Problem 55 - Algebra 1: Concepts and Skills

Question

What is the width of a rectangular soccer field that has an area of 9000 square feet and a length of 120 feet?

Step-by-Step Solution

Answer

The width of the soccer field is 75 feet.

Step 1: Recall the formula for the area of a rectangle

The formula for the area of a rectangle is given by \(Area = Length \times Width\). In this particular problem, the area and length are given, so the formula can be re-arranged to solve for width.

Step 2: Re-arrange the formula to solve for width

The formula can be re-arranged as follows: \(Width = \frac{Area}{Length}\). Now we can substitute the given dimensions into the formula.

Step 3: Substitute and solve

Substituting the given area of 9000 square feet and length of 120 feet, we get \(Width = \frac{9000}{120} = 75\). Therefore, the width of the soccer field is 75 feet.

Key Concepts

Rectangular Dimensions

When dealing with geometric figures like rectangles, it's essential to understand their basic properties. A rectangle is a four-sided shape with opposite sides that are equal in length. This means both pairs of opposite sides (the length and the width) are identical in measurement.

The area of a rectangle, which is a measure of the total surface the shape covers, is calculated by multiplying the length by the width. This simple multiplication informs us how much space is contained within the perimeter of the rectangle. For example, if you know a soccer field is rectangular and you have its area and one side's measurement, you can find its unknown dimension using these properties.

• The key dimensions are length and width.
• The relationship between them determines the overall size of the rectangle.
• Remember: all calculations are in units squared for area (like square feet).

Understanding these dimensions provides the foundation for working with this geometric shape efficiently.

Formula Rearrangement

Working with equations often involves rearranging them to solve for a specific variable. In the case of a rectangle's area, you normally encounter the formula:\[ Area = Length \times Width \]

However, when you need to find one of the dimensions, and you know the area and the other dimension, you can rearrange this formula. Rewriting the expression to solve for width, you get:

\[ Width = \frac{Area}{Length} \]
This approach allows you to isolate the width as the subject of the formula. Rearranging formulas is a valuable skill because it lets you manipulate the equation to directly compute the desired variable. This can be incredibly useful in various mathematical and real-world problems.

• Identify the known variables in your problem.
• Rearrange the formula as necessary to solve for the unknown variable.
• This method is universally applicable to similar problems.

The ability to rearrange formulas efficiently is a keystone in mathematical problem-solving.

Solve for Width

Let's apply the formula rearrangement to solve a real problem: finding the width of a rectangle. Imagine you have a rectangular soccer field with a known area of 9000 square feet and a length of 120 feet. To find the width, use the rearranged formula:

\[ Width = \frac{9000}{120} \]

Simplifying this division, you get:

\[ Width = 75 \text{ feet} \]
Solving for the width involves substituting the known values into your rearranged formula. And, **presto!** you find that the width is 75 feet.

• Always verify which values are given in a problem.
• Substitute these values correctly in the formula.
• Solve the equation to find your answer—double-check your work!

This method not only applies to geometry but is a handy technique across various science and engineering disciplines.