Question

A study of the exercise habits of 17,000 alumni from a certain college found that the death rate (deaths per 10,000 person-years) was approximately R(x) = 5x2 − 25x + 103, where x is the weekly amount of exercise in thousands of calories (0 ≤ x ≤ 4). Find the exercise level that minimizes the death rate.

Answer 1

The minimum death rate occurs at an exercise level of 2.5 thousand calories per week, which is found by setting the derivative of the death rate function to zero and solving for x.

Step-by-step explanation:

The exercise level that minimizes the death rate for the given study can be determined using calculus, specifically by finding the vertex of the parabolic function R(x) = 5x2 − 25x + 103. To find the minimum point, we take the derivative of R(x), set it to zero, and solve for x:

R'(x) = 10x - 25

Setting R'(x) to zero gives us:

0 = 10x - 25

Solving for x, we find:

x = 2.5

The weekly amount of exercise in thousands of calories that minimizes the death rate is 2.5 thousand calories or 2500 calories.