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Chapter 5: Q.80 (page 453)

An office building has a thermostat that keeps the temperature between \(20\) and \(26\) degrees Celsius for \(24\) hours a day. As the heating and cooling units alternate, the temperature starting from midnight is given by

\(T(t)=20\sin^3\left(\frac{\pi}{3}x\right)\cos^3\left(\frac{\pi}{3}x\right)+22.5\)

as shown in the figure. Find the average temperature in the office building between the hours of 9 AM and 5 PM.

Short Answer

Expert verified

The average temperature is \(22.67^\circ C \).

Step by step solution

01

Given Information

Consider the function \(T(t)=20\sin^3\left(\frac{\pi}{3}x\right)\cos^3\left(\frac{\pi}{3}x\right)+22.5\).

Since the temperature starts from mid-night, the value of \(x\) for 9 AM is \(x=9\) and for 5 PM is \(x=17\).

The average of the function \(f(x)\) on \([a,b]\).

\(f_{avg}=\frac{1}{b-a}\int_a^bf(x)dx\)

02

Find the average temperature

\(\begin{align*}T_{avg}&=\frac{1}{17-9}\int_9^{17}\left ( 20\sin^3\left(\frac{\pi}{3}x\right)\cos^3\left(\frac{\pi}{3}x\right)+22.5 \right )dx\\&=\frac{1}{8}\times(181.34)\\&=22.67\end{align*}\)

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