Chapter 4.4, Problem 84E

Solution

To identify: The functions that can be determined from the given information:

  $\left(\text{I}\right)f\left(0\right)\left(\text{II}\right)f\left(6\right)\left(\text{III}\right)f\left(96\right)$

The graph of $y=f\left(x\right)$ is a sinusoid with period 45 passing through the point $\left(6,0\right)$ .

The given options are:

  $\left(A\right)$ I only $\left(B\right)$ II only $\left(C\right)$ I and III only

  $\left(D\right)$ II and III only $\left(E\right)$ I, II, and III only

The correct option is $\left(D\right)$ II and III only.

Given information:

The graph of $y=f\left(x\right)$ is a sinusoid with period 45 passing through the point $\left(6,0\right)$ .

Explanation:

Consider the given function: $y=f\left(x\right)$

So, $y=a\sin \left(bx+c\right)$

So the period is 45 then:

  $\begin{array}{c}45=45\cdot \frac{\pi }{180}\\ =\frac{\pi }{4}\end{array}$

  $\begin{array}{c}\text{Period}=\frac{2\pi }{\left|b\right|}\\ \frac{\pi }{4}=\frac{2\pi }{\left|b\right|}\\ b=8\end{array}$

So, the function is $y=a\sin \left(8x+c\right)$ .

It is given that the function is passing through the point $\left(6,0\right)$ .

  $\begin{array}{c}0=a\sin \left(8\cdot 6+c\right)\\ 0=a\sin \left(48+c\right)\\ \sin \left(48+c\right)=0\\ 48+c=0\\ c=-48\cdot \frac{\pi }{180}\\ c=-\frac{4\pi }{15}\end{array}$

Thus, the function is $y=a\sin \left(8x-\frac{4\pi }{15}\right)$ .

Now find the value of $f\left(0\right)$ .

  $\begin{array}{c}y=a\sin \left(8x-\frac{4\pi }{15}\right)\\ =a\sin \left(8\left(0\right)-\frac{4\pi }{15}\right)\\ =a\sin \left(-\frac{4\pi }{15}\right)\\ =-a\sin \left(\frac{4\pi }{15}\right)\end{array}$

To find the value of $f\left(0\right)$ , the value of a must be needed. Thus, $f\left(0\right)$ cannot be determined by the given information.

Now find the value of $f\left(6\right)$ .

  $\begin{array}{c}y=a\sin \left(8x-\frac{4\pi }{15}\right)\\ y=a\sin \left(8\left(6\right)-\frac{4\pi }{15}\right)\\ y=a\sin \left(48-\frac{4\pi }{15}\right)\\ y=-a\sin \left(\frac{4\pi }{15}-\frac{4\pi }{15}\right)\\ y=a\sin \left(0\right)\\ y=0\end{array}$

Thus, the value of $f\left(6\right)$ can be determined by the given information.

Now find the value of $f\left(96\right)$ .

  $\begin{array}{c}y=a\sin \left(8x-\frac{4\pi }{15}\right)\\ y=a\sin \left(8\left(96\right)-\frac{4\pi }{15}\right)\\ y=a\sin \left(768-\frac{4\pi }{15}\right)\\ y=-a\sin \left(\frac{64\pi }{15}-\frac{4\pi }{15}\right)\\ y=a\sin \left(\frac{60\pi }{15}\right)\\ y=a\sin \left(4\pi \right)\\ y=0\end{array}$

Thus, the value of $f\left(96\right)$ can be determined by the given information.

Therefore, the correct option is $\left(D\right)$ II and III only.