Draw a right triangle with a tangent ratio of 3/2 for one of the acute angles.
Then find the measure of the other acute angle to the nearest tenth of a degree.
cosine

Step-by-step explanation:

Let's find out:

ax - bx + y = z        subtract 'y' from both sides of the equation

ax-bx = z-y             reduce L side

x ( a-b)  = z-y           divide both sides by (a-b)

x = (z-y) / (a-b)         Done.

Rounding to the nearest cent, Sasha will pay $35.10 in interest this month. Therefore, the correct answer is option A.

To calculate the interest that Sasha will pay, we need to use the following formula:

Interest = (Balance * APR * Days in a billing cycle) / 365

where Balance is the amount owed after the payment, APR is the annual percentage rate, and Days in the billing cycle are the number of days in the billing cycle.

Since we do not know the number of days in the billing cycle, we will assume it to be 30 days for simplicity. Therefore, the balance owed after the payment is:

Balance = $3200 - $300 = $2900

Substituting the values into the formula, we get:

Interest = ($2900 * 0.1875 * 30) / 365

= $35.09

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The diagram represents a circle with center O and a radius of 7 yards.

The point A is located on the circumference of the circle. The measure of angle AOB is 30 degrees, and the measure of angle AOC is 60 degrees.

Using the properties of circles, we can conclude that angle BOC measures 90 degrees.

We can also determine that the length of segment AB is 7√3 yards, and the length of segment AC is 7 yards.

Finally, we can conclude that triangle AOB is an equilateral triangle since each angle measures 60 degrees and each side has a length of 7 yards.

The length of the major arc LNM is approximately 7.33π yards.

The general formula for finding the length of a major arc is given the measure of the central angle and the radius of the circle.

The formula for the length of a major arc is:

Length of major arc = (central angle measure / 360°) x (2πr)

where r is the radius of the circle.

Using this formula, and assuming that LNM is a major arc, we can find its length if we know the central angle measure and the radius of the circle.

If m LKM = 60° and the radius of the circle is 7 yards, then the length of the major arc LNM would be:

Length of major arc LNM = (60° / 360°) x (2π x 7 yd)

Length of major arc LNM = (1/6) x (14π yd)

Length of major arc LNM = (7/3)π yd ≈ 7.33π yd

Therefore, the length of the major arc LNM is approximately 7.33π yards.

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The correct question is:

The circle below has center O and its radius is 7 yd. Given that m ∠LKM = 60°, find the length of the major arc LNM.

The series of transformations correctly show that △CAT≅△DOG is rotate ACAT 180° about the origin, the correct option is A.

We are given that;

△CAT≅△DOG

Now,

To show that ACAT and ADOG are congruent, we need to find a sequence of rigid transformations that maps one onto the other.

One possible sequence is:

This will map A to D, C to O, A to G, and T to O.

Translate the image 2 units left. This will align the image with ADOG.

Therefore, by transformation the answer will be rotate ACAT 180° about the origin.

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Answer:

center = -1, -3

radius = 6

Step-by-step explanation:

x² + y² + 2x + 6y = 26

x² + 2x + y² +6y = 26

equation of a circle is,

(x - h)² + (y - k)² = r²

where center of a circle is (h,k)

radius = r

x² + 2x + y² + 6y = 26

finding the middle point for mid term breaking of the equations,

(2/2)² = 1

(6/2)² = 9

x² + 2x + 1 + y² + 6y + 9 = 26 + 1 +9

to balance the equation we have to add the midpoints at both sides,

thus we have equation of a circle,

(x + 1)² + (y + 3)² = 36

so,

centre of a circle = -1, -3

radius = 6

The president of Doerman Distributors, Inc., believes that 27% of the firm's orders come from first-time cu:

sample of 100 orders will be used to estimate the proportion of first-time customers. Use z-table.

0.30. What is the sampling distribution of p for this study?

a. Assume that the president is correct and p

=

Hint(s) Che

A normal distribution because np and n(1-p) are both greater than 5 v

b. What is the probability that the sample proportion will be between 0.20 and 0.40 (to 4 decimals)?

c. What is the probability that the sample proportion will be between 0.25 and 0.35 (to 4 decimals)?

