A rectangle with base b and height h has an isosceles...

A rectangle with base b and height h has an isosceles triangle that has base b and height h on the top of the rectangle as shown below. The b is decreasing at a rate of 3 cm per minute, while h is increasing at a rate of 5 cm per minute. What is the rate of change of the area when the base is 73 and the height is 37? Is the area increasing or decreasing at the same time and how do you know which it is?

Transcript

00:01 All right, so we have a rectangle with base b and height h with this isosceles triangle sitting on top of it.

00:08 So a is a isosceles triangle.

00:09 So we can say this base is also b.

00:12 It's isosceles, so these sides are equal.

00:14 So as we're looking at this figure, we are given, what do we know? well, we are given a couple rates.

00:20 B is decreasing at a rate.

00:23 So as soon as we see it's a rate, that tells me i'm going to use my derivative notation.

00:27 So db, the rate of the base per unit of time, dbt is decreasing.

00:37 How am i going to show that a rate is decreasing? i would say that's a negative 3 centimeters per minute.

00:44 So that indicates that this rate, the value of b is getting shorter.

00:50 So the length of b is getting less every minute.

00:53 Now what else do i know? h, the height here, is increasing at a.

00:57 Rate.

00:58 So i'm not going to say h equals, i'm going to say again, the rate of h.

01:01 So d .h.

01:03 D .t.

01:05 Now, the height is increasing.

01:06 So i'm going to say that's a positive five centimeters per minute.

01:12 Now, as i'm looking at this, i want, what do i want? i want.

01:16 I don't know the rate of change of the area.

01:19 So again, i need a rate.

01:21 So it's going to be the difference of the difference of a over d t.

01:26 So d -a -d -t, that represents the rate of change of the area.

01:30 Per unit time.

01:32 Now, depending on what time it is, that area might be changing at different rates.

01:37 So that's why they say, what is the rate of change at this instant, at the moment that somebody uses a vertical line to say at the instant that b equals 73 and h equals 37.

01:52 So we're going to see that that comes into play in a little bit.

01:54 So this is what i don't know.

01:56 That's what i want.

01:57 So i have what i know, i have what i want to know.

02:00 I have these h's and these bs and i'm looking at a specific instance and now to need an equation that relates these.

02:07 So i'm dealing with areas.

02:09 There's my big hint that i need an area equation.

02:13 So i need to know what is the area of this figure.

02:16 Well, i see a rectangle and a triangle.

02:18 The area of a rectangle is just base times height.

02:22 And i'm going to add that to the area of this triangle, which is one -half base.

02:27 Times height.