REALLY EASY QUESTION!
The city council voted on a new tax. The council has 85 members and 20% of the council members voted in favor of the new tax. How many members voted in favor of the tax?

Answer:

$47

Step-by-step explanation:

First, we have to find 1% by taking

2.82 divided by 6 = $0.47

To find the original, we take

0.47 times 100 = $47

So, the original price of the sweatshirt is $47

The original price of the sweatshirt is, $47

A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.

To Calculate the percent of a number , divide the number by whole number and multiply by 100.

Given that;

Suppose you paid $2.82 sales tax for a sweatshirt in state where the sales tax is 6%.

Let the original price of the sweatshirt is, x

Hence, We can formulate;

6% of x = 2.82

6x / 100 = 2.82

6x = 2.82 x 100

6x = 282

x = 282 / 6

x = $47

Thus, The original price of the sweatshirt is, $47

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Answer:

3/14

Step-by-step explanation:

They traveled 5/7 of the distance on Saturday and 1/2 of the remaining

distance, 3/7, on Sunday so..

(1/2)(3/7) = 3/14

Hope this helps :)

Answer:Lupe's family traveled 5/7 of the total distance to her aunt's house on Saturday, so the remaining distance was 1 - (5/7) = 2/7.

On Sunday, they traveled 1/2 of the remaining distance, which was 2/7, so the total distance traveled on Sunday was 1/2 * 2/7 = 2/14 of the total distance.

Step-by-step explanation:

Lupe's family traveled 5/7 of the total distance to her aunt's house on Saturday, so the remaining distance was 1 - (5/7) = 2/7.

On Sunday, they traveled 1/2 of the remaining distance, which was 2/7, so the total distance traveled on Sunday was 1/2 * 2/7 = 2/14 of the total distance. :D

The composite functions f ∘ g and g ∘ f is defined as follows:

f ∘ g (x) = f(g(x))

g ∘ f (x) = g(f(x))

In other words, f ∘ g is the composition of two functions f and g, such that the output of g is used as the input to f. Similarly, g ∘ f is the composition of two functions g and f, such that the output of f is used as the input to g.

It's important to note that the order in which the functions are composed matters, as f ∘ g is not necessarily equal to g ∘ f. In general, the composite function of two functions is not commutative, meaning that the order in which the functions are composed affects the result.

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We can prove the equality using mathematical induction. We assume it holds for n and show it holds for n+1. Then, we add n+1 to both sides of the equation, proving the equality holds for n+1.

To prove the equality using mathematical induction, we must first assume it holds for n. Then, we can add n+1 to both sides of the equation:

∑ ― 1 = 1 ( 1 i(i 1)) + (n + 1) = 1 ― 1/ + (n + 1)

The right side of the equation is equal to 1 ― 1/ + n + 1, which simplifies to 1 ― 1/(n+1), proving the equality holds for n+1. Therefore, the equality holds for all n ≥ 2, and we have proven the equation using mathematical induction. Note: The expression in the equation should be n(n+1)/2, not 1/i(i+1).

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The triangle with the greater perimeter has angles 55, 15, and 112 degrees.

In a triangle, the sum of the interior angles is 180 degrees. Thus, if we have two non-congruent triangles with one angle equal (in this case, B = 55 degrees), we can set up an equation to solve for the third angle of the triangle with the greater perimeter.

Let's call the third angle in the first triangle x. Then, x + 55 + b = 180 degrees, or x + b = 125 degrees.

Since we know that a + b + B = 180 degrees for any triangle, we can set up a similar equation for the second triangle: y + 55 + a = 180 degrees, or y + a = 125 degrees.

Since the perimeter of a triangle is equal to the sum of its side lengths, and we know that a = 15 and b = 13, we can set up another equation to find the triangle with the greater perimeter:

Perimeter 1 = 15 + 13 + x and Perimeter 2 = 15 + 13 + y.

Solving for x, we find that x = 125 - b = 125 - 13 = 112 degrees. Solving for y, we find that y = 125 - a = 125 - 15 = 110 degrees.

Since 112 > 110, the triangle with the greater perimeter has angles 55, 15, and 112 degrees.

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The triangle with the greater perimeter is the one with the greater value of c, which we can calculate using the law of cosines and the law of sines.

Given two noncongruent triangles with angle B equal to 55 degrees and sides a and b, we need to find the measures of the angles of the triangle with the greater perimeter.

