Calculus related rates problem

Question

A 200 foot tree is falling in the forest; the sun is directly overhead. At the moment when the tree makes an angle of 30? with the horizontal, its shadow is lengthening at the rate of 50 feet/sec. How fast is the angle changing at that moment?

Answer 1

Hi Valentina.

 

This is a problem that's best solved by drawing a right triangle at the moment the problem considers. In this case, it'll be a right triangle with a hypotenuse of 200 ft and an angle with the ground of 30 degrees.

 

Let's consider the triangle ABC. Let A be the top of the tree, B be the base of the tree, and C be the tip of the tree's shadow on the floor.

 

The length AB, from the problem, is given as 200ft.

The length BC, from the base of the tree to the tip of the tree's shadow, is 200ft*cos(30) = 100*sqrt(3)ft

The length CA, from the tip of the tree's shadow to the tip of the tree, is 200ft*sin(30) = 100 ft

 

Here's where the question gets fun.

 

We want to know the rate of change of the angle when the shadow's length is increasing by 50ft/sec. We need to find a relationship between the angle and the known lengths of the triangle. Let's use cosine.

 

cos(theta) = adj/hyp

 

The adjacent side will be length of the shadow, BC.

The hypotenuse will be the tree itself, AB.

 

Let's take the implicit derivative of the entire equation:

 

d/dt(cos(theta) = BC/AB)

 

I never remember the quotient rule so I'm going to rewrite it as:

 

d/dt(cos(theta) = (BC)*(AB)^(-1))

 

Don't forget your chain rule!!!

 

-sin(theta)*dtheta/dt = dBC/dt*(AB^(-1)) + BC*(-1*(AB)^(-2))*dAB/dt

 

dBC/dt is the rate of change of the length of the shadow. We know from the problem that it's 50 ft/ sec

 

dAB/dt is the rate of change of the length of the tree. As far as I know, the tree isn't growing as it's falling. So that's going to be 0 ft/ sec. Thank goodness, too... That's an ugly term.

 

dtheta/dt is what we're after. That's the rate of change of the angle. Since the tree is falling, it should be negative, since the angle is getting smaller.

 

theta, from the problem, is 30 degrees.

 

Let's plug in some numbers:

 

-sin(30)*dtheta/dt = (50)*(1/200) + 0

 

-1/2*dtheta/dt = 1/4

 

dtheta/dt = -1/2 deg/ sec

 

The negative answer checks out.

 

If we had used sin(theta) instead of cosine, we would have wound up with +cos(theta)*dtheta/dt as the derivative. But that's okay, because had we chosen sine we would have used the opposite length, CA, which is obviously decreasing (therefore dCA/dt is negative) since the tree is falling. We just didn't know what dCA/dt was, so we went with cos(theta) instead. Easier to evaluate that way.

 

Hope this helps.