What two numbers add to make -17 and times to get 30

The value of system of equations are,

⇒ x = - 1 and y = 1

We have to given that;

The system of equations are,

x + 2y = 1  .. (i)

x = y - 2  .. (ii)

Now, We can plug the value of x in (i);

x + 2y = 1

(y - 2) + 2y = 1

3y - 2 = 1

3y = 3

y = 1

And, From (ii);

x = y - 2

x = 1 - 2

x = - 1

Thus, The value of system of equations are,

⇒ x = - 1 and y = 1

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The approximated value of the volume of the cone cup is 18.5 cubic cm

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

Diameter = 2.8 cm

Height = 9 cm

The volume of the cone cup is calculated as

Volume = 1/3 * 3.14 * r^2h

substitute the known values in the above equation, so, we have the following representation

Volume = 1/3 * 3.14 * (2.8/2)^2 * 9

Evaluate the products

So, we have

Volume = 18.4632

Approximate

Volume = 18.5

Hence, the volume is 18.5

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Answer:

c = 12.2

Step-by-step explanation:

a squared + b squared = c sqared

7 squared + 10 squared = 49+ 100 = 149 = [tex]\sqrt{x} 149[/tex] = 12.2

(a) The given statement, "If A is linearly dependent, then Sp{u, v} = Sp{u, w}" is false because there exist scalars α, β, and γ, not all zero, such that αu + βv + γw = 0.

(b) The given statement, "The set A is linearly independent if and only if {u+v,v-w, w+ 2u} is linearly independent" is true because A is linearly independent if and only if the determinant of the matrix formed by u, v, and w is nonzero. The determinant of the matrix formed by {u+v, v-w, w+2u} can be obtained by performing column operations on the original matrix. Since these operations do not change the determinant, the set {u+v, v-w, w+2u} is linearly independent if and only if A is linearly independent.

(c)The given statement, "Assume that A is linearly dependent. We define u₁ = 2u, v₁ = -3u + 4v, and W₁ = u + 2v - tw for some t E R. Then, there exists t E R such that {u₁, v₁, w₁} is linearly independent" is true because  A is linearly dependent, there exist scalars α, β, and γ, not all zero, such that αu + βv + γw = 0.

Let us discuss this in detail.

(a) False. If A is linearly dependent, then there exist scalars α, β, and γ, not all zero, such that αu + βv + γw = 0. Without loss of generality, assume α ≠ 0. Then we can solve for u: u = (-β/α)v + (-γ/α)w. Therefore, u is a linear combination of v and w, which means Sp{u, v} = Sp{u, w}.

(b) True. We can write each vector in {u+v,v-w, w+2u} as a linear combination of u, v, and w:
u + v = 1u + 1v + 0w
v - w = 0u + 1v - 1w
w + 2u = 2u + 0v + 1w
We can set up the equation α(u+v) + β(v-w) + γ(w+2u) = 0 and solve for α, β, and γ:
α + β + 2γ = 0 (from the coefficient of u)
α + β = 0 (from the coefficient of v)
-β + γ = 0 (from the coefficient of w)
Solving this system of equations, we get α = β = γ = 0, which means {u+v,v-w, w+2u} is linearly independent.

(c) True. Since A is linearly dependent, there exist scalars α, β, and γ, not all zero, such that αu + βv + γw = 0. Without loss of generality, assume α ≠ 0. Then we can solve for u: u = (-β/α)v + (-γ/α)w. Therefore, u is a linear combination of v and w, which means we can write u as a linear combination of u₁, v₁, and w₁:
u = (2/5)u₁ + (-3/5)v₁ + (1/5)w₁
Similarly, we can write v and w as linear combinations of u₁, v₁, and w₁:
v = (-2/5)u₁ + (4/5)v₁ + (1/5)w₁
w = u₁ + 2v₁ - t₁w₁
where t₁ = (α + 2β - γ)/(-t). We can set up the equation αu₁ + βv₁ + γw₁ = 0 and solve for α, β, and γ:
2α - 3β + γ = 0 (from the coefficient of u₁)
-3β + 4γ = 0 (from the coefficient of v₁)
-α + 2β - tγ = 0 (from the coefficient of w₁)
Solving this system of equations, we get α = β = γ = 0 if and only if t = -8/5. Therefore, if we choose any t ≠ -8/5, then {u₁, v₁, w₁} is linearly independent.

