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.
(a) Determine the speed at which the car gets its maximum mileage. Round your answer to the nearest mph.?
(a) Determine the speed at which the car gets its maximum mileage. Round your answer to the nearest mph.?
Answers
GPT-4o mini
To find the speed at which the car gets its maximum mileage, we need to determine the vertex of the quadratic function represented by \( m(x) = -0.03x^2 + 2.673x - 35.012 \). The quadratic function can be expressed in the form \( ax^2 + bx + c \), where:
- \( a = -0.03 \)
- \( b = 2.673 \)
- \( c = -35.012 \)
The x-coordinate of the vertex for a quadratic equation \( y = ax^2 + bx + c \) can be calculated using the formula:
\[
x = -\frac{b}{2a}
\]
Substituting in our values for \( a \) and \( b \):
\[
x = -\frac{2.673}{2 \times -0.03} = -\frac{2.673}{-0.06} = \frac{2.673}{0.06}
\]
Calculating the right-hand side:
\[
\frac{2.673}{0.06} = 44.55
\]
Rounding to the nearest mph, the speed at which the car gets its maximum mileage is:
\[
\boxed{45} \text{ mph}
\]
- \( a = -0.03 \)
- \( b = 2.673 \)
- \( c = -35.012 \)
The x-coordinate of the vertex for a quadratic equation \( y = ax^2 + bx + c \) can be calculated using the formula:
\[
x = -\frac{b}{2a}
\]
Substituting in our values for \( a \) and \( b \):
\[
x = -\frac{2.673}{2 \times -0.03} = -\frac{2.673}{-0.06} = \frac{2.673}{0.06}
\]
Calculating the right-hand side:
\[
\frac{2.673}{0.06} = 44.55
\]
Rounding to the nearest mph, the speed at which the car gets its maximum mileage is:
\[
\boxed{45} \text{ mph}
\]