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Partially Correct

Answer:

Step-by-step explanation:

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a. Since the sample size is large enough (n=100) and the expected number of first-time customers (np=27) and the expected number of repeating customers (n(1-p)=73) are both greater than 5, we can assume that the sampling distribution of the sample proportion p is approximately normal with mean μ=p=0.27 and standard deviation σ=sqrt[p(1-p)/n]=sqrt[(0.27)(0.73)/100]=0.046.

b. To find the probability that the sample proportion will be between 0.20 and 0.40, we need to standardize the values using the formula z = (x - μ) / σ, where x is the sample proportion, μ is the population proportion, and σ is the standard deviation of the sampling distribution of p. Then we can use a standard normal distribution table (z-table) to find the corresponding probabilities.

z(0.20) = (0.20 - 0.27) / 0.046 = -1.52

z(0.40) = (0.40 - 0.27) / 0.046 = 2.83

Using the z-table, the probability of getting a z-score between -1.52 and 2.83 is approximately 0.9747. Therefore, the probability that the sample proportion will be between 0.20 and 0.40 is 0.9747 (to 4 decimals).

c. To find the probability that the sample proportion will be between 0.25 and 0.35, we can follow the same steps as in part b:

z(0.25) = (0.25 - 0.27) / 0.046 = -0.43

z(0.35) = (0.35 - 0.27) / 0.046 = 1.74

Using the z-table, the probability of getting a z-score between -0.43 and 1.74 is approximately 0.5267. Therefore, the probability that the sample proportion will be between 0.25 and 0.35 is 0.5267 (to 4 decimals).

The equilibrium point is (1600, 80) in the form (x, y).

We need to find the point where the demand function D(x) is equal to the supply function S(x).

The functions are given as follows:
D(x) = 3200/√x
S(x) = 2x√

To find the equilibrium point, we need to set D(x) equal to S(x):

3200/√x = 2x√

Now, let's solve for x:

1. Isolate x by multiplying both sides by √x:
3200 = 2x√ * √x

2. Simplify by squaring both sides:
(3200)^2 = (2x√)^2

3. Perform the squaring:
10,240,000 = 4x^2

4. Divide both sides by 4 to isolate x^2:
2,560,000 = x^2

5. Take the square root of both sides:
x = √2,560,000
x = 1600

Now that we have x, we can find the corresponding price y by plugging x into either D(x) or S(x):

y = D(1600) = 3200/√1600
y = 3200/40
y = 80

So, the equilibrium point is (1600, 80) in the form (x, y).

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The inverse of f(x) is: [tex]f^{-1(x)}[/tex] = (x - 8)/5 switched the variables, so instead of [tex]f^{-1(x)}[/tex] , we wrote it as [tex]f^{-1(x)}[/tex] .

What is function?

In mathematics, a function is a relationship between two sets of elements, called the domain and the range, such that each element in the domain is associated with a unique element in the range.

To find the inverse of the function f(x) = 8 + 5x, we need to solve for x in terms of f(x).

Let y = f(x) = 8 + 5x

To find the inverse, we need to isolate x on one side of the equation:

y = 8 + 5x

y - 8 = 5x

x = (y - 8)/5

Therefore, the inverse of f(x) is:

[tex]f^{-1(x)}[/tex] = (x - 8)/5

Or, we can write it as:

[tex]f(x)^{-1}[/tex] = (x - 8)/5

Note that we switched the variables, so instead of [tex]f^{-1}[/tex] , we wrote it as [tex]f^{-1(x)}[/tex] .

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Developing a well-organized work schedule is crucial for the success of the facility and ensures that the business operations run smoothly, resulting in increased customer satisfaction, employee engagement, and profitability.

Developing the work schedule for the associates is the 2nd most impactful scenario as it directly affects the productivity and efficiency of the facility. A well-planned and balanced work schedule ensures that the right number of associates are scheduled at the right time to meet the demands of the business while minimizing unnecessary labor costs. Failure to create an effective schedule can result in understaffing, overstaffing, or scheduling conflicts, which can lead to a decrease in productivity and customer satisfaction.

The work schedule also affects employee satisfaction, as they need to balance their personal lives with their work schedules. A poorly managed schedule can lead to burnout, increased absenteeism, and high turnover rates. It is essential to consider associate availability, skill sets, and workload while developing the schedule to ensure that all tasks are completed efficiently and effectively.

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Based on the average daily balance, the finance charge for this credit card that charges 10.5% APR is $2.10.

The finance charge consists of the interest and other fees that lenders charge borrowers.

One of the methods for computing the finance charge is the average daily balance, which takes the sum of the daily balances and divides by the number of days in the billing cycle.

