Question

The gas mileage m (x) (in mpg) for a certain vehicle can be approximated by m(x)=-0.033x²+2.677x-35.017, where x is the speed of the vehicle in mph.

Determine the speed at which car gets its maximum mileage. Round your answer to the nearest mph. The gas mileage is at a maximum when the car travels at __ mph?

Answer 1

The gas mileage is at a maximum when the car travels at 40.56 mph?

Explanation:

The equation relating gas mileage and mpg is given as

`m(x)=-\;0.033x^(2)+2.677x-35.017`

where m(x) = mpg and x = miles driven

The maximum of this function can be found by taking the first derivative of this function, setting it equal to 0 and solving for x

`(d)/(dx)\left(-0.033x^2+2.677x-35.017\right) \\\\=(d)/(dx)\left(-0.033x^2\right)+(d)/(dx)\left(2.677x\right)-(d)/(dx)\left(35.017\right)\\\\= -0.066x + 2.677 + 0\\\\= -0.066x + 2.677`

Setting this expression equal to 0 gives:

`-0.066x+2.677=0`

Move 2.677 to the right:

`-0.066x = -2.677\\\\`

Divide by -0.066:

`x = (-2.677)/(-0.066)\\\\x = 40.56 \;mph`