Math CentralQuandaries & Queries


Question from Jeevitha, a student:

The side of an equilateral triangle decreases at the rate of 2 cm/s.
At what rate is the area decreasing when the area is 100cm^2?

Hi Jeevitha. The first step in a related rates question is to determine how the quantities are related.

You have the constant rate of change of the length of the sides of an equilateral triangle and are asked about the rate of change of the area. So the first question you should ask is "how is the side length of an
equilateral triangle related to its area?"

I'm going to leave that part to you. Let's say the answer is A = 4s2, where A = area and S = side length (it isn't, but I want to show you how to do the differentiation).

The second step is to determine the length when the area is 100. Just use the equation you wrote:

100 = 4s2

s2 = 25

s = 5.

The third step is differentiating both sides of the equation with respect to time. Pay close attention to the chain rule:

A = 4s2

dA/dt = d(4s2)/dt

dA/dt = 4 d(s2)/dt

dA/dt = 4 (2s) (ds/dt)

dA/dt = 8s (ds/dt)

Okay, that's the differentiation part of the solution. The question is asking for dA/dt (the rate of change of area with respect to time) and we know the rest. So the last step is plugging in the values we know:

dA/dt = 8(5) (2)

dA/dt = 80 cm2/s.

Now you try it after determining the REAL relationship between the side length and the area.

A review of the general approach:
With this kind of question, the first step is finding out the relationship between the two quantities (for us, area and length). The second step is finding out the value of the second quantity (length) when the first quantity is the value given in the question (area = 100). The third step is differentiating the relationship. The final step is substituting in the values you know and solving the problem.

Hope you find this helpful,
Stephen La Rocque.

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