What is said to occur when the effect of one input factor on the output depends on the level of another input factor?FactorsBlockingANOVAInteraction

The pairs of triangles that can be proven by SAS similarity is (c)

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

The list of options

The SAS similarity is used to check if two triangles are similar by comparing the lengths of the corresponding sides and the angle between the corresponding sides.

Using the above as a guide, we have the following:

The pair of triangles in (c) can be proved by SAS

This is because

Corresponding sides are similar and the angle between them are equal

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The value of x, considering the similar triangles in this problem, is given as follows:

x = 4.5.

Similar triangles are triangles that share these two features listed as follows:

The similar triangles for this problem are given as follows:

ABC and ADE.

Hence the proportional relationship is given as follows:

x/9 = 3/6.

x/9 = 1/2.

Hence the value of x is obtained applying cross multiplication as follows:

2x = 9

x = 4.5.

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Answer:16%

Step-by-step explanation:

The integral of 2^(x) t dt is: (2^(x) t^2)/(ln 2) + C where C is the constant of integration.

To evaluate this integral, we can use the power rule of integration, which states that the integral of x^n is (x^(n+1))/(n+1), where n is any real number except for -1. In this case, we have a product of 2^(x) and t, so we use the product rule of integration, which states that the integral of f(x)g(x)dx is f(x)∫g(x)dx + g(x)∫f(x)dx. We let f(x) = 2^(x) and g(x) = t, so that ∫g(x)dx = (t^2)/2, and we have:

∫2^(x) t dt = f(x)∫g(x)dx = 2^(x) (t^2)/2 + C

We then simplify this expression by multiplying the second term by ln 2/ln 2, which gives:

(2^(x) t^2)/(ln 2) + C

This is the final answer.

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Using division operation and unit conversion, the number of 2k stamps that can be bought for N5.28 is 264.

Division operation and multiplication operation are two of the mathematical operations used in unit conversions.

Division operation involves the dividend divided by the divisor, resulting in the quotient.

The total amount spend for stamps = N5.28

N1 = 100k

N5.28 = 528k (N5.28 x 100)

The unit price per stamp = 2k

2k = N0.02 (2 ÷ 100)

The number of stamps = 264 (528 ÷ 2) or (N5.28 ÷ N0.02)

Thus, one can comfortably buy 264 stamps of 2k each with N5.28, based on division operation for unit conversions.

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The statement is true. To understand why, we need to look at the relationship between the traveling salesman problem (TSP) and the 3-satisfiability problem (3-SAT).

The TSP is a well-known NP-hard problem, which means that there is no known algorithm that can solve it in polynomial time. On the other hand, 3-SAT is also an NP-hard problem, but it is known to be solvable in polynomial time by using algorithms such as the Davis-Putnam-Logemann-Loveland (DPLL) algorithm.

Now, if there exists a 7-approximation algorithm for the TSP, it means that we can find a solution that is at most 7 times the optimal solution. This is a significant improvement over not having any polynomial time algorithm at all. In fact, there are many approximation algorithms for the TSP that are used in practice, such as the Christofides algorithm and the Lin-Kernighan heuristic. The connection between the TSP and 3-SAT comes from the fact that we can reduce the TSP to a special case of 3-SAT known as the Hamiltonian cycle problem. This means that if we can solve the TSP using a 7-approximation algorithm, we can also solve the Hamiltonian cycle problem using the same algorithm.


In summary, if there exists a 7-approximation algorithm for the TSP, then there exists a polynomial time solution for the 3-SAT problem. This is because the TSP can be reduced to the Hamiltonian cycle problem, which in turn can be reduced to 3-SAT. However, it is worth noting that the constant factor of 7 in the approximation algorithm for the TSP may be very large, and may not be practical for solving large instances of 3-SAT.

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Therefore, the Jacobian matrix for the transformation is:

J =

[2u + v u]

[7v^2 14uv]

To find the Jacobian for the given transformation, we need to compute the partial derivatives of the new variables (x and y) with respect to the original variables (u and v).

Given:x = u^2 + uv

y = 7uv^2

We calculate the partial derivatives as follows:

∂x/∂u = 2u + v (partial derivative of x with respect to u)

∂x/∂v = u (partial derivative of x with respect to v)

∂y/∂u = 7v^2 (partial derivative of y with respect to u)

∂y/∂v = 14uv (partial derivative of y with respect to v)

The Jacobian matrix J is formed by arranging these partial derivatives:

J = [∂x/∂u ∂x/∂v]

[∂y/∂u ∂y/∂v]

Substituting the values we calculated, the Jacobian matrix J is:

J = [2u + v u]

[7v^2 14uv]

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The coefficient of OH- in the balanced equation is 1.

