A state department of public safety added 1/3 x² + 5x miles of highway to connect CoolTown and Bragville. It cost the department 3x² – 9 thousand dollars to complete each mile of interstate. Which expression shows the total cost, in thousands of dollars, of creating this highway?
x⁴ + 12x³ – 45x²
x⁴ – 45x²
x⁴ + 15x³ – 48x²
9x⁴ – 12x³ – 45x²
To find the total cost of creating the highway, we need to multiply the number of miles of highway by the cost per mile.
1. Miles of highway: Given as ( frac{1}{3}x^2 + 5x ).
2. Cost per mile: Given as ( 3x^2 – 9 ) thousand dollars.
The total cost (in thousands of dollars) is calculated as follows:
[
text{Total Cost} = left(frac{1}{3}x^2 + 5xright)(3x^2 – 9)
]
Now, let’s distribute:
[
= left(frac{1}{3}x^2 cdot 3x^2right) + left(frac{1}{3}x^2 cdot (-9)right) + left(5x cdot 3x^2right) + left(5x cdot (-9)right)
]
Calculating each part step by step:
– ( frac{1}{3}x^2 cdot 3x^2 = x^4 )
– ( frac{1}{3}x^2 cdot (-9) = -3x^2 )
– ( 5x cdot 3x^2 = 15x^3 )
– ( 5