You have a string with a length of 24 ft closed up in a loop.

Question

You make a square with part of it, and an equilateral triangle with the other part. Keeping these two basic shapes, we decrease the side of the square at a rate of 4 ft/s.

At what rate are the sides of the triangle increasing?

What rate is the area of the triangle increasing when the sides of it are at 4 ft. each?

What are the dimensions of the equilateral triangle and square constructed with the string that provide us the minimum and maximum total areas?

Answer 1

Hi Gracie, You need to know (or develop) a formula for the area of an equilateral triangle when you know the length of a side. Perhaps you already know A=s²√3/4 where s is the length of one side (of the equilateral triangle).

Let S = length of side of square, T = length of side of triangle.

For first question:

4S + 3T = 24 defines the relationship. Take the derivative with respect to "t" where t is time in seconds.

4 dS/dt + 3 dT/dt = 0. Substitute what you know (dS/dt = -4 ft/sec) and solve for what you are trying to find (dTdt).

How about the area of the triangle (2nd question).

A = T²√3/4.

So, dA/dt = 2T √3/4 dT/dt = T√3/2 dT/dt.

Substitute what you know from the answer to part a (dT/dt) and simplify to get dA/dt.

There are several ways to approach part c. Here is one.

Imagine the rope 24 feel long stretched out horizontally. Imagine x representing the coordinate of a point moving from 0 towards 24. That point divides the string into two parts one with length x and the other with length 24-x. Let the string with length x be shaped into a square and the 24-x into an equilateral triangle.

Then the side of the square will be (x/4) and a side of the triangle will be (24-x)/3.

Now we create a formula for the sum of the areas of the two shapes.

A = (x/4)² + [(24-x)/3]²√3/4

Find the derivative of A with respect to x (dA/dx) or A'. Set that equal to zero and solve for x to get the critical number(s). Convince yourself that this gives a minimum value for the total area. But the endpoints of the interval must also be considered. When x = 0 what is A? When x = 24 what is A? One of the three values will give you a max total area and one will give you a min total area.

Finding the critical number gets a bit messy and leads to an expression involving a not so pretty looking radical expression. Something like P√3/(Q +R√3)--where P,Q,R are integers. Remember to answer the question correctly (asking for dimensions of the shapes), you have to solve for x/4 (side of the square) and (24-x)/3 (side of the triangle.

If you need additional clarification on any of this let me know.