# What is the Derivative of a Swimming Pool with a Sloping Bottom?

• andrew.c
In summary, the water level is rising at a rate of 1/150 m/s when the water is 3m deep at the deep end.
andrew.c

## Homework Statement

A rectandular swimming pool is 10m wide and 20m long. The bottom of the pools is a sloping plane with the depth of the pool varying along the length of the pool from 1m at the shallow end to 5m at the deep end. Water is being pumped into the pool at the rate of $$30 m^3 / min$$. Shat that the total volume of water in the pook when the depth of thewater at the deep end is hm (where $$0 \leq h \leq 4$$) satisfies

$$V = 25h^2$$

Hence find the rate at which the water level is rising when the water is 3m deep at the deep end.

n/a

## The Attempt at a Solution

Equation to describe volume at h...

When 4m full, the 'length' filled by the water is 20m, therefore..

\begin{align*} V &= \frac{1}{2}LHW\\ &=\frac{1}{2}(5h)(h)(10)\\ &=(2.5h^2)(10)\\ &=25h^2\\ \end{align*}

Then, i tried to differentiate this expression to find h' (difference in height), but this came to 0.

Ideas?

Say what? If $V(h)=25h^2$ then $\frac{dV}{dh}=50h$. So $\frac{dV}{dt}=?$

Is the final answer 1/150 m/s?

I'm a newb but I'll try to help -_-.

Try differentiating $$V=25h^2$$ with respect to $$t$$. Since you know $$dV/dt$$ and $$h$$, you can figure out the answer.

Last edited:
$$V=25h^2$$

\begin{align*} differentiated &=\frac{dV}{dt} (25h^2)\\ &=\frac{dV}{dh}(25h^2) \frac{dh}{dt}\\ &=(50h) \frac{dh}{dt}\\ \frac{dh}{dt} &= 50h\\ sub. in value for h to find dh/dt\\ \frac{dt}{dh} &= 50(3)\\ \frac{dt}{dh} &= 150\\ \frac{dh}{dt} &= 1/150\\ \end{align*}

Does that look even a little bit right to anyone?

Not right.

For one thing, it looks like you are writing $$\frac{dV}{dt}(25h^2)$$ when you mean $$\frac{d}{dt}(25h^2)$$.

Here is better notation.

\begin{align*} \frac{dV}{dt}&=\frac{dV}{dh}\frac{dh}{dt}\\ &=\frac{d}{dh}(25h^2) \frac{dh}{dt} \end{align*}

What's next?

\begin{align*} &= 50h \frac{dh}{dt} \\ \end{align*}

Then move dh/dt over to the other side, where it is 'dividing', i.e $$\frac{1}{\frac{dh}{dt}} = 50h$$

so

$$\frac{dt}{dh} = 50h$$

but we want dh/dt, so needs to be rearranged,

$$\frac{dh}{dt} = \frac{1}{50h}$$

then I subbed in 3 for H to get 1/150

Yes, now write it as part of an equation. Then what?

I editted above post ^^ Accidently hit reply before I was finished.

No, in your first step, the other side is not "1." What is the other side?

dV/dt? and does that = 30?

Bingo

Thanks for all the help!

Ta muchly!
x

## 1. What are derivatives in relation to swimming pools?

Derivatives in relation to swimming pools are financial contracts that derive their value from underlying assets or financial instruments, such as a swimming pool. These contracts are used to manage risks associated with fluctuations in the value of the pool, such as changes in interest rates or foreign exchange rates.

## 2. How are derivatives used in the swimming pool industry?

Derivatives are used in the swimming pool industry to hedge against risks and protect against potential losses. For example, a company that manufactures and sells swimming pool equipment may use derivatives to protect against fluctuations in the price of raw materials or changes in interest rates that could impact their profitability.

## 3. What types of derivatives are commonly used in the swimming pool industry?

The most commonly used derivatives in the swimming pool industry are interest rate swaps, currency swaps, and options. Interest rate swaps help companies manage interest rate risks, while currency swaps help manage foreign exchange risks. Options provide the right, but not the obligation, to buy or sell an underlying asset at a specified price and date, allowing companies to manage various risks related to the swimming pool industry.

## 4. How do derivatives impact the swimming pool market?

Derivatives can have a significant impact on the swimming pool market. They can help companies manage risks, reduce volatility, and increase liquidity. However, if not used properly, derivatives can also contribute to market instability and potential financial crises.

## 5. What are the potential risks associated with using derivatives in the swimming pool industry?

There are several potential risks associated with using derivatives in the swimming pool industry. These include counterparty risk, market risk, liquidity risk, and credit risk. If not managed properly, these risks can lead to significant financial losses for companies.

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