Therefore, the correct answer is A) 1.

To balance the equation, we first balance the atoms other than hydrogen and oxygen. Then, we balance the oxygen atoms by adding water molecules (H₂O).

Next, we balance the hydrogen atoms by adding hydrogen ions (H+). Finally, we balance the charges by adding electrons (e-) to one side of the equation.

To balance the given redox equation in basic solution:

MnO₄- + H₂O → MnO₂ + OH-

Let's balance the oxygen atoms by adding water (H₂O) on the right side:

MnO₄- + H₂O → MnO₂ + OH- + H₂O

Now, let's balance the hydrogen atoms by adding hydrogen ions (H+) on the left side:

MnO₄- + 4H₂O → MnO₂ + OH- + 4H₂O

Next, let's balance the charge by adding electrons (e-) on the left side:

MnO₄ + 4H₂O + 8e- → MnO₂ + OH- + 4H₂O

Finally, let's check the balancing of the atoms:

Manganese (Mn): 1 Mn on each side

Oxygen (O): 4 O on each side

Hydrogen (H): 12 H on each side

Charge: -8e- on each side

Therefore,

The coefficient of OH- in the balanced equation is 1.

Therefore, the correct answer is A) 1.

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Note: The question would be as

What is the coefficient of OH when the equation is balanced using the set of smallest whole-number coefficients? MnO4-+「→ MnO2 + 103" (basic solution) A) 1 B)2 C)4 D) 10 E) None of these.

Answer:

3

Step-by-step explanation:

y + 2 = -5x

Substituting x = -1, we get:

y + 2 = 5

y = 5 - 2 = 3

Therefore, the value of y is 3. Answer: 3.

Answer: multiple of sum 2+5

Step-by-step explanation:

18/9=2

45/9=5

2+5=7

18+45=63

63/7=9

2+5

The experimental probability of winning a prize in the contest is 1/28 or approximately 0.0357.

To calculate the experimental probability of winning a prize in the contest, we need to divide the number of winning caps found by the total number of caps examined.

Here are the steps to follow:

Calculate the total number of caps examined:

Total number of bottles bought x Number of caps per bottle = Total number of caps examined

56 bottles x 1 cap per bottle = 56 caps examined

Calculate the number of winning caps found:

Given: 2 winning caps were found

Calculate the experimental probability of winning a prize:

Experimental probability = Number of winning caps found / Total number of caps examined

Experimental probability = 2 / 56

Experimental probability = 1 / 28

Explanation: Out of 56 caps examined, only 2 were found to be winning caps. Therefore, the probability of finding a winning cap is 2/56, which can be simplified to 1/28. This means that on average, for every 28 caps examined, one is expected to be a winning cap.

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The design parameters for the rapid-mix system are Number of tanks= 1, Water power input = 11.9 kW, Tank dimensions: depth = 2 m, width = 4.62 m, Diameter of the impeller is 1.2 m and Rotational speed of impeller is  50 rpm.

To design the rapid-mix system, we can use the following equations:

Number of tanks: n = Q t / V

where Q = flow rate = 0.168 m^3/s

t = detention time = 5 s

V = volume of one tank = [tex](depth)^{2}[/tex] × width

Water power input: P = ρ Q [tex]G^{2}[/tex] B / NP

where ρ = density of water = 1000 kg/[tex]m^{3}[/tex]

G = velocity gradient = 700 [tex]s^{-1}[/tex]

B = shape factor = 3/2

NP = power number = 5.7

Tank dimensions: depth = width / 2

Diameter of the impeller: D = 0.35 × width

Rotational speed of impeller: N = (P / 2π) × (NP / ρ [tex]D^{5}[/tex])

Using the above equations, we can solve for the design parameters as follows:

Volume of one tank:

V = Q t / n = (0.168)(5) / 1 = 0.84[tex]m^{3}[/tex]

Tank dimensions:

width = [tex](V/Depth^{2} )^{1/3}[/tex] = [tex](0.84/Depth^{2} )^{1/3}[/tex]

depth = width / 2

To find the width and depth of the tank, we need to try different values of depth and calculate the corresponding width using the above equation. We can start with a depth of 1 m and iterate until we get a width that is close to a square plan (i.e., width ≈  [tex](depth)^{2}[/tex] ). For example, if we try a depth of 1 m, we get:

width = [tex](0.84/1^{2} )^{1/3}[/tex] ≈ 0.96 m

This is not close to a square plan, so we can try a larger depth, say 2 m:

width = [tex](0.84/2^{2} )^{1/3}[/tex] ≈ 1.21 m

This is closer to a square plan, so we can use a depth of 2 m and a width of 4.62 m.

