Answer :
To find the total cost of creating the highway, we need to calculate the product of the expression for the miles of highway added and the expression for the cost per mile. Let's break down the solution step-by-step:
1. Expression for Miles of Highway Added:
- The state department added a highway described by the expression [tex]\(\frac{1}{3}x^2 + 5x\)[/tex].
2. Expression for Cost per Mile (in thousands of dollars):
- The cost to complete each mile is given by the expression [tex]\(3x^2 - 9x\)[/tex].
3. Total Cost Calculation:
- To find the total cost, multiply the expressions for miles of highway added and the cost per mile.
[tex]\[
\left(\frac{1}{3}x^2 + 5x\right) \times (3x^2 - 9x)
\][/tex]
4. Multiply the Two Expressions:
- Distribute each term in the first expression by each term in the second expression.
[tex]\[
\begin{align*}
\text{First term: } & \frac{1}{3}x^2 \times 3x^2 = x^4 \\
\text{Second term: } & \frac{1}{3}x^2 \times (-9x) = -3x^3 \\
\text{Third term: } & 5x \times 3x^2 = 15x^3 \\
\text{Fourth term: } & 5x \times (-9x) = -45x^2
\end{align*}
\][/tex]
5. Combine Like Terms:
- Combine all the terms to form a single expression.
[tex]\[
x^4 - 3x^3 + 15x^3 - 45x^2
\][/tex]
- Simplify by combining the [tex]\(x^3\)[/tex] terms:
[tex]\[
x^4 + 12x^3 - 45x^2
\][/tex]
6. Identify the Correct Expression:
- The expression that matches our simplified result is [tex]\(x^4 + 12x^3 - 45x^2\)[/tex].
Thus, the total cost, in thousands of dollars, for creating the highway is given by:
[tex]\(x^4 + 12x^3 - 45x^2\)[/tex].
1. Expression for Miles of Highway Added:
- The state department added a highway described by the expression [tex]\(\frac{1}{3}x^2 + 5x\)[/tex].
2. Expression for Cost per Mile (in thousands of dollars):
- The cost to complete each mile is given by the expression [tex]\(3x^2 - 9x\)[/tex].
3. Total Cost Calculation:
- To find the total cost, multiply the expressions for miles of highway added and the cost per mile.
[tex]\[
\left(\frac{1}{3}x^2 + 5x\right) \times (3x^2 - 9x)
\][/tex]
4. Multiply the Two Expressions:
- Distribute each term in the first expression by each term in the second expression.
[tex]\[
\begin{align*}
\text{First term: } & \frac{1}{3}x^2 \times 3x^2 = x^4 \\
\text{Second term: } & \frac{1}{3}x^2 \times (-9x) = -3x^3 \\
\text{Third term: } & 5x \times 3x^2 = 15x^3 \\
\text{Fourth term: } & 5x \times (-9x) = -45x^2
\end{align*}
\][/tex]
5. Combine Like Terms:
- Combine all the terms to form a single expression.
[tex]\[
x^4 - 3x^3 + 15x^3 - 45x^2
\][/tex]
- Simplify by combining the [tex]\(x^3\)[/tex] terms:
[tex]\[
x^4 + 12x^3 - 45x^2
\][/tex]
6. Identify the Correct Expression:
- The expression that matches our simplified result is [tex]\(x^4 + 12x^3 - 45x^2\)[/tex].
Thus, the total cost, in thousands of dollars, for creating the highway is given by:
[tex]\(x^4 + 12x^3 - 45x^2\)[/tex].
