The cost in dollars to manufacture x pairs of shoes is given by 12,000 + 19x. This month, the manufacturer produced 1000 more pairs of shoes than last month. The average cost per pair dropped by $0.43.

a) Write an expression for the average cost per pair of shoes. Use this expression to write an equation to represent there situation.

b) Solve your equation

c) Are there any mathematical restrictions on the domain? Explain.

d) Determine reasonable domain in the context of the problem. Use your answers to parts I and II to answer the question.

(a) The expression for the average cost per pair of shoes is [tex]A(x)=\frac{12000}{x}+19[/tex] and equation for the situation is [tex]\frac{12000}{x}+19-(\frac{12000}{x+1000}+19)=0.43[/tex].

(b) x=[tex]x\approx 4806[/tex]

(c) The average function is not defined for x=0, so the domain of the function is all real numbers except 0.

(d) The reasonable domain is all natural numbers.

Step-by-step explanation:

The cost in dollars to manufacture x pairs of shoes is given by

[tex]C(x)=12000+19x[/tex]

where, x is the pairs of shoes.

(a)

The expression for the average cost per pair of shoes.

[tex]A(x)=\frac{C(x)}{x}[/tex]

[tex]A(x)=\frac{12000+19x}{x}[/tex]

[tex]A(x)=\frac{12000}{x}+19[/tex]

This month, the manufacturer produced 1000 more pairs of shoes than last month. The average cost per pair dropped by $0.43.

[tex]A(x)-A(x+1000)=0.43[/tex]

[tex]\frac{12000}{x}+19-(\frac{12000}{x+1000}+19)=0.43[/tex]

Therefore the expression for the average cost per pair of shoes is [tex]A(x)=\frac{12000}{x}+19[/tex] and equation for the situation is [tex]\frac{12000}{x}+19-(\frac{12000}{x+1000}+19)=0.43[/tex].

(b)

On solving the above equation we get

[tex]\frac{12000}{x}-\frac{12000}{x+1000}=0.43[/tex]

[tex]\frac{12000000}{x^2 + 1000 x} = 0.43[/tex]

[tex]12000000=0.43(x^2 + 1000 x)[/tex]

[tex]12000000=0.43x^2 + 430x[/tex]

[tex]0=0.43x^2 + 430x-12000000[/tex]

Using graphing calculator we get

[tex]x\approx -5806.31,4806.31[/tex]

The pair of shoe can not be native and decimal value.

[tex]x\approx 4806[/tex]

Therefore the solution is [tex]x\approx 4806[/tex].

(c)

The average cost function is

[tex]A(x)=\frac{12000}{x}+19[/tex]

The function is not defined if the denominator is 0.

The above function is not defined for x=0, so the domain of the function is all real numbers except 0.

(d)

In the average function x represents the number of pair of shoe.

It means the value of x must be a positive integer.

Since the average function is not defined for x=0, So the reasonable domain of average function is

Domain={x : x∈Z⁺, x≠0}

Domain=N

Therefore the reasonable domain is all natural numbers.