Problem 13 - Beginning and Intermediate Algebra

Question

Write an equation and solve. The volume of a rectangular storage box is 1440 in \(^{3}\). It is 20 in. long, and it is half as tall as it is wide. Find the width and height of the box.

Step-by-Step Solution

Answer

The width and height of the rectangular storage box are 12 inches and 6 inches, respectively. This is found using the volume formula and the given relationship between height and width, solving for width, and then calculating the height.

Step 1: Define the variables

First, let's define the variables: - Let w represent the width of the box. - Let h represent the height of the box. According to the given condition, the height is half of the width, so we can write it as h = w/2.

Step 2: Use the volume formula and given values for the box

We know that the volume of the box is given by V = length × width × height. Using the given values for the problem: 1440 = 20 × w × h

Step 3: Substitute the relationship of height and width into the equation

Now, substitute the height (h = w/2) into the volume equation: 1440 = 20 × w × (w/2)

Step 4: Solve the equation for width

Now, we need to solve this equation for the width (w). Start by simplifying the equation: 1440 = 10w^2 Now, divide both sides by 10: 144 = w^2 To find the width, take the square root of both sides: w = \(\sqrt{144}\) w = 12 inches

Step 5: Calculate the height using the width

Now that we have found the width, we can use the relationship given (h = w/2) to find the height: h = (12)/2 h = 6 inches So, the width of the box is 12 inches, and the height of the box is 6 inches.

Key Concepts

Rectangular Storage Box

A rectangular storage box is a common shape used for many containers, characterized by its six faces, all of which are rectangles. This geometrical shape is prevalent because of its practicality in maximizing space in packaging and storage. When you think of a rectangular box, it might remind you of shoe boxes or cartons used for shipping. The simplicity of its dimensions allows it to be easily described by three parameters: length, width, and height.
- **Length** is the longest side of the rectangle base. - **Width** is the side perpendicular to the length on the base. - **Height** is the distance from the base to the top of the box.
Understanding these three dimensions is crucial because they are the foundation for calculating the box’s volume, which is key to determining how much it can hold.

Volume Formula

The volume of a box determines the capacity inside the box, and it’s a crucial aspect when designing or utilizing storage solutions. The formula to calculate the volume of a rectangular box is given by:\[ V = ext{length} imes ext{width} imes ext{height} \]This formula is derived from the space inside the box where the base area (length times width) is multiplied by the height.
In our exercise, we are given a certain volume of 1440 cubic inches. This means the space inside the box can hold 1440 cubic inches. By knowing the volume and one other dimension, we can find the remaining two dimensions using algebraic manipulation. This makes the volume formula a powerful tool for solving geometric problems involving boxes.

Width and Height Calculation

Finding the dimensions of a rectangular storage box, like width and height, often involves understanding relationships between these dimensions. In the given problem, the box's height is half of its width. This relationship is expressed as: - **Height** = *Width* / 2
Such expressions help us in reducing the number of unknowns, making the equation solving process simpler. By substituting this relationship into the main volume equation, we can transform complex geometry problems into manageable algebraic equations. This is particularly helpful when at least one dimension or relationship between dimensions is given.

Equation Solving

Solving the equation involves substituting known relationships and simplifying to find unknown values. Here's the step-by-step breakdown:

• Insert the relationship between height and width (\(h = w/2\)) into the formula for volume: \(1440 = 20 \times w \times (w/2)\).
• Simplify the equation: \(1440 = 10w^2\). We notice that we end up with a quadratic equation.
• Divide both sides by 10 to isolate \(w^2\): \(144 = w^2\).
• Take the square root of both sides to solve for \(w\): \(w = \sqrt{144} = 12 \text{ inches}\).

Now that we have the width, we can find the height using the relationship \(h = w/2\):- Height = 12/2 = 6 inches
Thus, solving equations is a crucial skill for tackling geometry problems involving dimensions and volumes, transforming theoretical problems into practical solutions.