Question

a state department of public safety added $\frac{1}{3}x^2 + 5x$ miles of highway to connect cooltown and bragville. it cost the department $3x^2 - 9x$ thousand dollars to complete each mile of interstate. which expression shows the total cost, in thousands of dollars, of creating this highway?
options:
$x^4 + 12x^3 - 45x^2$
$x^4 - 45x^2$
$x^4 + 15x^3 - 48x^2$
$9x^4 - 12x^3 - 45x^2$

Explanation:

Step1: Recall the formula for total cost

Total cost is the product of the number of miles and the cost per mile. So we need to multiply \(\frac{1}{3}x^{2}+5x\) (miles) by \(3x^{2}-9x\) (thousand dollars per mile).

Step2: Use the distributive property (FOIL method for polynomials)

First, multiply \(\frac{1}{3}x^{2}\) by each term in \(3x^{2}-9x\):
\(\frac{1}{3}x^{2}\times3x^{2}=\frac{1\times3}{3}x^{2 + 2}=x^{4}\)
\(\frac{1}{3}x^{2}\times(-9x)=\frac{1\times(-9)}{3}x^{2+1}=- 3x^{3}\)
Then, multiply \(5x\) by each term in \(3x^{2}-9x\):
\(5x\times3x^{2}=15x^{3}\)
\(5x\times(-9x)=-45x^{2}\)

Step3: Combine like terms

Now, combine the results from Step 2:
\(x^{4}-3x^{3}+15x^{3}-45x^{2}\)
Combine the \(x^{3}\) terms: \(-3x^{3}+15x^{3}=12x^{3}\)
So the total cost expression is \(x^{4}+12x^{3}-45x^{2}\)

Answer:

\(x^{4}+12x^{3}-45x^{2}\) (the first option among the given choices)