Question

The temperatue of a town months after January can be estimated by the function f(t) = - 26cos(pi/6 * t) + 48 .

Answer 1

ANSWER

$\frac{2\left(78+72\pi\right)}{3\pi}=64.55211$

EXPLANATION

Given;

$f(t)=\left(-26cos\left(\frac{\pi}{6}t\right)+48\right)\:$

The average temperature from month 3 to month 6 ;

This will be the average value of the function over the interval (a,b);

Hence;

$\begin{gathered} \frac{1}{b-a}\int_a^bf(t)dt \\ \\ \end{gathered}$

Substituting the values, we have;

$\begin{gathered} \frac{1}{6-3}\int _3^6\left(-26cos\left(\frac{\pi }{6}t\right)+48\right)\:dt \\ \frac{1}{3}\int_3^6\left(-26cos\left(\frac{\pi}{6}t\right)+48\right)\:dt \\ 2\left(-13\cos \left(\frac{\pi t}{6}\right)+24\right) \\ \int _3^62\left(-13\cos \left(\frac{\pi t}{6}\right)+24\right)dt \end{gathered}$

Take constant out;

$2\cdot \int _3^6-13\cos \left(\frac{\pi t}{6}\right)+24dt$

Apply sum rule;

$\begin{gathered} 2\left(-\int _3^613\cos \left(\frac{\pi t}{6}\right)dt+\int _3^624dt\right) \\ \int_3^613\cos\left(\frac{\pi t}{6}\right)dt=13\cdot\int_3^6\cos\left(\frac{\pi t}{6}\right)dt \\ \end{gathered}$

Using integral substitution;

$\begin{gathered} \:u=\frac{\pi t}{6} \\ \frac{du}{dt}=\frac{\pi}{6} \\ du=\frac{\pi }{6}dt \\ dt=\frac{6}{\pi }du \\ \int \cos \left(u\right)\frac{6}{\pi }du \end{gathered}$

At t=3;

$\begin{gathered} \frac{\pi 3}{6} \\ \frac{\pi }{2} \end{gathered}$

At t=6;

$\frac{\pi6}{6}=\pi$

Now, we have;

$\begin{gathered} \int _{\frac{\pi }{2}}^{\pi }\cos \left(u\right)\frac{6}{\pi }du \\ 13\cdot \int _{\frac{\pi }{2}}^{\pi }\cos \left(u\right)\frac{6}{\pi }du \\ 13\cdot \frac{6}{\pi }\cdot \int _{\frac{\pi }{2}}^{\pi }\cos \left(u\right)du \\ 13\cdot \frac{6}{\pi }\left[\sin \left(u\right)\right]^{\pi }_{\frac{\pi }{2}} \\ \frac{78}{\pi }\left[\sin \left(u\right)\right]^{\pi }_{\frac{\pi }{2}} \\ 2\left(-\left(-\frac{78}{\pi }\right)+72\right) \\ 2\left(\frac{78}{\pi }+72\right) \\ \frac{2\left(78+72\pi \right)}{3\pi } \end{gathered}$