#### To determine

**To find:**

The man’s total basal metabolism over an 24- hour time period

#### Answer

**2040 kcal**

#### Explanation

**1) Concept:**

i. The substitution rule:

If u=g(x) is a differentiable function whose range is I and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du. Here,g(x) is substituted as u and g’(x)dx =du.

ii. Indefinite integral

∫xn dx=xn+1n+1+C (n≠-1)

**2) Given:**

The basal metabolism rate of man is Rt=85-0.18cosπt12.

3) **Calculation:**

Given that the basal metabolism rate of man is Rt=85-0.18cosπt12

To find the man’s total basal metabolism over an 24- hour time period, we need to integrate the rate of basal metabolism of man over 0 to 24.

**∫024Rtdt=∫02485-0.18cosπt12dt**

Use the substitution method.

Substitute πt12=u. Differentiating with respect to t

π12dt=du

dt=12πdu

The limits change and the new limits of integration are calculated by substituting

for t=0, u=0& for t=24, u=2π

Therefore, the given integral becomes

∫02485-0.18cosπt12dt=12π∫02π85-0.18cosudu

=12π(85u-0.18sinu)02π

=12π85(2π)-0.18sin2π-85(0)-0.18sin0

Simplify.

=12π85(2π)

=2040

Therefore, total basal metabolism over an 24- hour time periodis 2040 kcal.

**Conclusion:**

Therefore, total basal metabolism over an 24- hour time periodis 2040 kcal.