Number of tanks:

n = Q t / V = (0.168)(5) / 0.84 ≈ 1.0

We can use one tank for this design.

Diameter of the impeller:

D = 0.35 × width = 0.35 × 4.62 m ≈ 1.62 m

We can choose the impeller diameter of 1.2 m from the available options.

Water power input:

P = ρ Q [tex]G^{2}[/tex] B / NP = (1000)(0.168)[tex]700^{2}[/tex](3/2) / 5.7 ≈ 11.9 kW

Rotational speed of impeller:

N = (P / 2π) × (NP / ρ [tex]D^{5}[/tex]) = (11.9 kW / (2π)) × (5.7 / (1000)[tex]1.2^{5}[/tex] ≈ 50 rpm

Therefore, the design parameters for the rapid-mix system are:

Number of tanks: 1

Water power input: 11.9 kW

Tank dimensions: depth = 2 m, width = 4.62 m

Diameter of the impeller: 1.2 m

Rotational speed of impeller: 50 rpm

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If a car is -0.67 standard deviations away from the mean speed of 50 miles per hour, it is traveling at approximately 43.3 miles per hour.

To find a particular value given the number of standard deviations away from the mean it falls, we can use the z-score equation:
z = (x - μ) / σ

where z is the number of standard deviations away from the mean, x is the value we want to find, μ is the mean, and σ is the standard deviation.

To rearrange this equation to find x, we can isolate it by multiplying both sides by σ and adding μ:
x = z * σ + μ

a. To find the value associated with a car that is 2.3 standard deviations above the z-score, we can use the above equation:
x = 2.3 * σ + μ

Since we don't have any specific values for μ and σ, we can't find an exact answer. However, we can make some generalizations based on the normal distribution.

For example, we know that about 2.3% of the area under the normal curve falls beyond 2.3 standard deviations above the mean.

So, if we assume that the data follows a normal distribution, we can say that the value associated with a car that is 2.3 standard deviations above the z-score is relatively rare and unlikely to occur.

b. To find how many miles per hour a car is traveling if it is -0.67 standard deviations away from the mean, we can use the same equation:

x = z * σ + μ

In this case, z = -0.67, and we don't have any specific values for μ and σ. Again, we can make some generalizations based on the normal distribution. For example, if we know that the mean speed of cars on a particular road is 50 miles per hour, and the standard deviation is 10 miles per hour, we can plug these values into the equation:

x = -0.67 * 10 + 50

x = 43.3 miles per hour

Therefore, if a car is -0.67 standard deviations away from the mean speed of 50 miles per hour, it is traveling at approximately 43.3 miles per hour.

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An adult ticket costs$ 22, while a child's ticket is priced at$ 14.

Let's assume the cost of an grown-up's ticket is A bones and the cost of a child's ticket is C bones .

According to the given information

9A + 2C = 226 .......(1)

C = A - 8.............(2)

We can break this system of equations to find the values of A andC.

Substituting equation 2 into equation 1

9A + 2(A - 8) = 226

9A + 2A - 16 = 226

11A = 242

A = 22

and, C = 22- 8

C = 14

Thus, the price of an grown-up's ticket is$ 22, and the price of a child's ticket is$ 14.

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Answer:

26, 29, 32

Step-by-step explanation:

Each term is 3 more than the previous term.

5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, ...

The statement "If ln(55x) = ln(y), then 55x = y" is True.

For the first statement:

log(55) + log(y) = log(z) can be rewritten as:

log(55y) = log(z)

By the logarithmic identity log(ab) = log(a) + log(b), we can simplify this to:

log(55y) = log(55) + log(y)

Therefore, if log(55) + log(y) = log(z), then 55y = z.

