Question

A lighthouse is 200ft from a long straight wall on the beach. The light in the lighthouse rotates through on complete rotation once every 4 sec. Find the equation that gives the distance d in terms of time t. Then, to the nearest tenth, find d when t is 1/3 sec. (It has something to do with velocities).

Answer 1

You want to find distance in terms of the angle first,
Then find the angle in terms of t

Answer 2

A lighthouse is 200ft from a long straight wall on the beach.

This tells you: \[
d\cos\theta = 200 \implies d = 200\cos(\theta)
\]

Answer 3

Whoops, I mean \[
d = \frac{200}{\cos(\theta)}
\]

Answer 4

Now you need to find \(\theta\) in terms ot \(t\).

Answer 5

"The light in the lighthouse rotates through on complete rotation once every 4 sec."

This means when \(\Delta t=4\), then \(\Delta \theta = 2\pi\)

Answer 6

Would it be \[D=200\tan \frac{ \pi }{ 2}t \] though?



The drawing looks like this

Answer 7

Ohhh, so not the distance from the lighthouse to the wall, but just the distance from the center of the wall?

Answer 8

In that case: \[
\tan(\theta) = \frac{d}{200} \implies d = 200\tan(\theta)
\]

Answer 9

Right next to the 'd' on the left is a right angle by the way, sorry. If that helps.

Answer 10

We know that \(\theta = ct\) and \(\theta(4)=2\pi = c(4)\) which means \(c = \pi/2\) and \(\theta(t) = \pi t/2\)

Answer 11

Meaning: \(d = 200\tan(\pi t/2)\)

Answer 12

Would t be on the outside?

Answer 13

No, it's in the function. Why would it be on the outside? \(\theta\) was on in inside.