Problem 50 - Calculus with Concepts in Calculus

Question

The volume \(V\) of a rectangular box with square base is 60 cubic centimeters. Express the length \(l\) (in centimeters) of a vertical side as a function of the length \(s\) of a side of the base.

Step-by-Step Solution

Answer

The length of the vertical side is given by \( l(s) = \frac{60}{s^2} \).

Step 1: Write the Formula for Volume

The volume of a rectangular box is given by the formula \( V = s^2 \times l \), where \( s \) is the length of each side of the square base, and \( l \) is the height or length of the vertical side.

Step 2: Substitute Volume Value

We are given that the volume \( V \) is 60 cubic centimeters. Substitute this value into the volume formula: \( 60 = s^2 \times l \).

Step 3: Solve for the Vertical Side Length

You want to solve for \( l \) in terms of \( s \). To do this, rearrange the equation from Step 2: \( l = \frac{60}{s^2} \).

Step 4: Express Length as a Function

Now, express \( l \) as a function of \( s \): \( l(s) = \frac{60}{s^2} \). This function tells us that the length of the vertical side is inversely proportional to the square of the side of the base.

Key Concepts

Volume

Understanding the concept of volume is crucial when working with three-dimensional shapes like a rectangular box. Volume measures the amount of space an object occupies.
When dealing with boxes, volume is calculated by multiplying the area of the base by the height.
In our case, the base is a square, meaning each side is equal. The formula for the volume of a rectangular box with a square base is:

• \[ V = s^2 \times l \]

Here, \( s \) represents the length of a side of the square base, and \( l \) is the height of the box.
This formula helps to calculate the volume of any box by substituting the known measurements.
In the exercise, the known volume is 60 cubic centimeters, meaning our formula becomes specific to this volume.

Functions

Functions are a fundamental concept in mathematics that describe the relationship between two quantities.
A function often expresses how one quantity changes in response to another.
For the rectangular box example, we are interested in how the length of the vertical side changes according to the base side length.

• The function derived from rearranging the volume formula is:\[ l(s) = \frac{60}{s^2} \]

This function captures the inverse relationship between the side length \( s \) of the box's base and its height \( l \).
As the length \( s \) of the base increases, the height \( l \) of the box decreases.
This type of function is useful because it allows us to predict the height of the box given any base length.
By expressing \( l \) as a function of \( s \), we gain insight into how changes to one dimension affect another.

Rectangular Box

A rectangular box, sometimes referred to as a cuboid, is a three-dimensional shape with six rectangular faces.
When its base is square, it simplifies the calculations related to volume.

• The defining features of a rectangular box include:
• A square or rectangular base determined by length \( s \) and height \( l \)
• All angles are right angles

Understanding how to manipulate the dimensions of a rectangular box has practical applications in fields such as packaging and architecture.
In this instance, recognizing that adjusting the side of the base affects the height helps solve problems where the volume is fixed, like in the exercise.
By grasping this concept, students learn to apply mathematical reasoning to real-world scenarios efficiently.