Question

The speed of a runner, in miles per hour, on a straight trail is modeled by f(m)=1/10(-2m³ 9m²-12m), where m is the runner's distance, in miles, from the start of the trail. What is the maximum speed of the runner for 0 ≤ m ≤ 3?

Answer 1

The maximum speed of the runner over the interval from 0 to 3 miles can be found using calculus to identify the critical points of the function representing the speed and evaluating the speed at those points and the interval’s endpoints.

Step-by-step explanation:

The maximum speed of the runner can be found using calculus, by finding the critical points of the function f(m) = 1/10(-2m³ + 9m² - 12m) and evaluating the end points, m = 0 and m = 3, since we are interested in the interval from 0 to 3 miles. To find the critical points, we first compute the derivative of the function with respect to m, set it to zero, and solve for m. The critical points are where the function's slope (speed) changes from increasing to decreasing or vice versa. We evaluate the function at the critical points and at the end points m = 0 and m = 3 to find the maximum speed. Given the cubic nature of the function, there could be up to two critical points within the given interval where local maxima might occur.