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The appropriate alternative hypothesis is: Hₐ: [tex]\mu_{sc[/tex] - [tex]\mu_{dc[/tex] does not equal 0, where [tex]\mu_{sc[/tex] is the mean time to complete all matches using the "shapes and cutouts are all the same color" matching scheme, and [tex]\mu_{dc[/tex] is the mean time to complete all matches using the "shapes and cutouts are different colors" matching scheme.

An assertion used in statistical inference experiments is known as the alternative hypothesis. It is indicated by Hₐ or H₁ and runs counter to the null hypothesis.

The appropriate alternative hypothesis is: Hₐ: [tex]\mu_{sc[/tex] - [tex]\mu_{dc[/tex] does not equal 0, where [tex]\mu_{sc[/tex] is the mean time to complete all matches using the "shapes and cutouts are all the same color" matching scheme, and [tex]\mu_{dc[/tex] is the mean time to complete all matches using the "shapes and cutouts are different colors" matching scheme.

This hypothesis is appropriate because it is testing whether there is a statistically significant difference in the mean time to complete all matches between the two matching schemes. The null hypothesis would be that there is no difference in the mean time between the two schemes, i.e., H₀: [tex]\mu_{sc[/tex] - [tex]\mu_{dc[/tex] = 0.

To test this hypothesis, we can use a paired t-test, which compares the mean difference between the two sets of measurements (in this case, the time to complete all matches using the two matching schemes) to the standard error of the mean difference. If the t-test results in a p-value that is smaller than the chosen significance level (typically 0.05), we reject the null hypothesis and conclude that there is a statistically significant difference between the mean times for the two matching schemes.

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The appropriate alternative hypothesis is: Hₐ:  - does not equal 0, where  is the mean time to complete all matches using the "shapes and cutouts are all the same color" matching scheme and  is the mean time to complete all matches using the "shapes and cutouts are different colors" matching scheme.

An assertion used in statistical inference experiments is known as the alternative hypothesis. It is indicated by Hₐ or H₁ and runs counter to the null hypothesis.

The appropriate alternative hypothesis is: Hₐ:  - does not equal 0, where  is the mean time to complete all matches using the "shapes and cutouts are all the same color" matching scheme and is the mean time to complete all matches using the "shapes and cutouts are different colors" matching scheme.

This hypothesis is appropriate because it is testing whether there is a statistically significant difference in the mean time to complete all matches between the two matching schemes. The null hypothesis would be that there is no difference in the mean time between the two schemes, i.e., H₀:  - = 0.

To test this hypothesis, we can use a paired t-test, which compares the mean difference between the two sets of measurements (in this case, the time to complete all matches using the two matching schemes) to the standard error of the mean difference. If the t-test results in a p-value that is smaller than the chosen significance level (typically 0.05), we reject the null hypothesis and conclude that there is a statistically significant difference between the mean times for the two matching schemes.

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Answer: Y-intercept:

Axis of symmetry: X = - 1

Vertex: Y = 2(X + 1)^2-5

Maximum: -1

Minimum: - 5

Domain:

(−∞,∞),{x|x∈R}

Range:

[−5,∞),{y|y≥−5}

Step-by-step explanation:

The Pythagorean Theorem is a fundamental concept that relates to the sides of a right-angled triangle, and it can also be used to understand the relationship between the areas of the squares constructed on the sides of the triangle.

