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A swimming pool is 20 $\mathrm{ft}$ wide, 40 $\mathrm{ft}$ long, 3 $\mathrm{ft}$ deep at theshallow end, and 9 ft deep at its deepest point. A cross-section is shown in the figure. If the pool is being filled at a rate of 0.8 $\mathrm{ft}^{3} / \mathrm{min}$ , how fast is the water level rising whenthe depth at the deepest point is 5 $\mathrm{ft}$ ?

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Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 9

Related Rates

Derivatives

Differentiation

Leo L.

December 10, 2021

Could someone else please record the explanation step by step

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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A swimming pool is $ 20 ft…

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A swimming pool is 20 ft w…

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A swimming pool is $20 \ma…

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swimming pool is 20 ft wid…

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A rectangular swimming poo…

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A public swimming pool is …

to figure out how fast the water level is rising. The first thing you want to do is write out our differential equation. You know, the volume of the swimming pool is the area of the cross section eight times the width of the pool W therefore we know differentiating we have DV over d t is w times d A over DT. This is our equation. Therefore, we now know that a is H and B is a h over three because remember, we had 16. Overby is six over h two. Just simplifying this therefore we know now, but we can write out our equation as 11 h squared over six plus 12 h which means D A over GT's. Now this is differentiating. There's gonna be 11 h plus 12 in the 11 is divided by three Then this is all times de age divide by DT. So now, given what we have in the problem, we can plug in 0.8 feet. You per minute is our devi over. DT is 20 times 11 times five over three because 12 times d h over DT, which gives us d h over DT is 1.32 times 10 to the negative three feet per minute. And because the answer is positive, we know the rate is increasing, therefore the water level's increasing.

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