Question

The gas mileage for a certain vehicle can be approximated by m=0.02x ²+39x where x is the speed of the vehicle in mph. Determine the speed(s) at which the car gets 20 mpg.

a) 49 mph and 14 mph
b) 20 mph and 19 mph
c) 18 mph and 21 mph
d) 16 mph and 23 mph

Answer 1

To determine the speeds at which the car gets 20 mpg, the quadratic equation 0.02x² + 39x = 20 needs to be solved. Using the quadratic formula yields speeds of approximately 14 mph and 49 mph where the car attains 20 mpg, aligning with option (a).

Step-by-step explanation:

The student's question involves determining the speeds at which a car gets 20 miles per gallon (mpg), given the formula m = 0.02x² + 39x, where x is the speed of the vehicle in mph and m is the gas mileage (mpg). To find the speeds, we need to solve the quadratic equation 0.02x² + 39x = 20 by setting m to 20 mpg.

1. Subtract 20 from both sides of the equation: 0.02x² + 39x - 20 = 0.
2. Use the quadratic formula to solve for x: x = (-b ± √(b² - 4ac)) / (2a), where a = 0.02, b = 39, and c = -20.
3. Calculate the determinant (√(b² - 4ac)), plug the values of a, b, and c into the formula, and simplify to find the two possible speeds for x.

After completing the calculations, we find that the car gets 20 mpg at speeds of approximately 14 mph and 49 mph, which corresponds to option (a).