Question

A searchlight revolving once a minute is located at a distance of 1/4 mile from a straight beach. How fast is the light moving along the beach when the beam makes an angle of 60 degrees with the shore line?

Answer 1

Ask a question, do not simply post a hw problem? This is a community for learning, not passing work onto others for ease.

Answer 2

will i use tangent?

Answer 3

\[\tan(\theta) = b/a , \theta=\pi/3, a=1/4mi\]

Answer 4

\[a=1/12\pi\] right?

Answer 5

Ultimately we need an equation that relates a to b, because we are trying to find the derivative of b with respect to time. We know that \[\tan(\theta)=\frac{b}{a}\]\[\frac{\sin(\theta)}{\cos(\theta)}=\frac{b}{a}\]\[asin(\theta)=bcos(\theta)\]\[b = a\frac{sin(\theta)}{cos(\theta)}=atan(\theta)\] Since b is in terms of a, taking the derivative yields an expression involving \[\frac{db}{dt},\frac{da}{dt},\frac{d\theta}{dt}\] since a is constant, its derivative is 0, leaving the rate of the angle which is a full rotation(2pi)/min, and the rate of b, the answer.