A public swimming pool is 30 m long and 5 m wide. Its depth is 3 m at the deep end and 1 m a

step 1 Let's start by drawing a diagram of the pool. So at the shallow end, it is one meter. And then a

A public swimming pool is 30 m long and 5 m wide. Its...

A public swimming pool is $30 \mathrm{~m}$ long and $5 \mathrm{~m}$ wide. Its depth is $3 \mathrm{~m}$ at the deep end and $1 \mathrm{~m}$ at the shallow end. If water is pumped into the pool at the rate of $15 \mathrm{~m}^{3} / \mathrm{min}$, how fast is the water level rising when it is $1 \mathrm{~m}$ deep at the deep end? Use Figure 4 as a guide.

Key Concepts

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to time by relating it to other quantities whose rates of change are known. In these problems, a formula that connects multiple variables is set up, and then each variable is differentiated with respect to time to establish a relationship between their rates of change.

Similar Triangles

When dealing with geometrical figures that change proportionally, such as a sloped surface, similar triangles are used to establish relationships between corresponding dimensions. In a variable-depth container, similar triangles help relate the geometry of the container to the water level, allowing one to express one dimension in terms of another.

Volume as a Function of Height

Calculating the volume of water in a container whose shape changes with the water level often requires expressing the volume as a function of the water height. This function is crucial as it links the physical dimensions of the container with the changing water depth, and it becomes the basis for differentiating with respect to time to find rates of change.

Chain Rule

The chain rule is a fundamental differentiation technique used to handle composite functions. In the context of related rates, it enables the differentiation of the volume function with respect to time by connecting the derivative of volume with respect to height to the derivative of height with respect to time, which is the quantity of interest.

Transcript

00:01 Let's start by drawing a diagram of the pool.

00:03 So at the shallow end, it is one meter.

00:06 And then at the deep end, it is three meters.

00:10 Okay, so we're assuming that it goes down at a constant, right? okay, so that's the top.

00:18 And then we'll just draw it in 3d, something that looks like this.

00:27 Okay, so this is our pool.

00:29 It is 30 meters long, so this is 30, 5 meters wide.

00:34 So here is five.

00:36 This here is one and this here is three.

00:43 So if the water is being pumped into the pool at a rate of 15 meters cubed per minute, so if we're pumping water in, then that's the volume that's changing.

00:54 Okay, so that means dvd t is equal to 15 meters cubed per minute.

01:03 So how fast is the water level rising when it is one meter.

01:08 Deep at the deep end.

01:09 Okay, so basically we're saying when the water is at this height, when it's one meter, okay, something like this.

01:29 That looks a little bit off.

01:30 I mean, it should be like this.

01:32 Yeah, there we go.

01:36 Let's draw that.

01:40 Something like that.

01:41 Okay.

01:43 So now we can see that this volume here, well, that's a triangular prism, right? because this here's a triangle and this here's a rectangle.

01:56 So if we're talking about the volume of the water here, it's going to be half of this side times this height, which will give us the area of the triangle, and then we multiply it by the width of the pool, which is five meters.

02:14 Okay, so let's go ahead and write down the formula for the for a triangular prism here.

02:19 So volume v is going to be one half of.

02:26 So let's call this water level here h.

02:32 So that's half of the height times this side.

02:37 Let's call that x times the depth, the width of the pool, which is a constant at 5.

02:46 So h and x are not constant.

02:50 So as you can see, if the water level is lower, i'm going to have a smaller h and a smaller x.

02:55 If the water level is higher, it's going to be a bigger h and a bigger x.

02:59 But this 5 here is always 5 because that's the width of the pool.

03:04 Okay, so that means my volume is going to be 5 over 2 times h times x...