college grades. These are the main focus of the study.

2. Determine the correlation: The study indicates that a strong correlation exists between SAT scores and college grades. This suggests that higher SAT scores tend to be associated with higher college grades.

3. Evaluate lurking variables: In this case, no additional variables are mentioned or implied. It is possible that the study accounted for other factors, such as student demographics or study habits, to ensure the correlation between SAT scores and college grades was not confounded by other variables.

4. Conclusion: Based on the information provided, there is no indication of a lurking variable. The study simply found a strong correlation between SAT scores and college grades, suggesting a direct relationship between the two variables.

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The probability that a product will function properly for a specified time under stated conditions is determined by its reliability, durability, and functionality.

These factors are influenced by the quality of materials used in manufacturing, the maintenance schedule of the product, and its fitness for use. The higher the reliability, durability, and functionality of a product, the higher the probability that it will function properly for the specified time under the stated conditions. Therefore, it is important to consider these factors when assessing the performance of a product and determining its fitness for use. The probability that a product will function properly for a specified time under stated conditions is referred to as its reliability. Reliability is an essential aspect of a product's overall quality, as it indicates the product's durability and fitness for use. In order to maintain a high level of reliability, proper functionality and maintenance must be ensured throughout the product's lifetime.

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The order of the product ab in the abelian group is lcm(m, n).

In an abelian group, the order of the product of two elements can be determined using the concept of the least common multiple (LCM) of their individual orders.

Let a and b be elements of an abelian group, where the order of a is m and the order of b is n. The order of an element in a group is defined as the smallest positive integer k such that the element raised to the power of k yields the identity element.

In this case, the order of the product ab can be determined by considering the LCM of m and n, denoted as lcm(m, n). The LCM is the smallest positive integer that is divisible by both m and n.

Therefore, the order of the product ab in the abelian group is lcm(m, n).

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To estimate the area under the graph of f(x) = 20x from x = 0 to x = 4, we can use the concept of numerical integration, specifically the trapezoidal rule.

The trapezoidal rule approximates the area under a curve by dividing the interval into small trapezoids and summing up their areas.

Here's how we can estimate the area using the trapezoidal rule:

Since we don't know the specific value of n, let's assume we divide the interval into 4 subintervals, resulting in Δx = (4 - 0) / 4 = 1.

Now, let's calculate the estimated area using the trapezoidal rule:

Area ≈ [(f(0) + f(1)) * Δx / 2] + [(f(1) + f(2)) * Δx / 2] + [(f(2) + f(3)) * Δx / 2] + [(f(3) + f(4)) * Δx / 2]

Substituting the values of f(x) = 20x:

Area ≈ [(20(0) + 20(1)) * 1 / 2] + [(20(1) + 20(2)) * 1 / 2] + [(20(2) + 20(3)) * 1 / 2] + [(20(3) + 20(4)) * 1 / 2]

= [(0 + 20) * 1 / 2] + [(20 + 40) * 1 / 2] + [(40 + 60) * 1 / 2] + [(60 + 80) * 1 / 2]

= [10] + [30] + [50] + [70]

= 160

Therefore, the estimated area under the graph of f(x) = 20x from x = 0 to x = 4 is approximately 160 square units.

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The Volume of sphere is 3,589.543 ft³.

We have,

Diameter of sphere = 19 ft

Radius of sphere= 19/2

So, the formula for Volume of sphere

= 4/3 πr³

= 4/3 x 3.14 x 19/2 x 19/2 x 19/2

= 86,149.04 / 24

= 3,589.543 ft³

Thus, the Volume of sphere is 3,589.543 ft³.

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Answer:

-2 slope

Step-by-step explanation:

when 2 lines are parallel they have the same slope. line a will have a slope of -2.

I and II are correct,  y = 0.10x and y = 0.12x are two equations  will Daniel need to use to calculate the amount of income tax on his taxable income.

We have information available from the question:

Daniel’s taxable income after taking the standard deduction is $33,914.

Let x is the amount of taxable income that’s taxed at the corresponding marginal tax rate.

and, y is the amount of taxes owed.

We need to find the equations will Daniel need to use to calculate the amount of income tax on his taxable income

Now, According to the question:

y = 0.10x and y = 0.12x are two equations  will Daniel need to use to calculate the amount of income tax on his taxable income.

Hence, I and II are correct,  y = 0.10x and y = 0.12x are two equations  will Daniel need to use to calculate the amount of income tax on his taxable income

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The simplified form of the given expression is (3x - 4x^2 - 15x - 20)/(7x^2 - 33x - 70).

o simplify the expression (-4)/(x-5) + 3x/(7 (x+2)), we need to find a common denominator and combine the fractions.

   The first term (-4)/(x-5) has the denominator (x-5), while the second term 3x/(7 (x+2)) has the denominator 7(x+2). To find a common denominator, we multiply the first term by 7(x+2) and the second term by (x-5).

   After multiplying, we get (-4)(7(x+2))/(7(x+2)(x-5)) + (3x)(x-5)/(7(x+2)(x-5)).

   Simplifying the numerator, we have -28x - 56 + 3x^2 - 15x.

   Combining like terms, the numerator becomes -4x^2 - 43x - 56.

   The denominator remains as 7(x+2)(x-5).

   The final simplified expression is (-4x^2 - 43x - 56)/(7(x+2)(x-5)).

Now, let's simplify the second expression: 1/(x^2+7x) + 2/(49-x^2).

   The denominators are x^2+7x and 49-x^2. To find a common denominator, we multiply the first term by (49-x^2) and the second term by (x^2+7x).

   After multiplying, we get (49-x^2)/(x^2+7x)(49-x^2) + (2)(x^2+7x)/(x^2+7x)(49-x^2).

