Problem 85 - Calculus Volume 1

Question

For the following exercises, use differentials to estimate the maximum and relative error when computing the surface area or volume. A pool has a rectangular base of 10 \(\mathrm{ft}\) by 20 \(\mathrm{ft}\) and a depth of 6 \(\mathrm{ft}\) . What is the change in volume if you only fill it up to 5.5 \(\mathrm{ft}\) ?

Step-by-Step Solution

Answer

The change in volume is 100 cubic feet.

Step 1: Understand the problem

We need to find the change in the volume of a rectangular pool when it is filled from a depth of 6 ft to 5.5 ft. We will use differentials to estimate this change in volume.

Step 2: Determine the volume formula

The volume of a rectangular box is given by the formula \( V = l \times w \times h \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height (or depth in this case). For our pool, \( l = 20 \) ft, \( w = 10 \) ft, and \( h = 6 \) ft for the full depth.

Step 3: Calculate initial volume at full depth

First, we calculate the initial volume when the pool is filled to the full depth of 6 ft: \( V = 20 \times 10 \times 6 = 1200 \) cubic feet.

Step 4: Determine the volume at reduced depth

Next, calculate the volume when the pool is filled to a reduced depth of 5.5 ft: \( V' = 20 \times 10 \times 5.5 = 1100 \) cubic feet.

Step 5: Calculate the change in volume

The change in volume \( \Delta V \) is the difference between the initial and reduced volume: \( \Delta V = 1200 - 1100 = 100 \) cubic feet.

Step 6: Use differentials to estimate change

With differentials, we consider the small change in depth \( dh = -0.5 \) ft (because the depth decreases) and use the differential formula \( dV = l \times w \times dh \). Thus, \( dV = 20 \times 10 \times (-0.5) = -100 \) cubic feet, confirming our calculation of the change in volume.

Key Concepts

Volume Estimation

Volume estimation is a technique used to determine how much space an object occupies. In the context of our exercise, we are interested in the volume of water that can fill a rectangular pool. For this, we use the formula for the volume of a rectangular prism:

• \( V = l \times w \times h \)
• where \( l \), \( w \), and \( h \) represent the length, width, and height of the pool, respectively.

To find the change in volume when the water level changes in the pool, we calculate the difference between the two volumes based on different heights. This helps us understand how changes in one dimension can affect the total volume occupied, which is crucial for applications in real-world scenarios like water conservation and capacity planning.
Differentials play a critical role in volume estimation as they provide a way to approximate small changes within our equation. The differential \( dV \) represents a tiny change in volume associated with a small change in depth \( dh \), allowing us to make quick and efficient estimations without extensive computation.

Rectangular Pool

A rectangular pool is a specific type of swimming pool whose shape is defined by three dimensions: length, width, and depth. This geometric shape is simple yet crucial for many calculations in mathematics and real-world applications.
Understanding the dimensions of a rectangular pool is key in solving problems related to volume and surface area.
For example, in our exercise, the rectangular pool has dimensions:

• Length: 20 ft
• Width: 10 ft
• Full Depth: 6 ft

These dimensions are used in the formula \( V = l \times w \times h \) to easily calculate the pool's volume at various depths.
This setting provides a straightforward model for learning about volume calculations and differential applications in a relatable context.

Relative Error

Relative error is a measure of the uncertainty or accuracy of a measurement compared to the true or accepted value. It is expressed as a ratio or a percentage, showing how much off a measurement might be.
It's important in calculations to understand how precise our estimations are when using differentials.
You can calculate relative error with the following formula:

• \( \text{Relative Error} = \frac{|\text{measured value} - \text{true value}|}{|\text{true value}|} \)

In terms of estimating volume changes, using differentials helps minimize relative error because they focus on small changes that are generally easier to measure or compute accurately, leading to reinforced confidence in our results. This practice is essential in fields like engineering, design, or physics, where precision is paramount.

Surface Area Calculation

Surface area calculation involves finding the total area covered by the surface of a three-dimensional object. For a rectangular pool, this requires calculating the areas of its individual sides and then summing those areas up.
Although our exercise primarily deals with volume, surface area calculations still play a critical role in the comprehensive understanding of geometry as they are often linked.
The surface area \( A \) of a rectangular box can be calculated using:

• \( A = 2lw + 2lh + 2wh \)
• where \( l \), \( w \), and \( h \) are the respective length, width, and height.

These calculations are valuable in various practical applications such as determining the amount of material required for lining a pool or the amount of paint needed for its surface. Understanding these concepts enriches one's ability to compute and visualize space in physical, real-world contexts.