To get 55 + y = z from this expression, we need to assume that y is a positive real number and take the antilogarithm (exponentiate) of both sides. This gives:

55y = z

y = z/55

Substituting this into 55 + y = z gives:

55 + z/55 = z

Multiplying both sides by 55 gives:

3025 + z = 55z

Subtracting z from both sides gives:

3025 = 54z

Dividing both sides by 54 gives:

z = 3025/54 ≈ 56.02

Substituting this value of z into 55 + y = z gives:

55 + y = 56.02

y ≈ 1.02

Therefore, the statement "If log(55) + log(y) = log(z), then 55 + y = z" is False.

For the second statement:

ln(55x) = ln(y) can be rewritten as:

ln(55x) - ln(y) = 0

Using the logarithmic identity ln(a/b) = ln(a) - ln(b), we can simplify this to:

ln(55x/y) = 0

Therefore, 55x/y = e^0 = 1.

So, if ln(55x) = ln(y), then 55x = y.

Therefore, the statement "If ln(55x) = ln(y), then 55x = y" is True.

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a) The standard form of the equation for this hyperbola is (x-1)²/4 - (y-1)²/21 = 1.

b) The standard form of the equation for this hyperbola is (y+6)²/4 - (x-1)²/3 = 1.

c) The standard form of the equation for this hyperbola is (x+3)²/5 - (y-2)²/6 = 1.

a. To find the standard form of the equation of a hyperbola with the given vertices and foci, we need to first determine the center of the hyperbola. The center of the hyperbola is the midpoint between the two vertices, which in this case is (1, 1).

In this case, a = 2. To find the value of b, we can use the equation b² = c² - a². Substituting the values we have found, we get b² = 21. The standard form of the equation of a hyperbola is

=>  (x-h)²/a² - (y-k)²/b² = 1,

where (h,k) is the center of the hyperbola.

Substituting the values we have found that h = 1 and the value of k as 1, we get the equation

=>  (x-1)²/4 - (y-1)²/21 = 1.

b. In this case, we can see that the vertices have the same x-coordinate but different y-coordinates, so the hyperbola is vertical. We can use the equation (y-k)²/a² - (x-h)²/b² = 1 for a vertical hyperbola.

We know that the center of the hyperbola is the midpoint between the vertices, which is (1, -6).

We can use the distance formula to find the value of a, which is the distance between the center and each vertex. In this case, a = 2.

To find the value of b, we can use the point given and the equation of the hyperbola. Substituting the values we have found, we get the equation

=> (y+6)²/4 - (x-1)²/3 = 1.

c. To find the standard form of the equation of a hyperbola with the given vertices and asymptotes, we need to determine the center of the hyperbola.

The center of the hyperbola is the midpoint between the vertices, which in this case is (-3, 2). We can use the equation (y-k)/(x-h) = ±a/b for the asymptotes.

Substituting the values we have found, we get the equations

=>  (y-2)/(x+3) = 6/5

and

=> (y-2)/(x+3) = -5/1.

We can solve for a and b by setting a/b equal to the slope of the asymptotes.

In this case, a/b = 6/5 or a/b = -5. We also know that a² - b² = c², where c is the distance between the center and each vertex. We can use the distance formula to find the value of c, which in this case is c = 3√5. Substituting the values we have found, we get two possible standard form equations for the hyperbola:

   • If a/b = 6/5, then a² = 36 and b² = 30.

The standard form of the equation for this hyperbola is

=> (x+3)²/36 - (y-2)²/30 = 1.

   • If a/b = -5, then a² = 5 and b² = 6.

The standard form of the equation for this hyperbola is

=> (x+3)²/5 - (y-2)²/6 = 1.

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To test the hypotheses H0: μ=800 vs Ha: μ>800 at an α level of significance, we reject the null hypothesis if the test statistic z is greater than or equal to the critical value zα.

In order to test the hypotheses H0: μ=800 vs Ha: μ>800 at an α level of significance, the null hypothesis will be rejected if the test statistic z is greater than or equal to the critical value zα.

This critical value is determined by the level of significance α and can be found using a z-table or statistical software.

If the test statistic z falls outside of the critical region (i.e. z < -zα or z > zα), then we fail to reject the null hypothesis.

On the other hand, if the test statistic falls within the critical region (i.e. -zα < z < zα), then we reject the null hypothesis and conclude that there is evidence to support the alternative hypothesis.

It is important to note that rejecting the null hypothesis does not necessarily mean that the alternative hypothesis is true.

It simply means that the observed data is unlikely to have occurred by chance alone assuming the null hypothesis is true.

The size of the test statistic relative to the critical value reflects the strength of the evidence against the null hypothesis.