   Simplifying the numerator, we have (49-x^2) + 2x^2 + 14x.

   Combining like terms, the numerator becomes 51 + x^2 + 14x.

   The denominator remains as (x^2+7x)(49-x^2).

   The final simplified expression is (51 + x^2 + 14x)/[(x^2+7x)(49-x^2)].

Therefore, the simplified form of the given expression is (3x - 4x^2 - 15x - 20)/(7x^2 - 33x - 70) + (51 + x^2 + 14x)/[(x^2+7x)(49-x^2)].

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Triangle KVM is similar to triangle BVG because angle M = angle G = 90° and angle V is common to both triangles.

Two triangles are similar if the angles are the same size or the corresponding sides are in the same ratio.

For two triangles to be similar, the corresponding angles must be congruent i.e equal.. Also the ratio of the corresponding sides of similar triangles are equal.

angle M and G are both 90° , this means they are equal.

angle KVM = BVG

therefore angle K = angle B

Since all the corresponding angles are equal, we can say triangle KVM is similar to triangle BVG

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The probability that the sample proportion will be between 0.9115 and 0.946 is 0.1496.

To calculate the probability that the sample proportion will be between 0.9115 and 0.946, we can use the sampling distribution of the sample proportion, assuming that the sample is taken from an infinite population.

The standard deviation of the sample proportion is given by:

σ_p = sqrt((p * (1 - p)) / n)

where p is the population proportion and n is the sample size.

In this case, p = 0.85 and n = 51. Plugging these values into the formula, we get:

σ_p = sqrt((0.85 * (1 - 0.85)) / 51)

= sqrt(0.127275 / 51)

≈ 0.092

Now, we can standardize the interval (0.9115, 0.946) using the sample proportion distribution:

z1 = (0.9115 - p) / σ_p

= (0.9115 - 0.85) / 0.092

≈ 0.667

z2 = (0.946 - p) / σ_p

= (0.946 - 0.85) / 0.092

≈ 1.043

Next, we can calculate the probability using the standard normal distribution:

P(0.9115 < p < 0.946) = P(z1 < Z < z2)

Looking up the values in the standard normal distribution table, we find:

P(0.9115 < p < 0.946) ≈ P(0.667 < Z < 1.043)

≈ 0.1496

Therefore, the probability that the sample proportion will be between 0.9115 and 0.946 is approximately 0.1496.

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Answer:

Step-by-step explanation:

Which ones have four sides?

A

B

D

In testing the hypotheses H0: p = 0.5 vs Ha: p > 0.5, the test statistic is found to be 1.83. We need to determine the correct p-value. From the options provided, the correct p-value is d) 0.0336

The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic, assuming that the null hypothesis is true. Since this is a right-tailed test (Ha: p > 0.5), we are interested in the probability of observing a test statistic larger than 1.83. Looking at the given options, the correct p-value would be the smallest value that corresponds to a probability larger than 1.83. From the options provided, the correct p-value is d) 0.0336, as it represents a probability smaller than 1.83. Therefore, 0.0336 is the correct p-value for this hypothesis test.

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The answer is 1.
The expected value of the random variable Y = 2Z + 1, where Z has a standard normal distribution, can be calculated as follows:

First, we need to find the expected value of Z, which is 0 since Z follows a standard normal distribution with a mean of 0 and a standard deviation of 1.

Next, we substitute the value of Z into the expression for Y: Y = 2(0) + 1 = 1.

Therefore, the expected value of Y is 1.

In this case, since Z has a standard normal distribution, it has a mean of 0. When we transform Z by multiplying it by 2 and adding 1, the mean is also shifted by the same amount. The mean of Y is given by E(Y) = E(2Z + 1) = 2E(Z) + 1 = 2(0) + 1 = 1. Thus, the expected value of Y is 1. This means that, on average, the value of Y is expected to be 1 when Z follows a standard normal distribution.

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A recurrence relation is a mathematical equation or formula that defines a sequence or series of values based on one or more previous terms in the sequence. The recurrence relation here is t(n) = n!

To solve the given recurrence relation t(n)=t(n-1) n, we can start by finding some initial values. Let's consider the base case t(1) = 1.

Now, we can use the recurrence relation to find t(2), t(3), t(4), and so on:

t(2) = t(1) * 2 = 1 * 2 = 2

t(3) = t(2) * 3 = 2 * 3 = 6

t(4) = t(3) * 4 = 6 * 4 = 24

We can see a pattern emerging: t(n) = n!.

So, the solution to the recurrence relation t(n) = t(n-1) * n is t(n) = n!, where n is a positive integer. This means that the value of t(n) is the product of all positive integers from 1 to n.

For example, t(5) = 5! = 1 * 2 * 3 * 4 * 5 = 120.

Therefore, the solution to the recurrence relation is t(n) = n!

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Answer:

Step-by-step explanation:

The number of boys with heights between 122 and 162 cm  is  499.

We first find  the z-scores for these heights using the formula:

z = (x - μ) / σ

where x =  height,

μ = mean height,

σ =  standard deviation.

case where x = 122 cm:

z = (122 - 148) / 12 = -2.1667

case where  x = 162 cm:

z = (162 - 148) / 12 = 1.1667

We then make use of a standard normal distribution table and determine  area under the curve between these z-scores:

Area under  z = -2.1667 and z = 1.1667 is 0.8315.

Hence, the number of boys with heights between 122 cm and 162 cm is:

600 * 0.8315 = 498.9 or 499 boys.

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#complete question

The heights of 600 boys are found to approximately follow such a distribution, with a mean height of 148 cm and a standard deviation of 12 cm. Find the number of boys with heights between: 122 cm and 162 cm

Step-by-step explanation:

Just by inspection ( counting the squares) you can see it is more than3  and less than 11 or 15    so   area = 7 square units