# Related Rates - Cylindrical Pools

• DrummingAtom
In summary, the given information in the problem does not match up and the math does not work out for the smaller pool. It is unclear how fast the water level is rising in the larger pool as the rate of change of water level in the smaller pool does not match the given information.
DrummingAtom

## Homework Statement

2 Cylindrical pools are filled simultaneously at the same rate, 1m3/min. The smaller pool has radius 5m and the water level rises at a rate of 0.5m/min. The larger pool has a radius 8m. How fast is the water level rising in the larger pool?

V = pi(r2)h

## The Attempt at a Solution

I took the derivative of V with respect to h and got:

dV/dt = pi(r2)(dh/dt)

Which dV/dt = 1 and then I just solve for dh/dt. But, if I'm doing this right, the smaller pool should yield the same answer and it doesn't. Because for the smaller pool I would have:

1/pi(52) = dh/dt and this doesn't equal 0.5m3/min.

Thanks for any help.

DrummingAtom said:

## Homework Statement

2 Cylindrical pools are filled simultaneously at the same rate, 1m3/min. The smaller pool has radius 5m and the water level rises at a rate of 0.5m/min. The larger pool has a radius 8m. How fast is the water level rising in the larger pool?

V = pi(r2)h

## The Attempt at a Solution

I took the derivative of V with respect to h and got:

dV/dt = pi(r2)(dh/dt)

Which dV/dt = 1 and then I just solve for dh/dt. But, if I'm doing this right, the smaller pool should yield the same answer and it doesn't. Because for the smaller pool I would have:

1/pi(52) = dh/dt and this doesn't equal 0.5m3/min.

Thanks for any help.

I partly agree with you. As stated, the problem doesn't make sense as far as the small pool is concerned. If the pool is being filled at a rate of 1 m^3/min, the rate of change of the water height is 1/(25pi) m/min, which is at odds with the given information.

Where you say that the water in both pools should rise at the same rate, I disagree. Given that both pools are being filled at a rate of 1 m^3/min, the water level in the smaller pool will rise more quickly than it will in the larger pool.

## What is a cylindrical pool?

A cylindrical pool is a type of swimming pool that has a circular cross section and a constant depth. It is often used in residential or commercial settings and can vary in size and volume.

## How are related rates used in cylindrical pools?

Related rates are used in cylindrical pools to calculate the rate at which certain variables, such as the water level or volume, are changing over time. This can help pool owners or designers monitor and maintain the pool's water level and ensure it is safe for use.

## What are some common related rates problems related to cylindrical pools?

Some common related rates problems related to cylindrical pools include calculating the rate of change of the pool's water level when water is being added or drained, or determining the rate at which the pool's volume is increasing or decreasing due to evaporation or other factors.

## What variables are involved in related rates problems for cylindrical pools?

The variables involved in related rates problems for cylindrical pools may include the pool's radius, height, volume, water level, and the rates at which these variables are changing over time. Other factors such as the flow rate of water into or out of the pool may also be considered.

## How can related rates be applied to real-life situations involving cylindrical pools?

Related rates can be applied to real-life situations involving cylindrical pools by helping pool owners or designers monitor and maintain the pool's water level and volume, ensuring it is safe for use. They can also be used to optimize the pool's filling or draining process, or to calculate the impact of environmental factors on the pool's water level and volume.

### Similar threads

• Calculus and Beyond Homework Help
Replies
11
Views
2K
• Calculus and Beyond Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
4
Views
3K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
3K
• Calculus and Beyond Homework Help
Replies
2
Views
3K
• Calculus and Beyond Homework Help
Replies
3
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
3K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
5K