Question

The gas mileage m (x) (in mpg) for a certain vehicle can be approximated by m (x) = -0.03x^2+2.673x- 35.012, where x is the speed of the vehicle in mph.
(b) Determine the maximum gas mileage. Round your answer to one decimal place.

Answer 1

To find the maximum gas mileage from the function \( m(x) = -0.03x^2 + 2.673x - 35.012 \), we need to determine the vertex of this quadratic function, since it opens downward (the coefficient of \( x^2 \) is negative).

The vertex \( x \) for a quadratic function in the form \( ax^2 + bx + c \) can be found using the formula:

\[
x = -\frac{b}{2a}
\]

In this function:
- \( a = -0.03 \)
- \( b = 2.673 \)

Now, let's calculate \( x \):

\[
x = -\frac{2.673}{2 \cdot -0.03} = -\frac{2.673}{-0.06} = \frac{2.673}{0.06}
\]

Calculating further:

\[
x \approx 44.55
\]

Now, we'll substitute this value of \( x \) back into the gas mileage function to find the maximum gas mileage:

\[
m(44.55) = -0.03(44.55^2) + 2.673(44.55) - 35.012
\]

Calculating \( 44.55^2 \):

\[
44.55^2 \approx 1986.9025
\]

Now, substituting this value back into \( m(x) \):

\[
m(44.55) \approx -0.03(1986.9025) + 2.673(44.55) - 35.012
\]
\[
\approx -59.607075 + 118.99115 - 35.012
\]
\[
\approx -59.607075 + 118.99115 - 35.012
\]
\[
\approx 24.372075
\]

Rounding this to one decimal place, the maximum gas mileage is:

\[
\boxed{24.4} \text{ mpg}
\]