First, let's use the law of cosines to find the measure of angle A in each triangle:

[tex]c^2 = a^2 + b^2 - 2ab * cos(A)\\\\c = \sqrt{(a^2 + b^2 - 2ab * cos(A))} \\\\cos(A) = (a^2 + b^2 - c^2) / 2ab[/tex]

For the first triangle, we can use the given values:

[tex]cos(A) = (15^2 + 13^2 - c^2) / 2 * 15 * 13\\\\c = \sqrt{(15^2 + 13^2 - 2 * 15 * 13 * cos(A))}[/tex]

For the second triangle, we can use a and b as the lengths of the sides opposite angles A and C, respectively. Then we can use the law of sines to find the lengths of the other sides:

[tex]sin(A) / a = sin(C) / c\\\\c = a * sin(C) / sin(A)[/tex]

Substituting values and solving, we get:

[tex]c = 15 * sin(C) / sin(A)\\\\cos(A) = (15^2 + b^2 - c^2) / 2 * 15 * b\\\\c = \sqrt{(15^2 + b^2 - 2 * 15 * b * cos(A))}[/tex]

Now we can compare the two perimeters (the sum of the lengths of all sides) to determine which triangle has the greater perimeter. The perimeter of each triangle will be a + b + c.

Whichever triangle has the larger value of c will have the greater perimeter, as a and b are the same in both triangles.

So, we can use the value of c that we just calculated for each triangle to determine which triangle has the greater perimeter, and therefore, which triangle has the greater measure of its angles.

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If we know the parameters a, b, and c in the first equation, we can find the equivalent parameters A, B, and C in the second equation by using the above relationships.

What is the logarithmic equation?

A logarithmic equation is an equation that involves the logarithm of an expression containing a variable. To solve exponential equations, first, see whether you can write both sides of the equation as powers of the same number.

The parameters in the two equations are related as follows:

A = log10(Psat/kPa) / ln(10)

B = b * ln(10)

C = c

T = t + 273.15 (to convert from Celsius to Kelvin)

Therefore, the relationship between the parameters in the two equations is:

A = log10(Psat/kPa) / ln(10)

B = b * ln(10)

C = c

T = t + 273.15

Hence, if we know the parameters a, b, and c in the first equation, we can find the equivalent parameters A, B, and C in the second equation by using the above relationships.

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The integral which represents the area between f(x) = x^4 is [tex]A = \int\limits^2_0 {x^4} \, dx[/tex] and the solution is A = 32/5.

A theorem that connects the ideas of integrating and differentiating functions is known as the fundamental theorem of calculus. By calculating the difference between the antiderivative at the higher and lower limits of the integration process, the fundamental theorem of calculus supports the method.

The given function of the area is:

f(x) = x^4.

The integral which represents this area can be written as:

[tex]A = \int\limits^2_0 {x^4} \, dx[/tex]

Using the fundamental Theorem of Calculus we have:

[tex]A = \int\limits^2_0 {x^4} \, dx\\\\A = [\frac{x^5}{5} ]_0^2[/tex]

Substituting the value of limits we have:

A = [2^5 / 5] = 32/5

Hence, the integral which represents the area between f(x) = x^4 is [tex]A = \int\limits^2_0 {x^4} \, dx[/tex] and the solution is A = 32/5.

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The length of the ramp is solved to be 15.33 feet

The length of the ramp is worked using SOH CAH TOA

Sin = opposite / hypotenuse - SOH

Cos = adjacent / hypotenuse - CAH

Tan = opposite / adjacent - TOA

The direction of movements describes a right triangle of

adjacent = 15 feet

hypotenuse = length of the ramp = ?

The length of the ramp is calculated using cos, CAH

let the angle be x

cos x = adjacent / hypotenuse

cos 12 =  15 / hypotenuse

hypotenuse = 15 / cos 12

hypotenuse = 15.334

length of ramp = 15.33 feet

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The line representing the equation part of the constraint [tex]$7 x_1+4 x_2 \leqslant 28$[/tex] goes through (4,0) & (0,7).

Hence, option (c) is the correct choice.

The constraint equation is g(x,y)=c, and we state that x and y are constrained by g(x,y)=c.

Constrained maximum and constrained minimum points are locations (x,y) that are maxima or minima of f(x,y) with the condition that they meet the constraint equation g(x,y)=c.

A constraint function can be translated into a new form that is equal to the original function; that is, the constraint boundary and feasible set for the problem remain unchanged, but the function's form does.

