Filling a pool A swimming pool is 50 m long and 20 m wide....

Filling a pool A swimming pool is $50 \mathrm{m}$ long and $20 \mathrm{m}$ wide. Its depth decreases linearly along the length from $3 \mathrm{m}$ to $1 \mathrm{m}$ (see figure). It is initially empty and is filled at a rate of $1 \mathrm{m}^{3} / \mathrm{min}$. How fast is the water level rising 250 min after the filling begins? How long will it take to fill the pool?

Key Concepts

Integration for Volume Calculation

Integration is used to compute volumes of objects with varying cross-sectional areas. When the shape of an object changes along a certain dimension, integration sums up the contributions of infinitesimal slices (or elements) across that dimension to determine the overall volume. This is especially important in cases where the cross-sectional area is not constant.

Geometric Similarity

Geometric similarity refers to the property of similar figures where corresponding dimensions maintain a constant ratio. This concept is essential when dealing with objects that have dimensions which change linearly, as it allows one to establish relationships between different parts of a shape, such as the varying depths of a tank or pool.

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to time by relating it to another quantity whose rate of change is known. This concept is fundamental in calculus when two or more variables are interconnected through a formula, and the chain rule is used to connect their derivatives with respect to time.

Chain Rule

The chain rule is a principle in calculus that is used to differentiate composite functions. In the context of related rates, it allows one to differentiate an equation that relates various quantities, thereby linking the rate of change of one variable with the rate of change of another variable that indirectly depends on time.

Transcript

00:01 Here we have a swimming pool that is being filled with water.

00:04 The dimensions is 50 meters long, 20 meters wide.

00:08 It's three meters high in the deep end of the pool and one meter on the shallow end.

00:15 Okay.

00:15 So we're being told that the pool is being filled at a rate of one meter cube per minute.

00:25 What we're trying to find is how fast the height is rising of the water level after 200, 50 minutes.

00:37 All right? so first thing i want to know is how far up the pool or what would be the height of the water in the pool after 250 minutes.

00:48 So i want to consider whether or not it has reached the two meter mark yet.

00:56 Okay.

00:57 So do i need to consider the rectangular prism part of the pool or can i just consider the triangular prism.

01:06 So let's see what the volume is of the triangular prism.

01:11 So it's one half height times length times width.

01:15 So i have one half.

01:17 My height would be two.

01:19 So this is going to be two meters.

01:25 My length is going to be 50.

01:29 And my width is going to be 20.

01:32 So if i multiply that all out, i end up with 1 ,000.

01:36 Okay, so if i fill the pool up to the two meter mark, okay, i'm going to have 1 ,000 cubic meters of water.

01:47 Well, if we're filling the pool at a rate of one cubic meter per minute, after 250 minutes, we're going to have 250 cubic meters of water.

01:58 So i want to know how high is that water after 250 minutes or when the volume is 250 cubic meters.

02:08 So i'm going to have one half times height.

02:12 That's what i'm trying to find, times length.

02:15 Well, do we know what our length is going to be? so let's just say it's right about here.

02:21 This is how high my water level is.

02:23 Okay? i'm just going to kind of shade that in a little bit.

02:28 If we think about this a little, we know that looking here, we have two similar triangles.

02:37 Okay...

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