The area of a square is given by the formula A = s², where s is the length of one of its sides. Therefore, the areas of the three squares are:

Area of the square with side a = a²

Area of the square with side b = b²

Area of the square with side c = c²

Now, let's compare the areas of the squares. We can start by subtracting the area of the square with side a from the area of the square with side c:

c² - a²

Using the Pythagorean Theorem, we know that c² = a² + b². Substituting this into the above expression, we get:

c² - a² = (a² + b²) - a² = b²

This tells us that the difference between the area of the square with side c and the area of the square with side a is equal to the area of the square with side b. In other words:

c² - a² = b²

This is known as the Pythagorean identity. It states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. We can also rearrange this identity to obtain the following:

c² = a² + b²

This is the Pythagorean Theorem that we are familiar with. Therefore, we can conclude that the relationship between the areas of the squares constructed on the sides of a right-angled triangle is given by the Pythagorean identity: the difference between the area of the square on the hypotenuse and the area of the square on the shorter side is equal to the area of the square on the other shorter side.

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Check the picture below.

Answer: B.146 feet

Step-by-step explanation:

Answer: B

Step-by-step explanation:

Hope this helps! :)

The total number of pairs (f, g) ∈ F × F such that g∘f(1) ≠ 1 is 4 * 4 * 4 * 4 = 256.

(a) To find the number of pairs (f, g) ∈ F × F such that g∘f(1) = 1, we need to count the possible functions f and g that satisfy this condition.

Since f is a function from A to A, there are 4 choices for f(1) since f(1) can take any value from A. However, in order for g∘f(1) to be equal to 1, there is only one choice for g(1), which is 1.

For the remaining elements in A, f(2), f(3), and f(4) can each take any value from A, giving us 4 choices for each element. Similarly, g(2), g(3), and g(4) can also take any value from A, giving us 4 choices for each element.

Therefore, the total number of pairs (f, g) ∈ F × F such that g∘f(1) = 1 is 4 * 4 * 4 * 4 = 256.

(b) To find the number of pairs (f, g) ∈ F × F such that g∘f(1) = 1 and g∘f(2) = 2, we need to consider the additional condition of g∘f(2) = 2.

Similar to the previous part, there are 4 choices for f(1) and only one choice for g(1) in order to satisfy g∘f(1) = 1.

For f(2), there is only one choice as well since it must be mapped to 2. This means f(2) = 2.

Now, for the remaining elements f(3) and f(4), each can take any value from A, giving us 4 choices for each element.

Similarly, g(2), g(3), and g(4) can also take any value from A, giving us 4 choices for each element.

Therefore, the total number of pairs (f, g) ∈ F × F such that g∘f(1) = 1 and g∘f(2) = 2 is 1 * 1 * 4 * 4 * 4 * 4 = 256.

Note that the answers for both (a) and (b) are the same since the additional condition of g∘f(2) = 2 does not affect the number of possible pairs.

(c) To find the number of pairs (f, g) ∈ F × F such that g∘f(1) = 1 or g∘f(2) = 2, we need to consider the cases where either g∘f(1) = 1 or g∘f(2) = 2.

For g∘f(1) = 1:

As discussed in part (a), there are 4 choices for f(1) and 1 choice for g(1). For the remaining elements f(2), f(3), and f(4), each can take any value from A, giving us 4 choices for each element. Similarly, g(2), g(3), and g(4) can also take any value from A, giving us 4 choices for each element.

Therefore, the total number of pairs (f, g) ∈ F × F such that g∘f(1) = 1 is 4 * 4 * 4 * 4 = 256.

For g∘f(2) = 2:

As discussed in part (b), there is only one choice for f(2) and one choice for g(2) since f(2) = 2 and g(2) = 2.

For the remaining elements f(1), f(3), and f(4), each can take any value from A, giving us 4 choices for each element. Similarly, g(1), g(3), and g(4) can also take any value from A, giving us 4 choices for each element.

Therefore, the total number of pairs (f, g) ∈ F × F such that g∘f(2) = 2 is 1 * 4 * 4 * 4 * 4 = 256.

Now, to find the total number of pairs (f, g) ∈ F × F such that g∘f(1) = 1 or g∘f(2) = 2, we need to consider the sum of the counts from the two cases. Since these cases are mutually exclusive, we can simply add the counts:

Total number of pairs = 256 + 256 = 512.