In summary, to test the hypotheses H0: μ=800 vs Ha: μ>800 at an α level of significance, we reject the null hypothesis if the test statistic z is greater than or equal to the critical value zα.

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The Cost of water used in 2021 is $76.

To calculate the cost of water used, we need to know the rate or price of water per kiloliter. Once we have that information, we can multiply the rate by the amount of water used to find the cost.

Let's assume that the rate of water is $2 per kiloliter.

The amount of water used is given as 38 kiloliters.

Cost of water used = Rate * Amount of water used

Plugging in the values:

Cost of water used = $2 * 38 kiloliters

Calculating the multiplication:

Cost of water used = $76

Therefore, the cost of water used in 2021 is $76.

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A sequential multiplier is a type of digital multiplier that utilizes a sequential circuit to perform multiplication. One of the main advantages of using a sequential multiplier is that it can operate at higher speeds than a combinational multiplier, which uses a purely combinational circuit to perform multiplication.

This is because a sequential multiplier can be designed to perform multiplication using a pipeline architecture, where multiple multiplication operations are performed simultaneously, resulting in faster computation. Additionally, a sequential multiplier can be more efficient in terms of circuit size and power consumption than a combinational multiplier for larger operands.

However, there are also some disadvantages to using a sequential multiplier. One of the main drawbacks is that it introduces latency or delay into the system due to the need for a sequential circuit. This can result in slower computation times for smaller operands or when the multiplication operation needs to be performed quickly. Additionally, a sequential multiplier can be more complex to design and implement than a combinational multiplier, which can result in longer development times and higher costs.

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Answer:

Set your calculator to Degree mode.

[tex] \alpha = {cos}^{ - 1} \frac{7}{8} [/tex]

[tex] \cos(2 \alpha ) = \frac{7}{x} [/tex]

[tex] \cos(2 {cos}^{ - 1} \frac{7}{8} ) = 2 {cos}^{2} ( {cos}^{ - 1} \frac{7}{8} ) - 1 = \frac{7}{x} [/tex]

[tex]2( { \frac{7}{8}) }^{2} - 1 = \frac{7}{x} [/tex]

[tex] \frac{17}{32} = \frac{7}{x} [/tex]

[tex]17x = 224[/tex]

[tex]x = \frac{224}{17} = 13.176[/tex]

[tex] \alpha = {cos}^{ - 1} \frac{7}{8} = 28.955 \: degrees[/tex]

So x = 224/17 = 13.176 and theta = 28.955°.

The formula for the number of ways to seat r of n people around a circular table is given by (n-1) choose (r-1), where "choose" denotes a binomial coefficient.

When we arrange the people around a table, we can fix one person's position, for instance, at the top of the table. Then, we can arrange the other (n-1) people in a line, and there are (n-1) choose (r-1) ways to pick r-1 people from the remaining (n-1) to sit with the fixed person. This is because we are essentially choosing r-1 positions in a line to be filled by people, and there are (n-1) positions to choose from.

Since the table is circular, there is only one way to rotate the arrangement, which gives us (n-1) different arrangements for each arrangement of the chosen r people. Therefore, the total number of arrangements is (n-1) choose (r-1).

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The Wechsler IQ test is scored with a mean of 100 and a standard deviation of 15. Therefore, a score of 125 falls 1.67 standard deviations above the mean. This is considered a very high score, as it falls within the top 5% of the population.

The standard deviation range in which your score of 125 on the Wechsler IQ test falls, follow these steps:
1. Identify the mean and standard deviation for the Wechsler IQ test. The mean (average) is 100, and the standard deviation is 15.

2. Calculate the range for each standard deviation:
  - 1 standard deviation above the mean: 100 + 15 = 115
  - 2 standard deviations above the mean: 100 + (15 * 2) = 130

3. Compare your score (125) to these ranges. Since your score is between 115 and 130, it falls within 1 and 2 standard deviations above the mean.
In conclusion, on a Wechsler IQ test, if you received a score of 125, your score would fall between 1 and 2 standard deviations above the mean.

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Answer:

98

Step-by-step explanation:

If you times 98 by 4, you get 392, and three lots of 98 are cheeseburgers, with one lot being hamburgers.

Answer:

C. 7 because 7 x 7 = 49

An exponential function is a function of the form f(x) = ab^x, where a is the initial value and b is the base.

To find the equation of an exponential function that passes through two points, we need to use the given points to solve for a and b. In this case, the formula for the exponential function that passes through the points (0, 7000) and (3, 7) is f(x) = 7000 * (1/10)^(x/3).

