WEBVTT
00:00:02.040 --> 00:00:09.360
A triangular lamina with a base twice its height expands while maintaining its shape.
00:00:10.200 --> 00:00:20.280
Find the average rate of change in its area when its height changes from 14 centimetres to 23 centimetres.
00:00:21.520 --> 00:00:27.040
The question wants us to find the average rate of change in the area of a triangular lamina.
00:00:27.800 --> 00:00:37.720
Weβre told this triangular lamina has a base which is twice its height and it expands while maintaining its shape.
00:00:38.840 --> 00:00:57.440
We recall, to find the average rate of change of a function π of π₯ from π₯ is equal to π to π₯ is equal to π, we use that π average is equal to π evaluated at π minus π evaluated at π all divided by π minus π.
00:00:58.520 --> 00:01:15.040
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 β.
00:01:15.840 --> 00:01:29.080
And in our average rate of change formula, we want π equal to 14, π equal to 23, and π equal to the area in terms of the height β.
00:01:30.480 --> 00:01:39.240
If we were to sketch our triangular lamina with a height of β, then we know the base would be two β.
00:01:39.960 --> 00:01:48.360
In fact, since this triangle is expanding while maintaining its shape, this will always be the case.
00:01:49.520 --> 00:01:55.680
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.
00:01:56.680 --> 00:02:08.600
So we can calculate the area of this triangle as a half multiplied by the height β multiplied by the length of the base two β, which, of course, simplifies to give us β squared.
00:02:09.600 --> 00:02:14.120
So weβve found the area of our triangle in terms of β.
00:02:14.600 --> 00:02:16.680
Itβs just equal to β squared.
00:02:18.040 --> 00:02:21.400
We can now use our average rates of change formula.
00:02:22.320 --> 00:02:44.800
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 π΄ average, is equal to π΄ evaluated at 23 minus π΄ evaluated at 14 divided by 23 minus 14.
00:02:45.840 --> 00:03:03.280
Using the fact that the area of our triangular lamina with the height of β is equal to β squared, we can evaluate this to give us 23 squared minus 14 squared divided by nine.
00:03:04.040 --> 00:03:06.640
We can then evaluate this to be 37.
00:03:07.720 --> 00:03:10.080
We could leave our answer like this.
00:03:10.280 --> 00:03:20.040
However, weβre told that the height is measured in centimetres, which means the area is measured in centimetres squared.
00:03:21.040 --> 00:03:33.120
And since our height is measured in centimetres, weβre measuring the rate of change in the area as we change the height, we can have the units of centimetres squared per centimetre.
00:03:34.080 --> 00:04:01.320
Therefore, weβve 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.