APR interest rate = 10.5%

Monthly period days = 30

Days 1-3: $200 (initial balance)        3 days      $600 ($200 x 3)

Days 4-20: $300 ($100 purchase)  17 days   $5,100 ($300 x 17)

Days 21-30: $150 ($150 payment)  10 days   $1,500 ($150 x 10)

Total balances = $7,200

Average daily balance = $240 ($7,200 ÷ 30)

Finance charge = $2.10 ($240 x 10.5% x 30/360)

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It will take God approximately 32 days to use up all the gas in his tanker and need a refill.

If God filled his gas tanker with 19/5/9 tank of gas and uses 1 5/6 gallons of gas each day, we can calculate how many days it will take for him to need a refill.

First, we need to convert the mixed number 19/5/9 to an improper fraction:

19/5/9 = (19 * 9 + 5) / 9 = 176/9

So God has 176/9 tanks of gas in his tanker.

Next, we can calculate how much gas God uses each day:

1 5/6 = (6 * 1 + 5) / 6 = 11/6

So God uses 11/6 gallons of gas each day.

To find out how many days it will take for God to need a refill, we can divide the amount of gas in his tanker by the amount of gas he uses each day:

(176/9) / (11/6) = (176/9) * (6/11) = 32

Therefore, it will take God approximately 32 days to use up all the gas in his tanker and need a refill.

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In scenario a, the function c(x) represents the density of traffic on a straight road, measured in cars per mile, where x is the number of miles east of a major interchange. The product c(x) · Ax has units of cars, as it represents the number of cars in a certain segment of the road. The definite integral ∫ c(x) dx and its Riemann sum approximation given by 1/n ∑ c(xi) · Δx have units of cars per mile, as they represent the average density of traffic over a certain distance. When evaluating the definite integral ∫ c(x) dx, we get a value that represents the total number of cars on the road between two given points.

In scenario b, the function B(x) represents the distribution of the weight of books on a shelf, in pounds per inch, where x is the number of inches from the left end of the shelf. The product B(x) · Ax has units of pounds, as it represents the weight of books in a certain segment of the shelf. The definite integral ∫ B(x) dx and its Riemann sum approximation given by 1/n ∑ B(xi) · Δx have units of pounds, as they represent the total weight of books on the shelf. When evaluating the definite integral ∫ B(x) dx, we get a value that represents the total weight of books on the shelf.
a. i. The units on the product c(x) · Δx are cars per mile (from c(x)) multiplied by miles (from Δx), resulting in cars.

a. ii. The units on the definite integral and its Riemann sum approximation are the same as the units on the product c(x) · Δx, which are cars.

a. iii. To evaluate the definite integral, we have:

∫(200 + 100e^(-0.12x)) dx

Using the integral rules, we get:

[200x - (100/0.12)e^(-0.12x)] (evaluate this from 0 to a specific value to find the total cars between 0 and that value)

The meaning of the value is the total number of cars on the road between 0 miles and the specified value of x miles east of the major interchange.

b. i. The units on the product B(x) · Δx are pounds per inch (from B(x)) multiplied by inches (from Δx), resulting in pounds.

b. ii. The units on the definite integral and its Riemann sum approximation are the same as the units on the product B(x) · Δx, which are pounds.

b. iii. To evaluate the definite integral, we have:

∫(0.5 + (x+1)^2) dx

Using the integral rules, we get:

[0.5x + (1/3)(x+1)^3] (evaluate this from 0 to 72 to find the total weight of books on the shelf)

The meaning of the value is the total weight of the books on the 6-foot-long shelf.

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Answer:

1, 768

2, 2088

3, 7789

4, 1828

5, 1435.

We cannot rule out the null hypothesis that the population means the weight is equal to 100 lb based on the critical value rule at = 0.01.

A one-sample t-test can be used to evaluate whether we can rule out the null hypothesis that the population's average weight is 100 lb.

First, we need to calculate the test statistic t:

[tex]$t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$[/tex]

[tex]$t = \frac{(102 - 100)}{\frac{10}{\sqrt{25}}} = 2$[/tex]

The critical value for the t-distribution with 24 degrees of freedom and a significance level of 0.01 must then be determined. To determine this value, we can utilize a t-table or statistical software.

We establish the critical value to be 2.492 using a t-table. We are unable to reject the null hypothesis because our calculated t-value (2) is smaller than the crucial value (2.492).

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5, instances of the number 5 will be stored in this 2D array. Option B.

To count the instances of the number 5 in the 2D array, we need to iterate through all the elements of the array and count the occurrences of 5.

