Video Transcript
A triangular lamina with a base twice its height expands while maintaining its shape. Find the average rate of change in its area when its height changes from 14 centimetres to 23 centimetres.
The question wants us to find the average rate of change in the area of a triangular lamina. We’re told this triangular lamina has a base which is twice its height and it expands while maintaining its shape. We recall, to find the average rate of change of a function 𝑓 of 𝑥 from 𝑥 is equal to 𝑎 to 𝑥 is equal to 𝑏, we use that 𝑓 average is equal to 𝑓 evaluated at 𝑏 minus 𝑓 evaluated at 𝑎 all divided by 𝑏 minus 𝑎. Since the question wants us to find the average rate of change of the area of our triangular lamina when the height is changing from 14 to 23, we want the area in terms of the height ℎ. And in our average rate of change formula, we want 𝑎 equal to 14, 𝑏 equal to 23, and 𝑓 equal to the area in terms of the height ℎ.
If we were to sketch our triangular lamina with a height of ℎ, then we know the base would be two ℎ. In fact, since this triangle is expanding while maintaining its shape, this will always be the case. And we recall we can calculate the area of any triangle as a half times the height of the triangle times the length of the base. So we can calculate the area of this triangle as a half multiplied by the height ℎ multiplied by the length of the base two ℎ, which, of course, simplifies to give us ℎ squared. So we’ve found the area of our triangle in terms of ℎ. It’s just equal to ℎ squared.
We can now use our average rates of change formula. We have the average rate of change in the area of our triangle as the height changes from 23 centimetres to 14 centimetres, which we will call 𝐴 average, is equal to 𝐴 evaluated at 23 minus 𝐴 evaluated at 14 divided by 23 minus 14. Using the fact that the area of our triangular lamina with the height of ℎ is equal to ℎ squared, we can evaluate this to give us 23 squared minus 14 squared divided by nine. We can then evaluate this to be 37.
We could leave our answer like this. However, we’re told that the height is measured in centimetres, which means the area is measured in centimetres squared. And since our height is measured in centimetres, we’re measuring the rate of change in the area as we change the height, we can have the units of centimetres squared per centimetre. Therefore, we’ve shown if you have a triangular lamina with a base twice its height which expands while maintaining its shape, then the average rate of change in its area when the height changes from 14 centimetres to 23 centimetres is 37 centimetres squared per centimetre.