Therefore, there are 512 pairs (f, g) ∈ F × F such that g∘f(1) = 1 or g∘f(2) = 2.

(d) To find the number of pairs (f, g) ∈ F × F such that g∘f(1) ≠ 1 or g∘f(2) ≠ 2, we need to consider the cases where neither g∘f(1) = 1 nor g∘f(2) = 2.

For g∘f(1) ≠ 1:

As discussed in part (a), there are 4 choices for f(1) and 1 choice for g(1). For the remaining elements f(2), f(3), and f(4), each can take any value from A, giving us 4 choices for each element. Similarly, g(2), g(3), and g(4) can also take any value from A, giving us 4 choices for each element.

Therefore, the total number of pairs (f, g) ∈ F × F such that g∘f(1) ≠ 1 is 4 * 4 * 4 * 4 = 256.

For g∘f(2) ≠ 2:

As discussed in part (b), there is only one choice for f(2) and one choice for g(2) since

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Answer:

a first

Step-by-step explanation:

because I just don't

Answer: -12

Step-by-step explanation:

Hi! To determine which data set is less variable, we can compare their ranges. The range is calculated by subtracting the minimum value from the maximum value in the data set.

a. 4 - 1 = 3
b. 8 - 1 = 7
c. 1 - (-1) = 2
e. 4.5 - 1 = 3.5
f. 8 - 1 = 7

The data set with the least variability is option C, with a range of 2.

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We have sufficient evidence to conclude that the mean is less than 126 at the 1% level of significance.

The null hypothesis is that the mean is equal to 126.

H0: μ = 126

The alternative hypothesis is that the mean is less than 126.

Ha: μ < 126

We will use a one-tailed t-test for this problem, with a significance level of 0.01 and 100 degrees of freedom (n-1).

a) State the hypotheses.

H0: μ = 126

Ha: μ < 126

b) Calculate the test statistic.

t = (x - μ) / (s / √n)

where x is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.

Plugging in the values given, we get:

t = (120.96 - 126) / (21.42 / √101) = -2.96

c) State the rejection criterion for the null hypothesis.

We will reject the null hypothesis if the test statistic is less than the critical value from the t-distribution with 100 degrees of freedom and a one-tailed test at a significance level of 0.01.

Using a t-table or a calculator, we find the critical value to be -2.364.

d) Draw your conclusion.

Since our test statistic (-2.96) is less than the critical value (-2.364), we reject the null hypothesis. We have sufficient evidence to conclude that the mean is less than 126 at the 1% level of significance.

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To diagonalize C, we first need to find its eigenvalues and eigenvectors. The characteristic equation for C is det(C -

                                λI)  =  0, which gives us (1 - λ)(7 - λ)  -  6  =                                

                                   0. Solving for λ, we get λ1  =  1 and λ2  =                                    

               7. To find the eigenvector corresponding to λ1, we solve the system of equations (C -                

                   λ1I)x  =  0, which gives us the equation  - x1  +  2x2  =  0. Choosing x2  =                    

                                           1, we get the eigenvector v1  =                                          

                            [2,1]. Similarly, for λ2 we get the eigenvector v2  =  [1, -                            

                              1]. We can then diagonalize C by forming the matrix P  =                              

                         [v1, v2] and the diagonal matrix D  =  [λ1 0; 0 λ2]. We have C  =                        

                      -                                    -                                                        

                   PDP 1. To compute A5, we first compute C 1 as [7  - 2;  - 3 1] / 4. Then, A  =                    

                         -         -   -                                  5       5       5                          

                      CDC 1  =  PDP 1DC 1P. We have D  =  [1 0; 0 7], so D   =  [1  0; 0 7 ]  =                      

                                           5       5 -                                                              

                    [1 0; 0 16807]. Thus, A   =  PD P 1  =  [2 1; 1  - 1][1 0; 0 16807][1 / 3  -                    

                              1 / 3; 1 / 3 2 / 3]  =  [11203 11202; 16804 16805] / 9.                                