The given constraint is:

[tex]7 x_1+4 x_2 \leq 2[/tex]

If x_1=0

So, we will get:

4x_2=28

=>x=7

If x_2=0

7 x_1 =28

x_1 =4

So, the points are: (4,0)

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The bond yield that comes closest to 7.75%, the  amount yield to maturity on an annual-pay bond, is 7.61%.

A. The bond yield that comes closest to 7.75%, the yield to maturity on an annual-pay bond, is 7.61%. A measure of return that considers the market parameters used to determine the return from an investment in a bond is called the bond-equivalent yield. It accounts for the time value of money, the coupon rate, and the current market rate to determine the yearly rate of return that would be generated if the bond were kept for a year. When comparing bonds with differing coupon rates and maturities, the bond-equivalent yield is utilised. It is a helpful tool for investors who want to compare various bonds and decide on their investments with knowledge.

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The check for independence in a one- sample is

The population size is further than 10 times the sample size.

The Correct option is D.

What is Z- test?

A Z- test is a statistical thesis test used to determine whether a sample mean significantly differs from a given population mean when the population standard divagation is known.

For a one- sample z- test for a population proportion, the sample is named without relief from a finite population.

In order for the test to be valid, it's necessary to insure that the sample is small relative to the population size.

still, it can be considered large enough to satisfy the independence supposition, If the population size is further than 10 times the sample size. This supposition is necessary for the validity of statistical conclusion in this environment.

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Women Emma Watson spoke intelligently, significantly, and powerfully about gender inequity and how to combat it.

She did this in order to begin the HeForShe campaign, which tries to recruit men and boys to the feminist movement's struggle for gender parity.

The purpose of this campaign is to encourage as many men and boys to support gender equality as they can. The terrible reality that "feminism" is all too frequently associated with "man hate," effectively stopping the progress toward attaining gender equality, was brought to light by Emma Watson in her powerful statement.

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Tt is also possible neither y nor z is x.

Now, According to the question;

The x and y-axis are two important lines of the coordinate plane. The x-axis is a horizontal number line and the y-axis is a vertical number line. These two axes intersect perpendicularly to form the coordinate plane. The x-axis is also called the abscissa and the y-axis is called the ordinate.

If x is true, a non true statement cannot be the same as it.

Now, In symbolic logic, = doesn't mean two statements have the same value—it means they are the same literal statement.

So, if x is true, and y and z contradict, at least one of y and z is false, and that false statement cannot possibly be x.

Hence,  it is also possible neither y nor z is x.

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Note that the missing values in the table are:

1) A - 28 : 20

2) B: - 63 : 49.

Note that this is a test of ratios and proportionality.

Note that the first tape comprises 7 boxes for the Fruit Mass and

the second tape 5 boxes for the Buiscuit mass.

So we know that the the figures represent Lowest Common Multiples when we are presented with the table.

From the table, we see that Fruitmass has a related number which is 28. Immediately we can see the relationship.

7 x 4 = 28

This means that the missing number can be arrived at by saying:

5 x 4 = 20

Repeat this same logic for the missing number in B to obtain: 63: 49.

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It is not possible to guarantee the exact number of points within a certain distance of each other in a square with 42 points placed inside.

Since the side length of the square is 2, the area of the square is 2 x 2 = 4. If 42 points are placed inside the square, the maximum distance between any two points will be equal to the diagonal of the square, which is equal to the square root of 2 times the side length of the square. The diagonal of the square is 2 * √(2), or approximately 2.82.

The number of points within a certain distance of each other depends on the size of the distance, as well as the density of the points in the square. However, it is not possible to guarantee that all 42 points will be within a certain distance of each other. The best we can say is that most of the points will be relatively close to each other, but some may be farther apart. The exact number of points that are within a certain distance of each other will depend on the specific arrangement of the points.

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It is not possible to guarantee the exact number of points within a certain distance of each other in a square with 42 points placed inside.

Since the side length of the square is 2, the area of the square is 2 x 2 = 4. If 42 points are placed inside the square, the maximum distance between any two points will be equal to the diagonal of the square, which is equal to the square root of 2 times the side length of the square. The diagonal of the square is 2 * √(2), or approximately 2.82.

The number of points within a certain distance of each other depends on the size of the distance, as well as the density of the points in the square. However, it is not possible to guarantee that all 42 points will be within a certain distance of each other. The best we can say is that most of the points will be relatively close to each other, but some may be farther apart. The exact number of points that are within a certain distance of each other will depend on the specific arrangement of the points.

