Question

As a wave passes by, the height of the water is modeled by h(t)=−9cos(5π/9t), where h(t) is the height in feet above mean sea level. Find the height of the wave. (That is, find the vertical distance between the trough and the crest of the wave.)
a) 9

b)10π/9

c) 18

d) -9

e) 5π/9

Answer 1

Answer:

(c)  18

Step-by-step explanation:

To find the maximum height of this trigonometric function is considering some of its features; amplitude, period, and phase, thus;

$h(t)=Acos(B(t+C))+D$

"A" stands for the amplitude, i.e, from the maximum value to a node ( the crest). The period is given by $\frac{2\pi}{B}$. C stands for the phase (shift at x-axis) and D stands for the vertical shift at y-axis. Given the previous criteria, a question must be solved, what does "-A" stand for? It stands for the value from a node to  minimum value (the trough).

$h(t)=-9cos(\frac{5\pi}{9t})$

For the exercise, A = -9, i.e, the trough and $\frac{5\pi}{9}$ is nothing else the period. There is no information of vertical/horizontal shift, i.e., the function has an origin at x=0, and y=0.  Now, focusing on the meaning of "A", the amplitude,  +A and -A (positive A and negative A) represent how much the function increases or decreases, respectively. To find the vertical distance between the trough and the crest is nothing else the double either of the crest or trough. Such values, numerically speaking, are the same but with different sign. Analyzing each given answer

(a) 9 represents the positive amplitude from the node to the maximum -  the crest and does not represent the total height of the wave (this is the half of the wave)

(b)  $\frac{10\pi}{9}$ represent the almost "the half" of the crest and does not provide information about the height.

(c) 18.  Owing to the period of the function is - 9, it is only important to have its magnitude, i.e, the value 9. The cosine function is a symmetric function in the point of view of the value of the crest and trough. Thus, the crest has the value of 9 and the trough the value of 9, then it is possible to sum both values and the height of the wave is therefore 18 (Right answer!!)

(d)  -9 represents the trough and it is the half of the total height

(e) $\frac{5\pi}{9}$ represents almost "quarter" of the crest