Question Video: Finding the Average Rate of Change of the Area of an Equilateral Triangle When Its Side Length Changes between Two Given Values Mathematics • Second Year of Secondary School

Video Transcript

A lamina in the shape of an equilateral triangle expands whilst maintaining its shape. Find the average rate of change of its area when its sides change from 12 centimetres to 14 centimetres.

In this question, we’re looking to find the average rate of change of the area of the equilateral triangle. Now, for a function 𝑓 of 𝑥 that varies from 𝑥 equals 𝑎 to 𝑥 equals 𝑎 plus ℎ, the average rate of change is given by 𝑓 of 𝑎 plus ℎ minus 𝑓 of 𝑎 over ℎ. But what is 𝑓 of 𝑥 here? Well, remember, we’re interested in the rate of change of the area. So, we need a function that describes the area of our triangle. So, let’s sketch the triangle out. We can define the side length to be 𝑥 or 𝑥 centimetres. This is our variable. We know the triangle is equilateral, so its interior angles are all 60 degrees. And then, we can use the formula the area of a triangle is a half 𝑎𝑏 sin 𝐶. And then, in this case, the area function will be a half times 𝑥 times 𝑥 times sin 60.

Well, we know that sin of 60 degrees is equal to the square root of three over two. So, this becomes the square root of three over four times 𝑥 squared. We’re told that the side length changes from 12 centimetres to 14 centimetres. So, we let 𝑎 be equal to 12, and then ℎ is the amount 𝑥 varies by; it’s 14 minus 12, which is equal to two. And so, now we can substitute everything we have into our formula for the rate of change. It’s 𝐴 of ℎ, so here that’s 𝐴 of two, and of course this is going to be equal to 𝑓 of 12 plus two minus 𝑓 of 12 all over two. We simplify 𝑓 of 12 plus two to 𝑓 of 14.

And now we need to work out 𝑓 of 14 minus 𝑓 of 12. It’s the square root of three over four times 14 squared minus the square root of three over four times 12 squared. Those values are obtained simply by substituting 𝑥 equals 14 and 𝑥 equals 12 into our function. We factor root three over four and then divide that by two to get root three over eight. 14 squared is 196, and 12 squared is 144. And so, this becomes root three over eight times 52. And then, we simplify by dividing through by four to give us 13 root three over two. And so, the average rate of change of its area is 13 root three over two. And we might say that’s 13 root three over two centimetres squared per centimetre.