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Step-by-step explanation:

For two triangles to be congruent, they either need to have three respectively equal sides (SSS), two respectively equal signs with the same angle between them (SAS), two same angles with a respectively equal side between them (ASA), or for a right-angled triangel, a respectively equal hypotenuse and side (RHS).

Here we have the triangles ΔABC and ΔCDE. We know DC is equal to BC because they are marked as such, same for AC and EC. We also know <ACB = <DCE because they are vertically opposite angles. Since we have two triangles with equal sides and the same angle between these sides (SAS), we can say these triangles are congruent.

The time taken by both of them to work on the same page is 2 hours. The number of pages written by Emmy is 53 and the number of pages written by Warren is 53.

Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.

Given that Emmy has already completed 27 pages and will continue working at a rate of 13 pages per hour. Warren has completed 33 pages and can work at a rate of 10 pages per hour.

The time taken to reach the same page is,

13t + 27 = 33 + 10t

13t - 10t = 33 - 27

3t = 6

t = 2 hours

The number of pages written by Emmy is,

N = 13t + 27

N = ( 13 x 2) + 27 = 53

The number of pages written by Warren is,

N = 33 + 10t

N = 33 + 20

N = 53

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Answer:

-117

Step-by-step explanation:

2x1+48-5x48+96-23=-117

First multiplication

2×1=2

5×48=240 ->

2+48-240+96-23

2+48=50

50-240=-190

-190+96=-94

-94-23=-117

Hope that helped! Let me know if you have any further questions.

It should be noted that 100,000 attendees for 10 bands a reasonable prediction as D. Yes, the pattern is linear. The average rate of change is about 10,000.

A straight-line link between two variables is referred to statistically as a linear relationship (or linear association).

A positive slope indicates a positive linear relationship, meaning that as one increases, the other also increases.

From the information, the year’s festival will feature 10 bands and the local TV station reports that the expected attendance is 100,000. The average rate will be:

= 100000 / 10

= 10000

The correct option is D.

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The composite function f (ƒ-¹ (5)) is 5

From the question, we have the following parameters that can be used in our computation:

f(x) = (x + 10)/5

Also, we have

f-¹ (x) = 5x - 10

The composite function f (ƒ-¹ (5)) is calculated as

f (ƒ-¹ (x)) = (5x - 10 + 10)/5

Evaluate the difference

f (ƒ-¹ (x)) = 5x/5

So, we have

f (ƒ-¹ (x)) = x

Substitute 5 for x

f (ƒ-¹ (5)) = 5

Hence, the solution is 5

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Complete question

Use the given function to find the following.

f(x) = (x + 10)/5

f-¹ (x) = 5x - 10

Find f (ƒ-¹ (5)). (no spaces)

The solution to the system of equations is (0.839, 1.129).

What is a graph ?

Graph can be defined as visual representation of the data through the given ordered pairs .

Given ,

The graph of a line passes through the points (0, 0.5) and (6, 5). A second i has the equation 5y = 7-6x.

The equation of the first line can be found using the two given points, using the slope-intercept form:

y = mx + b

where m is the slope and b is the y-intercept. To find the slope, we can use the formula:

m = (y2 - y1) / (x2 - x1)

Plugging in the values from the two points, we have:

m = (5 - 0.5) / (6 - 0) = 4.5 / 6 = 0.75

So the equation of the first line is:

y = 0.75x + 0.5

To find the solution to the system of equations, we need to find the point where the two lines intersect, which means finding the values of x and y that satisfy both equations. To do this, we can substitute the equation of the first line into the second equation:

5y = 7 - 6x

0.75x + 0.5 = 7 - 6x

Solving for x, we get:

7.75x = 6.5

x = 6.5 / 7.75 = 0.839

Substituting x back into either equation to find y, we get:

y = 0.75x + 0.5 = 0.75 * 0.839 + 0.5 = 0.629 + 0.5 = 1.129

So the solution to the system of equations is (0.839, 1.129).

Therefore, The solution to the system of equations is (0.839, 1.129).