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The 90% confidence interval for the proportion of Americans who decide not to go to college because they cannot afford it can be calculated using a statistical formula. The formula for a confidence interval is: CI = p ± zsqrt((p(1-p))/n)

Where CI is the confidence interval, p is the proportion of interest (in this case, 0.48 or 48%), and z is the critical value from the standard normal distribution for the desired level of confidence (in this case, 1.645 for 90% confidence), sqrt is the square root function, and n is the sample size (in this case, 331).

Plugging in the values, we get:

CI = 0.48 ± 1.645sqrt((0.48(1-0.48))/331)

CI = 0.48 ± 0.062

Thus, the 90% confidence interval for the proportion of Americans who decide not to go to college because they cannot afford it is (0.418, 0.542). This means that we can be 90% confident that the true proportion of Americans who decide not to go to college because they cannot afford it falls between 41.8% and 54.2%.

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Answer:

142 ft

Step-by-step explanation:

We have to find the perimeter of the rectangular garden.

     length = 30 ft

      Width = 41 ft

        [tex]\sf \boxed{\text{\bf Perimeter of rectangle =2*( length + width)}}[/tex]

                                                  = 2 * (30 + 41)

                                                  = 2 * 71

                                                  = 142 ft

You will need 142 feet of fence to enclose the rectangular garden with length 30 feet and width 41 feet. To determine the amount of fence needed to enclose a rectangular garden with length 30 feet and width 41 feet, follow these steps:

1. Identify the dimensions of the rectangular garden. In this case, the length is 30 feet and the width is 41 feet.
2. Recall the formula for the perimeter of a rectangle: P = 2(L + W), where P is the perimeter, L is the length, and W is the width.
3. Plug in the given dimensions: P = 2(30 + 41).
4. Calculate the sum inside the parentheses: P = 2(71).
5. Multiply by 2 to find the perimeter: P = 142 feet.

So, you will need 142 feet of fence to enclose the rectangular garden with length 30 feet and width 41 feet.

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Since the dimension of H and V is the same (n), it means that their bases have the same number of linearly independent vectors. Consequently, the basis of H can also serve as a basis for V.

Given that H is an n-dimensional subspace of an n-dimensional vector space V, where n is a natural number, we want to prove that H = V.
Since H is a subspace of V, it must satisfy the following properties:
1. H is closed under addition.
2. H is closed under scalar multiplication.
3. The zero vector (0) is in H.
We know that H is an n-dimensional subspace, which means it has a basis with n linearly independent vectors. Similarly, V is an n-dimensional vector space, so it also has a basis with n linearly independent vectors.
Since H and V share the same basis, they span the same space. Thus, we can conclude that H = V.

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Answer:

The correct answer is:

a. There were 77% of the entries with a score less than 68.

Step-by-step explanation:

The 77th percentile of the distribution of all the scores means that 77% of the entries had a score lower than 68, and 23% had a score equal to or greater than 68.

So option a is the best description of the 77th percentile. Option b is incorrect because it describes the complement of the 77th percentile (i.e., the percentage of entries with a score greater than 68). Option c is incorrect because it describes a single score,

whereas the percentile refers to a percentage of the distribution. Option d is incorrect because it provides a specific number of entries with a score less than 68, which may or may not be true,

but it doesn't address the percentile of the distribution.

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(a) The multiplicative inverse of 36 mod 45 is 4.

(b) The multiplicative inverse of 22 mod 35 is 4.

(c) The multiplicative inverse of 158 mod 331 is 201.

(d) The multiplicative inverse of 331 mod 158 is 119.

To find the multiplicative inverse of a number, we use the following formula:

[tex]a^-1 ≡ b (mod n)[/tex]

Where a is the number whose inverse is to be found, b is the multiplicative inverse of a and n is the modulus.

In this case, we have:

[tex]36^-1[/tex] ≡ b (mod 45) = 4

The multiplicative inverse of 22 mod 35 is 4. To find the multiplicative inverse of a number, we use the formula a * x ≡ 1 mod m where a is the number whose inverse we want to find, x is the inverse of a and m is the modulus.