To find the equation of an exponential function that passes through two points, we first need to determine the values of a and b in the general form of an exponential function, f(x) = ab^x. To do this, we can use the two given points (x1, y1) and (x2, y2) and solve for a and b simultaneously.

Using the point (0, 7000), we know that f(0) = 7000, so we can substitute x=0 and y=7000 into the equation to get:

7000 = ab^0 = a

Using the point (3, 7), we know that f(3) = 7, so we can substitute x=3 and y=7 into the equation to get:

7 = ab^3

Since we know that a = 7000, we can substitute this value into the second equation to get:

7 = 7000b^3

Solving for b, we get:

b = (1/10)^(1/3)

Now that we have found the values of a and b, we can substitute them back into the general form of the exponential function to get:

f(x) = ab^x = 7000 * (1/10)^(x/3)

This is the equation of the exponential function that passes through the points (0, 7000) and (3, 7). The base of the function, (1/10)^(1/3), is less than 1, which means that the function will approach 0 as x approaches infinity. This reflects the fact that the function is decreasing exponentially. The value of a, 7000, represents the initial value of the function when x = 0.

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[tex]\sin( 23^o )=\cfrac{\stackrel{opposite}{19}}{\underset{hypotenuse}{x}} \implies x=\cfrac{19}{\sin(23^o)}\implies x\approx 48.6[/tex]

Make sure your calculator is in Degree mode.

The statement that is true about the slope of both line segments is: "B. the slopes are the same because the small triangle and the large triangle are similar."

The Slope of a line is simply the ratio of the vertical distance over the horizontal distance along the line, which is calculated as: m = change in y / change in x or rise / run of the line. Also, note that the corresponding sides of two triangles will have the same slope.

Slope of line segment GH (m) = rise/run = -2/1 = -2

Slope of line segment HI (m) = rise/run = -4/2 = -2

This means that the slope of the hypotenuse of both triangles are the same because they are similar. The answer is: "B. the slopes are the same because the small triangle and the large triangle are similar."

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The correct combination of x and y that minimizes the cost Z is option (a), x = 0 and y = 3.

To solve this problem, we can use the method of linear programming. First, we need to convert the inequalities into equations by using slack variables. Thus, the two constraints become: (1) 2x - 4y + s1 = 12 and (2) 5x - 2y + s2 = 10.

Next, we create a table of values for the coefficients of x, y, and the slack variables, as well as the values of the objective function Z for each combination of x and y. Using this table, we can graph the feasible region and find the corner points. Evaluating Z for each corner point gives us the minimum and maximum values.

In this case, the corner points are (0,3), (2,2), and (2,0). Evaluating Z for each point gives us Z = 45 for (0,3), Z = 51 for (2,2), and Z = 48 for (2,0). Therefore, the minimum value of Z is 45, which occurs when x = 0 and y = 3. The overall cost for this solution is $45.

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The correct combination of x and y that minimizes the cost Z is option (a), x = 0 and y = 3.

To solve this problem, we can use the method of linear programming. First, we need to convert the inequalities into equations by using slack variables. Thus, the two constraints become: (1) 2x - 4y + s1 = 12 and (2) 5x - 2y + s2 = 10.

Next, we create a table of values for the coefficients of x, y, and the slack variables, as well as the values of the objective function Z for each combination of x and y. Using this table, we can graph the feasible region and find the corner points. Evaluating Z for each corner point gives us the minimum and maximum values.

In this case, the corner points are (0,3), (2,2), and (2,0). Evaluating Z for each point gives us Z = 45 for (0,3), Z = 51 for (2,2), and Z = 48 for (2,0). Therefore, the minimum value of Z is 45, which occurs when x = 0 and y = 3. The overall cost for this solution is $45.

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The size of angle QSR in the given triangle QSR is determined as 110 degrees.

The size of angle QSR is calculated by applying the following principle as shown below.

If  the internal bisectors of the angles at Q and R meet at S, as shown, the value of angle QSR is calculated as follows;

P = 180 - (Q + R)

Q + R = 180 - P

Q + R = 180 - 40

Q + R = 140 ------- (1)

S = 180 - (0.5Q + 0.5R)

S = 180 - 0.5(Q + R)

Substitute the value of Q + R into the equation;

S = 180 - 0.5 (Q + R )

S = 180 - 0.5(140)

S = 180 - 70

S = 110⁰

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