Starting from the top-left element, we see that it is not a 5. Moving to the right, we find a 5 in the second column. Continuing to the right, we find another 5 in the same row. So far, we have counted 2 instances of 5.

Moving to the next row, we find no 5s in the first column. In the second column, we find two 5s in the same row. This brings our total count to 4. Moving to the last row, we find a single 5 in the second column, bringing our final count to 5.

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a) The accumulated sales for the first 9 days is approximately $9,359.49.

b) The sales from the 2nd day through the 5th day is approximately $6,022.25.

To find the accumulated sales for the first 9 days, we need to integrate the given rate of change of sales with respect to time:

S'(t) = [tex]23e^t[/tex]

Integrating both sides with respect to t, we get:

S(t) = ∫S'(t) dt = ∫[tex]23e^t[/tex]dt = [tex]23e^t[/tex] + C

where C is the constant of integration.

To find the value of C, we use the initial condition that the sales at day 0 (i.e., the starting point) is $0:

S(0) = 0 = 23e^0 + C

Therefore, C = -23.

Substituting this value of C, we get:

S(t) = [tex]23e^t[/tex] - 23

a) To find the accumulated sales for the first 9 days, we need to evaluate S(9) - S(0):

[tex]S(9) - S(0) = (23e^9 - 23) - (23e^0 - 23) = 23(e^9 - 1) ≈ $9,359.49[/tex]

Therefore, the accumulated sales for the first 9 days is approximately $9,359.49.

b) To find the sales from the 2nd day through the 5th day, we need to evaluate S(5) - S(2):

[tex]S(5) - S(2) = (23e^5 - 23) - (23e^2 - 23) = 23(e^5 - e^2) ≈ $6,022.25[/tex]

Therefore, the sales from the 2nd day through the 5th day is approximately $6,022.25.

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The proportion of individuals with the common attribute in the first sample is 0.525, while in the second sample, it is 0.728.

We have,
To analyze these samples, we can calculate the proportion of individuals with a common attribute in each sample.

Step 1: Calculate the proportion for the first sample
Divide the number of people with the common attribute (21) by the total number of people in the sample (40).
Proportion 1 = 21/40 = 0.525

Step 2: Calculate the proportion for the second sample
Divide the number of people with the common attribute (1528) by the total number of people in the sample (2100).
Proportion 2 = 1528/2100 = 0.728

Thus,

The proportion of individuals with the common attribute in the first sample is 0.525, while in the second sample, it is 0.728.

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8.19 divided by 4.2 is approximately equal to 1.94047624, which can be rounded to 1.94 (to two decimal places).

In mathematics, division is a basic arithmetic operation that involves separating a quantity or a number into equal parts or groups. The division operation is denoted by the symbol "/", or in some cases, the symbol "÷"

When we divide one number by another, we are essentially finding out how many times the second number "fits into" the first number

To divide 8.19 by 4.2, we can use long division as follows:

     1.9 4 0 4 7 6 2 4 3 3 3...

  --------------------------

4.2| 8.1 9 0 0 0 0 0 0 0 0 0

    8 4

    ----

    2 6 0

    2 5 2

    -----

      8 0 0

      7 1 4

      -----

      8 5 0

      8 4 8

      -----

        2 0 0

        1 6 8

        -----

        3 1 0

        2 5 2

        -----

          5 8

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The quadratic equation with zeros at -6 and 2 is y² + 4y - 12 = 0. This is in standard form, which is ax² + bx + c = 0, with a = 1, b = 4, and c = -12.

To find the quadratic equation with zeros at -6 and 2, we can start by using the fact that if a quadratic equation has roots x₁ and x₂, then it can be written in the form

(y - x₁)(y - x₂) = 0

where y is the variable in the quadratic equation.

Substituting the given values of the zeros, we get

(y - (-6))(y - 2) = 0

Simplifying this expression, we get

(y + 6)(y - 2) = 0

Expanding this expression, we get

y² - 2y + 6y - 12 = 0

Simplifying this expression further, we get

y² + 4y - 12 = 0

So the quadratic equation with zeros at -6 and 2 is

y² + 4y - 12 = 0

This is the standard form of a quadratic equation, which is

ax² + bx + c = 0

where a, b, and c are constants. In this case, a = 1, b = 4, and c = -12.

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If T is an unbiased estimator, then the MSE can be decomposed as follows: MSE(T) = Var(T) + [E(T)-0]^2 = Var(T). Therefore, (d) is true.