We can solve this equation using the extended Euclidean algorithm1.

In this case, we have 22 * x ≡ 1 mod 35. Using the extended Euclidean algorithm, we can find that x = 41.

Therefore, the multiplicative inverse of 22 mod 35 is 4.

The multiplicative inverse of 158 mod 331 is 201. The modular multiplicative inverse of an integer a modulo m is an integer b such that the product ab is congruent to 1 with respect to the modulus m 1.

The multiplicative inverse of 331 mod 158 is 119.

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(a) The multiplicative inverse of 36 mod 45 is 4.

(b) The multiplicative inverse of 22 mod 35 is 4.

(c) The multiplicative inverse of 158 mod 331 is 201.

(d) The multiplicative inverse of 331 mod 158 is 119.

How to find the multiplicative inverse?

To find the multiplicative inverse of a number, we use the following formula:

a⁻¹  = b (mod n)

Where a is the number whose inverse is to be found, b is the multiplicative inverse of a and n is the modulus.

a) In this case, we have:

36⁻¹ ≡ b (mod 45) = 4

b) The multiplicative inverse of 22 mod 35 is 4.

To find the multiplicative inverse of a number, we use the formula

a * x ≡ 1 mod m

where a is the number whose inverse we want to find, x is the inverse of a and m is the modulus.

We can solve this equation using the extended Euclidean algorithm1.

In this case, we have 22 * x ≡ 1 mod 35. Using the extended Euclidean algorithm, we can find that x = 41.

Therefore, the multiplicative inverse of 22 mod 35 is 4.

c) The multiplicative inverse of 158 mod 331 is 201.

The modular multiplicative inverse of an integer a modulo m is an integer b such that the product ab is congruent to 1 with respect to the modulus m 1.

d) The multiplicative inverse of 331 mod 158 is 119.

hence, (a) The multiplicative inverse of 36 mod 45 is 4.

(b) The multiplicative inverse of 22 mod 35 is 4.

(c) The multiplicative inverse of 158 mod 331 is 201.

(d) The multiplicative inverse of 331 mod 158 is 119.

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This is the equation in terms of w that models the situation:[tex]w^2 - 10w + 18.75 = 0[/tex].

Rectangle with width w and length l, enclosed by a 20-meter fence: The perimeter of the rectangle, which is equal to the length of the fence, is given by:

2w + 2l = 20

We can simplify this equation by dividing both sides by 2:

w + l = 10

We also know that the area of the rectangle is 18.75 square meters:

w * l = 18.75

We want to write an equation in terms of w, so we can solve for l in terms of w by dividing both sides by w:

l = 18.75 / w

This expression for l into the equation for the perimeter, we get:

w + (18.75 / w) = 10

Multiplying both sides by w, we get:

[tex]w^2 + 18.75 = 10w[/tex]

Rearranging this equation, we get:

[tex]w^2 - 10w + 18.75 = 0[/tex]

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Answer:

Step-by-step explanation:

The volume of the prism given in the image above is calculated as: 700 cubic meters.

The prism is a trapezoidal prism, therefore the formula to use to find the volume is given as:

Volume (V) = (Base Area) × Length of prism

Base area of the prism = 1/2 * (a + b) * h

a = 10 m

b = 25 m

h = 5 m

Base area = 1/2 * (10 + 25) * 5

Base area = 87.5 m²

Length of the prism = 8 m

Therefore, we have:

Volume of the prism (V) = 87.5 * 8 = 700 cubic meters.

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The statement which correctly describes the data set include the following:

D. the data is bivariate and numerical.

In Mathematics, a bivariate data can be defined as a type of data set which comprises information that are based on two (2) variables, usually two types of related data.

In Mathematics and statistics, a numerical data can be defined as a type of data set that is primarily expressed in numbers only. This ultimately implies that, a numerical data simply refers to a type of data set consisting of numbers (numerals), rather than words or letters.