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Answer:

ar many types the total bar o

Answer:

I'm not sure if this is what you meant but if it's not, put a picture of the equation and I'll edit my answer

[tex]\frac{21}{670}[/tex]

Step-by-step explanation:

Assuming the equation is:

[tex]\frac{3}{5} * \frac{7}{2*67}[/tex]

The first thing you have to do is simplify:

2 * 67 = 134

[tex]\frac{3}{5} * \frac{7}{134}[/tex]

Multiply the numerators:

3 * 7 = 21

Then multiply the denominators:

5 * 134 = 670

Then simplify again:

[tex]\frac{21}{670}[/tex]

The friend bought 4.725 pounds of grape, which is 3/8 of the amount that Martin bought

An equation is an expression that shows the relationship between two or more numbers and variables. An equation can either be linear, quadratic, cubic and so on

Mixed fraction is a fraction that is made up of a whole number and a fraction combined together.

Martin bought 12 3/5 pounds of grapes for a picnic.

12 3/5 pound is a mixed fraction

12 3/5 pound = 63/5 pounds

His friend bought 3/8 of that amount. Hence:

Amount of grapes the friend bought = (3/8) * 63/5 pounds = 4.725 pounds

The friend bought 4.725 pounds of grape

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23. g(x)=3(2x+3) has a slope of 6 and y-intercept of 9.

24. g(x)=2(0.5x) +3 has a slope of 1 and y-intercept of 3.

25. g(x)=1(2x+3) has a slope of 2 and y-intercept of 3.

26. g(x)=2(1)x+3 has a slope of 2 and y-intercept of 3.

23. g(x)=3(2x+3) can be written as g(x)=6x + 9. This shows that the slope of the function is 6 and the y-intercept is 9.

24. g(x)=2(0.5x) +3 can be written as g(x)=x + 3. This shows that the slope of the function is 1 and the y-intercept is 3.

25. g(x)=1(2x+3) can be written as g(x)=2x + 3. This shows that the slope of the function is 2 and the y-intercept is 3.

26. g(x)=2(1)x + 3 can be written as g(x)=2x + 3. This shows that the slope of the function is 2 and the y-intercept is 3.

The value of k in the linear function g(x)=k(mx+b) affects the slope of the function by multiplying the slope of the original function m by k. The y-intercept of the function remains unchanged.

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The covariance between the number of coins and the value of the coins is 240.

The covariance between the number of coins (N) and the value of the coins (V) can be calculated as

Cov(N, V) = Cov(n nickels + n dimes, 5n nickels + 10n dimes)

= 5 Cov(n nickels, n nickels) + 10 Cov(n nickels, n dimes) + 5 Cov(n dimes, n nickels) + 10 Cov(n dimes, n dimes)

= 5 * Var(n nickels) + 10 * Cov(n nickels, n dimes) + 5 * Cov(n dimes, n nickels) + 10 * Var(n dimes)

= 5 * 8 + 10 * 5 + 5 * 5 + 10 * 10

= 40 + 50 + 50 + 100

= 240

So the covariance between the number of coins and the value of the coins is 240.

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To develop a 95% confidence interval estimate of the population mean rating for Miami International Airport, the student would need to use the sample data obtained from the 50 business travelers. The steps to develop the confidence interval would include the following:

Calculate the sample mean: The mean of the sample data would be calculated by summing all of the ratings and dividing by the sample size (n = 50).

Calculate the standard deviation: The standard deviation of the sample data would be calculated to estimate the variability of the data.

Calculate the standard error: The standard error of the mean would be calculated by dividing the standard deviation by the square root of the sample size (n = 50).

Calculate the critical value: The critical value would be calculated based on the desired confidence level (95%) and the degrees of freedom (n - 1 = 49). The critical value would be used to determine the width of the confidence interval.

Develop the confidence interval: The lower and upper bounds of the confidence interval would be calculated by subtracting and adding the critical value to the sample mean, respectively.

The resulting confidence interval would give an estimate of the population mean rating for Miami International Airport with a 95% level of confidence. The interpretation of the confidence interval would be that if the sampling process were repeated many times, 95% of the resulting confidence intervals would contain the true population mean.

It is important to note that this process assumes that the sample was taken using simple random sampling techniques and that the data follows a normal distribution.

Therefore, The interpretation of the confidence interval would be that if the sampling process were repeated many times, 95% of the resulting confidence intervals would contain the true population mean.

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To develop a 95% confidence interval estimate of the population mean rating for Miami International Airport, the student would need to use the sample data obtained from the 50 business travelers. The steps to develop the confidence interval would include the following:

Calculate the sample mean: The mean of the sample data would be calculated by summing all of the ratings and dividing by the sample size (n = 50).

Calculate the standard deviation: The standard deviation of the sample data would be calculated to estimate the variability of the data.

