Section II Part B — Full Practice Exam 3

Questions in this quiz (2)

1. A rotating spotlight is mounted on a tower and sweeps around at a constant speed. The light completes one full rotation every $$40$$ seconds. The light is mounted $$60$$ feet above the ground, and the beam has a length of $$20$$ feet. The height of the tip of the beam above the ground varies periodically. At time $$t=0$$, the beam is pointing straight upward.(A) This situation can be modeled by a sinusoidal function of the form $$h(t)=a\cos(bt)+d,$$ where $$t$$ is time in seconds and $$h(t)$$ is the height of the tip of the beam in feet. Find the function $$h(t).$$(B) On the interval $$0$$<$$t$$<$$40,$$ when is the height of the tip equal to $$60+10\sqrt{2}$$ feet? Give your answers in exact form.(C) i. Determine all times on the interval $$0\le t$$<$$40$$ at which the graph of $$h(t)$$ has a point of inflection.ii. Explain the meaning of these points in the context of the problem, in terms of how the height is changing and how the rate of change is changing.(D) Find the average rate of change of the height from $$t=10$$ to $$t=30,$$ and interpret its meaning in context.

2. The functions $$g$$ and $$h$$ are given by$$g(x)=3\log a-2\log b+\frac{1}{2}\log a+\log_{10}c$$$$h(x)=\sec^2 x\tan x-\tan x$$(A) i. Rewrite $$g(x)$$ as a single logarithm with base $$10$$ without negative exponents. Your answer should be in the form $$\log(\text{expression})$$ii. Rewrite $$h(x)$$ as an expression in which $$\tan x$$ appears once and no other trigonometric functions are involved.(B) The functions $$j$$ and $$k$$ are given by$$j(x)=2\csc^2 x-3$$$$k(x)=5e^{4-2x}+1$$i. Solve $$j(x)=1$$ for all values of $$x$$ in the interval $$[0,2\pi)$$.ii. Solve $$k(x)=6$$ for all real values of $$x$$. Express your answer exactly.(C) Two functions $$a$$ and $$b$$ are given by$$a(x)=4-2\sec(2x)$$ $$b(x)=1+\sec(2x)$$Find all values of $$x$$ in the interval $$[0,2\pi)$$ for which $$a(x)=b(x)$$.