Thus, In conclusion, we can logically deduce that the given data set is both bivariate and numerical.

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George's total dividends for the year as a percentage of the price he paid for each share is 380%, and the new percentage return for the year, assuming the stock price increases to $100, is 3.8%.

To calculate the total dividends and percentage return for the year, follow these steps:

1. Find the total dividend per share: $1 + $1 + $1 + $0.80 = $3.80

2. Find the price George paid for each share: Since the dividend is the same for all shares, we'll use the highest dividend of $1 as the price he paid for each share.

3. Calculate the total dividends for the year as a percentage of the price he paid for each share:

[tex](\frac{3.8}{1})(100)[/tex] = 380%

Now, let's find the new percentage return for the year, assuming the stock price increases to $100 and the company pays the same dividend:

4. Calculate the new percentage return for the year: [tex](\frac{3.8}{100})(100)[/tex]= 3.8%

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The component form and magnitude of the vector are;

v = ⟨-5, 3⟩ and ||v|| = √(34)

The difference between the points on the graph can be used to express the vector in component form as follows;

The component of the vectors are the horizontal and the vertical component

The horizontal component is; -(2 - (-3))·i = -5·i·

The vertical component is; ((5 - 2)·j = 3·j

The magnitude of the vector is; ||v|| = √((-3 - 2)² + (5 - 2)²) = √(34)

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The minimum cost of production is $278.46.

The base width of the box be x, then the base length is 2x. Let the height of the box be h.

The volume of the box is given by:

V = base area × height

[tex]70 = x[/tex] × [tex]2x[/tex] × [tex]h[/tex]

[tex]h = 35/x^2[/tex]

The cost of producing the box is given by:

[tex]C = 2[/tex] ×[tex](cost of top/bottom) + 4[/tex] × [tex](cost of side)[/tex]

[tex]C = 2[/tex]× [tex]9[/tex]× [tex](2x[/tex] × [tex]x) + 4[/tex] × [tex]6[/tex] × [tex](2x + 2h)[/tex]

[tex]C = 36x^2 + 48xh[/tex]

Substituting the expression for h obtained above:

[tex]C = 36x^2+ 48x(35/x^2)[/tex]

[tex]C = 36x^2+ 1680/x[/tex]

To minimize C, we take the derivative with respect to x and set it to zero:

[tex]dC/dx = 72x - 1680/x^2 = 0[/tex]

[tex]72x = 1680/x^2[/tex]

[tex]x^3 = 1680/72[/tex]

[tex]= 23.33[/tex]

x = 2.82 m (rounded to two decimal places)

Substituting this value of x in the expression for h, we get:

[tex]h = 35/(2.82)^2[/tex]

[tex]= 4.34 m[/tex] (rounded to two decimal places)

Therefore, the dimensions of the box that minimize the cost of production are:

[tex]Base width = x = 2.82 m[/tex]

[tex]Base length = 2x = 5.64 m[/tex]

[tex]Height = h = 4.34 m[/tex]

To find the minimum cost, we substitute these values of x and h in the expression for C:

[tex]C = 36(2.82)^2 + 1680/(2.82)[/tex]

[tex]C = $278.46[/tex] (rounded to two decimal places)

Therefore, the minimum cost of production is $278.46.

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The table should be completed with the correct key features as follows;

Axis of symmetry (1st graph): x = 1.

Vertex (1st graph): (1, -9).

Minimum (1st graph): -9.

y-intercept (1st graph): (0, -8).

Axis of symmetry (2nd graph): x = 2.

Vertex (2nd graph): (2, 16).

Maximum (2nd graph): 16.

y-intercept (2nd graph): (0, 12).

In Mathematics and Geometry, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. Based on the first graph of a quadratic function, we can logically deduce that the graph is an upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero (0).

Based on the second graph of a quadratic function, we can logically deduce that the graph is a downward parabola because the coefficient of x² is negative and the value of "a" is less than zero (0).

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Answer:b

Step-by-step explanation:

b