Calculate the standard error: The standard error of the mean would be calculated by dividing the standard deviation by the square root of the sample size (n = 50).

Calculate the critical value: The critical value would be calculated based on the desired confidence level (95%) and the degrees of freedom (n - 1 = 49). The critical value would be used to determine the width of the confidence interval.

Develop the confidence interval: The lower and upper bounds of the confidence interval would be calculated by subtracting and adding the critical value to the sample mean, respectively.

The resulting confidence interval would give an estimate of the population mean rating for Miami International Airport with a 95% level of confidence. The interpretation of the confidence interval would be that if the sampling process were repeated many times, 95% of the resulting confidence intervals would contain the true population mean.

It is important to note that this process assumes that the sample was taken using simple random sampling techniques and that the data follows a normal distribution.

Therefore, The interpretation of the confidence interval would be that if the sampling process were repeated many times, 95% of the resulting confidence intervals would contain the true population mean.

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The volume V of the solid obtained by rotating the region bounded by the given curves y = 2[tex]e^{-x}[/tex], y = 2, x = 4; about y = 4 is 256π.

The volume of the solid obtained by rotating the region bounded by the given curves about the line y=4 is given by

V = 2π ∫⁴₀ (4-2[tex]e^{-x}[/tex])² dx

We can calculate the integral using integration by parts.

Let u = 4-2[tex]e^{-x}[/tex]

⇒ du = 2d[tex]e^{-x}[/tex]

and dv = 2dx

⇒ v = x

So,

V = 2π ∫⁴₀ (4-2[tex]e^{-x}[/tex])² dx

 = 2π (x (4-2[tex]e^{-x}[/tex])²  - 0 (4-2[tex]e^{-x}[/tex]})²  - 2∫⁴₀ x d (4-2[tex]e^{-x}[/tex]))

 = 2π (2×4³ - 4∫⁴₀ x [tex]e^{-x}[/tex] dx)

Using integration by parts again

Let u = 4-2[tex]e^{-x}[/tex] ⇒ du = -2d[tex]e^{-x}[/tex]

and dv = xdx ⇒ v = x²/2

So,

V = 2π (2×4³ - 4∫⁴₀ x [tex]e^{-x}[/tex] dx)

 = 2π (2×4³ - 2×2∫⁴₀ x² [tex]e^{-x}[/tex] dx)

Again we can calculate the integral using integration by parts.

Let u = x² ⇒ du = 2xdx

and dv = [tex]e^{-x}[/tex] dx ⇒ v = -[tex]e^{-x}[/tex]

So,

V = 2π (2×4³ - 2×2∫⁴₀4 x^2 [tex]e^{-x}[/tex]} dx)

 = 2π (2 × 4³-2 × 2 × ([tex]-e^{-4}[/tex] + 2∫⁴₀  [tex]e^{-x}[/tex]dx))

 = 2π (2×4³ - 2×2 (1+4))

 = 2π (2×4³ - 16)

 = 2π (128-16)

 = 256π

Therefore, the volume V of the solid obtained by rotating the region bounded by the given curves y = 2[tex]e^{-x}[/tex], y = 2, x = 4; about y = 4 is 256π.

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The unit vector that is parallel to a⃗ is (6/√74)i^−(4/√74)k^.

1. Calculate magnitude of a⃗:

a⃗ = 6.0i^−4.0k^

|a⃗| = √(6.0^2+4.0^2) = √74

2. Construct unit vector, u⃗= (6/√74)i^−(4/√74)k^

u⃗ = (6/√74)i^−(4/√74)k^

3. The unit vector that is parallel to a⃗ is (6/√74)i^−(4/√74)k^.

The unit vector that is parallel to a⃗ is (6/√74)i^−(4/√74)k^.

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Answer: There are 8 11-year-old students and 22 12-year-old students in Liam's class.

Step-by-step explanation:

Let's call the number of 11-year-old students in class x. Then, the number of 12-year-old students in the class would be 30 - x.

The total number of years that the 11-year-old students represent is 11 * x.

The total number of years that the 12-year-old students represent is 12 * (30 - x).

The sum of all the years is 352, so we can set up an equation based on the above information:

11x + 12(30 - x) = 352

Expanding the second term on the left-hand side:

11x + 360 - 12x = 352

Combining like terms:

-x + 360 = 352

Subtracting 360 from both sides:

-x = -8

Dividing both sides by -1:

x = 8

So, there are 8 11-year-old students and 30 - 8 = 22 12-year-old students in Liam's class.