### 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, weβre looking
to find the average rate of change of the area of the equilateral triangle. Now, for a function π of π₯
that varies from π₯ equals π to π₯ equals π plus β, the average rate of change
is given by π of π plus β minus π of π over β. But what is π of π₯ here? Well, remember, weβre
interested in the rate of change of the area. So, we need a function that
describes the area of our triangle. So, letβs sketch the triangle
out. We can define the side length
to be π₯ or π₯ 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 ππ sin πΆ. And then, in this case, the
area function will be a half times π₯ times π₯ 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 π₯ squared. Weβre told that the side length
changes from 12 centimetres to 14 centimetres. So, we let π be equal to 12,
and then β is the amount π₯ varies by; itβs 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. Itβs π΄ of β, so here thatβs π΄
of two, and of course this is going to be equal to π of 12 plus two minus π of
12 all over two. We simplify π of 12 plus two
to π of 14.

And now we need to work out π
of 14 minus π of 12. Itβs 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 π₯ equals 14 and π₯ 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 thatβs 13 root
three over two centimetres squared per centimetre.