A model for the basal metabolism rate, in kcal/h, of a young man is R(t) = 75 − 0.19 cos(𝜋t/12), where t is the time in hours measured from 5:00 AM. What is the total basal metabolism of this man, 24R(t) dt,0 over a 24-hour time period?
Question
A model for the basal metabolism rate, in kcal/h, of a young man is R(t) = 75 − 0.19 cos(𝜋t/12), where t is the time in hours measured from 5:00 AM. What is the total basal metabolism of this man, 24R(t) dt,0 over a 24-hour time period?
Solution
To solve this problem, we need to integrate the function R(t) over the interval from 0 to 24. This will give us the total basal metabolism over a 24-hour period.
The integral of R(t) from 0 to 24 is:
∫ from 0 to 24 [75 - 0.19 cos(𝜋t/12)] dt
We can break this integral into two parts:
= ∫ from 0 to 24 75 dt - ∫ from 0 to 24 0.19 cos(𝜋t/12) dt
The first integral is easy to compute:
∫ from 0 to 24 75 dt = 75 * (24 - 0) = 1800
The second integral requires a substitution. Let u = 𝜋t/12. Then du = 𝜋/12 dt, and dt = 12/𝜋 du. The limits of integration also change: when t = 0, u = 0, and when t = 24, u = 2𝜋.
So the second integral becomes:
- ∫ from 0 to 2𝜋 0.19 cos(u) * 12/𝜋 du
This integral can be computed as:
- [0.19 * 12/𝜋 * sin(u)] from 0 to 2𝜋 = -0.19 * 12/𝜋 * [sin(2𝜋) - sin(0)] = 0
So the total basal metabolism over a 24-hour period is 1800 + 0 = 1800 kcal.
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