(a) E(T-0)^2=Var(T) + [E(T)-0]^2, which is always greater than or equal to 0, but it may not necessarily be 0 unless T is a constant function. Therefore, (a) is false in general.

(b) If E(T-0)=0, then T is an unbiased estimator of 0. This statement is true.

(c) E(T-ET)^2=Var(T) is always greater than or equal to 0, but it may not necessarily be 0 unless T is a constant function. Therefore, (c) is false in general.

(d) The Mean Squared Error (MSE) of T is defined as MSE(T) = E[(T-0)^2]. If T is an unbiased estimator, then the MSE can be decomposed as follows: MSE(T) = Var(T) + [E(T)-0]^2 = Var(T). Therefore, (d) is true.

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Check the picture below.

so the circle has a radius of 3 and a center at (-2 , 1)

[tex]\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{-2}{h}~~,~~\underset{1}{k})}\qquad \stackrel{radius}{\underset{3}{r}} \\\\[-0.35em] ~\dotfill\\\\ ( ~~ x - (-2) ~~ )^2 ~~ + ~~ ( ~~ y-1 ~~ )^2~~ = ~~3^2\implies (x+2)^2 + (y-1)^2 = 9[/tex]

Answer:

y = 8√2

Step-by-step explanation:

First, we can use the 30-60-90 triangle rule on the rightmost triangle. This states that:

So, we can solve for the middle line (I will call it L) on this triangle:

L = 16 / 2

L = 8

Next, we can solve for y using the rule with 45-45-90 triangles:

So, we can solve for y:

y = 8 · √2

y = 8√2

The equation stating requirement for goal is 250 = 25t and value of variable or shirts is 10.

The amount remaining to be raised = 370 - 120

Remaining amount = $250

The number of t-shirts need to be sold to meet the goal will be given by the formula -

Amount required = number of shirts × cost of each shirt

Keep the values in formula to find the expression and value of variable

250 = 25t

Solving the equation for the value of t

t = 250/25

Divide the values

t = 10

Hence, the expression is 250 = 25t and value of variable is 10.

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f(0)(-1) = 2

f(1)(-1) = 4

f(2)(-1) = 6

First, let's find the first three derivatives of f(x):

f(x) = 3x^2 + 2x + 2

f'(x) = 6x + 2

f''(x) = 6

Now, we can use the formula for the degree 2 Taylor polynomial at x = a = -1:

p2(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2

Plugging in a = -1 and the derivatives we found above, we get:

p2(x) = f(-1) + f'(-1)(x+1) + f''(-1)(x+1)^2/2

p2(x) = (3(-1)^2 + 2(-1) + 2) + (6(-1) + 2)(x+1) + 6(x+1)^2/2

p2(x) = 3 - 4(x+1) + 3(x+1)^2

Therefore, the degree 2 Taylor polynomial of f(x) at x = -1 is p2(x) = 3 - 4(x+1) + 3(x+1)^2.

To find the desired derivatives at x = -1:

f(0)(-1) = 2

f(1)(-1) = 4

f(2)(-1) = 6

Therefore, the degree 2 Taylor polynomial of f(x) at x = -1 is:

p2(x) = 3 - 4(x+1) + 3(x+1)^2

And the derivatives at x = -1 are:

f(0)(-1) = 2

f(1)(-1) = 4

f(2)(-1) = 6

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1. I imagine buying 100 shares of Amazon.com Inc. on January 3, 2020, when the stock price was $93.75, investing $9,375.  

Today, October 31, 2022, the stock price of Amazon.com Inc. is $102.44.

2. The real return on my investment in Amazon.com Inc was a net loss of  7.12% or $667.60.

3. The company I selected to buy its stock three years ago was Amazon.com Inc.

4. I decided on Amazon.com Inc., hoping to earn spectacular returns since it is a multinational technology company.

5. When I consider the actual return on the stock investment in Amazon.com Inc., I think it was an unwise investment.

6. The investment returned a negative real value because I realized less than I initially invested; I actually lost about $667.60 overall.

Stock investment is the purchase of shares for an ownership interest in a publicly-listed company.

The investor makes the investment with the hope that the investee will grow and perform well over some period, enabling the investor to earn some real returns (in the form of dividends and capital appreciation).

