Question

11 a company makes different sizes of boxes. the widest box has a width, w, of 27 inches, a depth, d, of 10 inches and a height, h, of 8 inches. other boxes the company makes decrease in width by increments of 3 inches. for each decrease of 3 inches in width, however, both d and h increase by 1 inch. which function represents the volume of each box the company makes? a. v(x)=(27x - 3)(10x + 1)(8x + 1) b. v(x)=(3 - 27x)(1 + 10x)(1 + 8x) c. v(x)=(3x - 27)(x + 10)(x + 8) d. v(x)=(27 - 3x)(10 + x)(8 + x) 12 the graph shown represents the function h(x)=p(x)-q(x)-r(x) which functions could be p(x), q(x), and r(x)? a. p(x)=x + 2; q(x)=-x + 1; r(x)=-5x - 1 b. p(x)=x - 2; q(x)=-x + 1; r(x)=5x - 1 c. p(x)=4x - 2; q(x)=-3x + 1; r(x)=-x - 1 d. p(x)=4x + 2; q(x)=3x + 1; r(x)=x + 2 13 which function has a zero with a multiplicity of 2? a. f(x)=(x - 1)(x + 3) b. f(x)=(x + 2)(x - 2) c. f(x)=(x - 5)(x + 1)(x - 5) d. f(x)=(x - 2)(x - 2)(x - 2) 14 the function f(x)=-4x²+10,000x represents a companys income on a product, where x is the price of the product in dollars. what is the maximum income the company could make selling the product? a. $781,250 b. $1,562,500 c. $3,125,000 d. $6,250,000

Explanation:

11.

Step1: Analyze the change in width

The initial width is 27 inches and it decreases by 3 - inch increments. If we let \(x\) be the number of increments, the width \(w = 27-3x\).

Step2: Analyze the change in depth

The initial depth is 10 inches and it increases by 1 - inch for each increment of \(x\), so \(d = 10 + x\).

Step3: Analyze the change in height

The initial height is 8 inches and it increases by 1 - inch for each increment of \(x\), so \(h = 8 + x\).

Step4: Recall the volume formula

The volume of a box \(V=w\times d\times h\). Substituting the expressions for \(w\), \(d\), and \(h\) we get \(V(x)=(27 - 3x)(10 + x)(8 + x)\).

We know that \(h(x)=p(x)-q(x)-r(x)\). First, find \(p(x)-q(x)-r(x)\) for each option.

Option a:

\(p(x)-q(x)-r(x)=(x + 2)-(-x + 1)-(-5x-1)=x + 2+x - 1 + 5x+1=7x+2\).

Option b:

\(p(x)-q(x)-r(x)=(x - 2)-(-x + 1)-(5x - 1)=x - 2+x - 1-5x + 1=-3x-2\).

Option c:

\(p(x)-q(x)-r(x)=(4x - 2)-(-3x + 1)-(-x - 1)=4x - 2 + 3x-1+x + 1=8x-2\).

Option d:

\(p(x)-q(x)-r(x)=(4x + 2)-(3x + 1)-(x + 2)=4x + 2-3x - 1-x - 2=-1\).
By analyzing the \(y\) - intercept and the general shape of the graph of \(h(x)\) and substituting \(x = 0\) into the functions from the options, we can see that the functions in option c can form a function that matches the general shape of \(h(x)\) (by considering the degree and leading - coefficient of the resulting polynomial).

The multiplicity of a zero is the number of times the corresponding factor appears in the factored form of the polynomial.

Option a:

\(f(x)=(x - 1)(x + 3)\) has zeros \(x = 1\) and \(x=-3\) with multiplicity 1 each.

Option b:

\(f(x)=(x + 2)(x - 2)\) has zeros \(x=-2\) and \(x = 2\) with multiplicity 1 each.

Option c:

\(f(x)=(x - 5)(x + 1)(x - 5)=(x - 5)^2(x + 1)\) has a zero \(x=-1\) with multiplicity 1 and a zero \(x = 5\) with multiplicity 2.

Option d:

\(f(x)=(x - 2)(x - 2)(x - 2)=(x - 2)^3\) has a zero \(x = 2\) with multiplicity 3.

Answer:

d. \(V(x)=(27 - 3x)(10 + x)(8 + x)\)

12.