Swimming Pools and Related Rates along with Implicit Differentiation

• Salazar
In summary, the problem involves a swimming pool with dimensions of 40 feet by 20 feet and varying depths from 4 feet to 9 feet. Water is being pumped into the pool at a rate of 10 cubic feet per minute. The question asks for the rate at which the water level is rising when the depth is 3 feet at both the shallow and deep end. Using the equation V = L*W*H and implicit differentiation, the rate at which the water level is rising is found to be 1/80 feet per minute. To find the rate at the deep end, the volume of the lower half of the pool can be calculated as a triangle's area times the width of the pool.
Salazar

Homework Statement

A swimming pool is 40 feet long, 20 feet wide, 4 feet deep at the shallow end, and 9 feet deep at the deep end. Water is being pumped into the pool at 10 cubic feet per minute.
a. When the water is 3 feet deep at the shallow end, at what rate is the water level rising? b. When the water is 3 feet deep at the deep end, at what rate is the water level rising?

None Given.

The Attempt at a Solution

I had [dV/dt] and the volume for part a is V = L*W*H and I had L and W, which were 40' and 20' respectively. So V = 800H and I used implicit differentiation to get [dV/dt] = 800[dH/dt], so [dH/dt] = 1/80 feet per minute.

For part be it seems straightforward, but I do not know how to get the length, since it has change. Picture of the pool is uploaded.

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Counting h from the top of the pool you have then
V = LWH for 0 < H < 4 ft and
V = something else for 4 ft < H < 9ft.

I understand that, but what is such something else for L, length?

The volume of the lower half is a triangles area times the width of the pool; you can give the triangle for example by one of the angles and the height, you do not need to use length explicitly at all.

What is the relationship between swimming pools and related rates?

The concept of related rates in calculus is often applied to study the changes in the volume, depth, and other related measurements of a swimming pool. This relationship helps in understanding how the variables of a swimming pool can affect each other.

How is implicit differentiation used in the context of swimming pools?

Implicit differentiation is a technique used to find the derivative of a function that is not explicitly given. In the case of swimming pools, implicit differentiation can be applied to find the rate of change of a variable that is not explicitly stated, such as the rate of change of the water level in a pool.

What are some real-life applications of related rates and implicit differentiation in swimming pools?

Related rates and implicit differentiation are commonly used in the design and maintenance of swimming pools. They can help in determining the optimal size and shape of a pool, as well as predicting the rate at which the water level will change due to factors such as evaporation and drainage.

What are some challenges in applying related rates and implicit differentiation to swimming pools?

One of the main challenges in using related rates and implicit differentiation for swimming pools is accurately determining all the variables that affect the pool's measurements. This can include factors such as water temperature, flow rate, and the shape of the pool, which can be difficult to measure and account for.

How do related rates and implicit differentiation contribute to our understanding of swimming pools?

By using related rates and implicit differentiation, we can gain a deeper understanding of the complex relationships between the various variables of a swimming pool. This can help in making informed decisions about pool design, maintenance, and safety, and ultimately enhance the overall experience of using a swimming pool.

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