Purchase of 100 shares Jan. 3, 2020 = $9,375 (100 x $93.75)

Sales of 100 shares Oct. 31, 2022 = $10,244 (100 x $102.44)

Tax (10%) = $1,024.40 ($10,244 x 10%)

Inflation (3%) = $307.32 ($10,244 x 3%)

Administration fee on sales (2%) = $204.88 ($10,244 x 2%)

Real Returns in dollars = $8,707.40 ($10,244 - $1,024.40 - $307.32 - $204.88)

Loss on returns = $667.60 ($8,707.40 - $9,375)

Loss percentage = 7.12% ($667.60/$9,375 x 100)

Unfortunately, Amazon.com Inc. did not pay any dividends during the period of my investment, and I really lost funds to taxes, inflation, and administration fees when I sold it.

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To find the maximum distance between the point (1,3) and a point on the circle of radius 4 centered at the origin, we can use the distance formula. Let (x,y) be a point on the circle, then the distance between (1,3) and (x,y) is given by:

d = √((x-1)^2 + (y-3)^2)

Since the point (x,y) lies on the circle of radius 4 centered at the origin, we have:

x^2 + y^2 = 16

We can solve for y in terms of x:

y = ±√(16 - x^2)

Substituting into the distance formula, we get:

d = √((x-1)^2 + (±√(16 - x^2) - 3)^2)

Simplifying and squaring, we get:

d^2 = (x-1)^2 + (±√(16 - x^2) - 3)^2

d^2 = x^2 - 2x + 1 + (16 - x^2 - 6√(16 - x^2) + 9)  (or d^2 = x^2 - 2x + 1 + (16 - x^2 + 6√(16 - x^2) + 9))

d^2 = -x^2 - 2x + 26 ± 6√(16 - x^2)

To maximize the distance, we want to maximize d^2. Note that the maximizing distance should be at least 4, which means that we only need to consider the positive root of d^2. The critical points of d^2 occur when the derivative is zero, so we differentiate with respect to x:

d(d^2)/dx = -2x - 2(±3x/√(16 - x^2))

Setting this equal to zero, we get:

x = ±4/√5, ±2√2/√5, 0

Note that x = 0 corresponds to the point (0,4) on the circle, which has distance 5 from (1,3), so it is not a critical point. The other critical points correspond to the points where the circle intersects the x-axis and the y-axis. Evaluating d^2 at these critical points, we get:

d^2 = 18 ± 6√6

The maximum distance is therefore √(18 + 6√6), which occurs when x = ±4/√5.

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The calculated value of the variable x in the figure is 6 degrees

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

The figure

By the given congruent angles, we have the following equation

9x - 14 = 6x + 4

Collect the like terms in the equation

so, we have the following representation

9x - 6x = 14 + 4

Evaluate the like terms

So, the equation becomes

3x = 18

Divide both sides of the equation by 3

x = 6

Hence, the value of the variable x in the figure is 6

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The function graphed on the axes above should be expressed as a piecewise function as follows;

f(x) = -3x - 8    {x ≤ -2}

    = 6x - 17     {x > 3}

In order to determine the piecewise function, we would determine an equation that represent each of line shown on the graph. Therefore, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (4 + 2)/(-4 + 2)

Slope (m) = 6/-2

Slope (m) = -3.

At data point (-2, -2) and a slope of -3, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y + 2 = -3(x + 2)  

y = -3x - 8

For the second line, we have:

Slope (m) = (7 - 1)/(4 - 3)

Slope (m) = 6/1

Slope (m) = 6.

At data point (3, 1) and a slope of 6, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 1 = 6(x - 3)  

y = 6x - 17

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The line integral of the given function f(x, y) along the given curve r(t) is 0.8404.

First, we need to parameterize the curve by substituting x = t³ and y = [tex]t^4[/tex] into the function f(x, y) to get:

f(t) = √([tex]t^4[/tex]/t³) = [tex]t^{1/2}[/tex]

Next, we need to find the derivative of r(t) with respect to t:

r'(t) = 3t²i + 4t³j

Then, we can compute the line integral using the formula:

∫f(r(t))|r'(t)|dt from ½ to 1

Substituting the values, we get:

∫[tex]t^{1/2}[/tex] |3t²i + 4t³j| dt from ½ to 1

= ∫[tex]t^{1/2}[/tex] |t²(3i + 4tj)| dt from ½ to 1

= ∫[tex]t^5[/tex] (9 + 16t²) dt from ½ to 1

This integral is not easy to solve analytically, so we can use numerical methods to find an approximate value. Using a numerical integration method such as Simpson's rule, we get:

≈ 0.8404

Therefore, the line integral of f(x, y) = √y/x along the curve r(t) = t³i + [tex]t^4[/tex]j, ½ ≤ t ≤ 1 is approximately 0.8404.

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