The base area of a certain rectangular pool is 20 m
2
. The pool is being filled with water at a constant rate of 0.4 m
3
/s. How fast is the water level rising?

Hint: Follow these steps, based on the problem solving strategy in the textbook:

Step 1. Assign symbols to all variables involved in the problem. Draw a picture of the bathtub if you want.

For example, let:

ℎ
=
the water level in the pool. (Is this a function of time
𝑡
, or is it constant all the time?)

𝐴
=
the base area of the pool. (Is this a function of time
𝑡
, or is it constant all the time?)

𝑉
=
the amount of water in the pool (the volume). (Is this a function of time
𝑡
, or is it constant all the time?)

𝑡
=
time (in seconds)

What are the other units?

Step 2. State, in terms of the variables, the information that is given and the rate to be determined.

What is
𝐴
? What is the rate (derivative) that the question is asking you to find? How can you write "The pool is being filled with water at a constant rate of 0.4 m
3
/s" in terms of
ℎ
,
𝐴
and/or
𝑉
, or perhaps derivatives of these?

Step 3. Find an equation relating the variables introduced in step 1.

Find an equation that relates two or more of
ℎ
,
𝐴
and
𝑉
.

Step 4. Using the chain rule, differentiate both sides of the equation found in step 3 with respect to the independent variable. This new equation will relate the derivatives.

Differentiate both sides of your equation with respect to
𝑡
. This means all your derivatives should be of the form
𝑑
″
𝑠
𝑜
𝑚
𝑒
𝑡
ℎ
𝑖
𝑛
𝑔
″
𝑑
𝑡
.

Step 5. Substitute all known variables into the equation from step 4, then solve for the unknown rate of change.

Are there any known numbers that were given in the question? If so, you can insert those now. (If a certain quantity does not change over time i.e. is constant, then those values can be inserted near the beginning of the problem. But those that change over time must be inserted after you differentiate. ) Finally, you may need to rearrange in order to finish solving the problem.

The final answer is:

Question

The base area of a certain rectangular pool is 20 m
2
.  The pool is being filled with water at a constant rate of 0.4 m
3
/s. How fast is the water level rising?

Hint: Follow these steps, based on the problem solving strategy in the textbook:

Step 1. Assign symbols to all variables involved in the problem. Draw a picture of the bathtub if you want.

For example, let:

ℎ
=
 the water level in the pool. (Is this a function of time 
𝑡
 , or is it constant all the time?)

𝐴
=
  the base area of the pool. (Is this a function of time 
𝑡
 , or is it constant all the time?)

𝑉
=
 the amount of water in the pool (the volume). (Is this a function of time 
𝑡
 , or is it constant all the time?)

𝑡
=
 time (in seconds)

What are the other units?

Step 2. State, in terms of the variables, the information that is given and the rate to be determined.

What is 
𝐴
 ? What is the rate (derivative) that the question is asking you to find? How can you write "The pool is being filled with water at a constant rate of 0.4 m
3
/s" in terms of 
ℎ
,
𝐴
  and/or 
𝑉
 , or perhaps derivatives of these?

Step 3. Find an equation relating the variables introduced in step 1.

Find an equation that relates two or more of 
ℎ
,
𝐴
 and 
𝑉
 .

Step 4. Using the chain rule, differentiate both sides of the equation found in step 3 with respect to the independent variable. This new equation will relate the derivatives.

Differentiate both sides of your equation with respect to 
𝑡
 . This means all your derivatives should be of the form 
𝑑
″
𝑠
𝑜
𝑚
𝑒
𝑡
ℎ
𝑖
𝑛
𝑔
″
𝑑
𝑡
 .

Step 5. Substitute all known variables into the equation from step 4, then solve for the unknown rate of change.

Are there any known numbers that were given in the question? If so, you can insert those now. (If a certain quantity does not change over time i.e. is constant, then those values can be inserted near the beginning of the problem. But those that change over time must be inserted after you differentiate. ) Finally, you may need to rearrange in order to finish solving the problem.

The final answer is:

BlackTom AI · Accepted Answer

To tackle this problem, consider the pool as a rectangular prism with a constant base area and a changing water height. 
Option 1: 'The water level is rising at 0.06 m/s.' Here, converting 0.06 m/s to cm/s gives 6 cm/s. Since the base area is 20 m^2, the corresponding dh/dt would imply dV/dt = A * dh/dt = 20 * 0.06 = 1.2 m^3/s, which does not match the given fill rate of 0.4 m^3/s. So this value is inconsistent with the data. 
Option 2: 'The water level is rising at 0.015 m/s.' This would yield dV/dt = 20 * 0.015 = 0.3 m^3/s, which is smaller than the provided 0.4 m^3/s, so this does not fit the given rate. 
Option 3: 'The water level is rising at 8 m/s.' Such a rate would produce a volume change dV/dt = 20 * 8 = 160 m^3/s, vastly larger than 0.4 m^3/s, making it clearly impossible in this scenario. 
Option 4: 'The water level is rising at 0.4 m/s.' If this were true, the volume rate would be dV/dt = 20 * 0.4 = 8 m^3/s, which contradicts the stated filling rate of 0.4 m^3/s, so this is incorrect. 
Option 5: 'The water level is rising at 2 centimeters per second.' Converting to meters per second gives 0.02 m/s. Then dV/dt = A * dh/dt = 20 * 0.02 = 0.4 m^3/s, which exactly matches the given fill rate. This option is consistent with all givens. 
Option 6: 'The water level is rising at 3 centimeters per second.' This is 0.03 m/s, which would give dV/dt = 20 * 0.03 = 0.6 m^3/s, not equal to the provided 0.4 m^3/s, so it does not align with the data.

In summary, each wrong choice either overestimates or underestimates the rate of volume increase, leading to a mismatch with the given dV/dt of 0.4 m^3/s. Only the option expressing a height rise of 2 cm/s yields a volume rate that matches the input data exactly through the simple relation V = A h with A constant. The core idea is that the base area is constant, so dV/dt = A * dh/dt, and solving for dh/dt gives the correct rate when you substitute the numbers.

MAT135H5_F25_ALL SECTIONS 4.1 Preparation Check

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