Problem 26 - Algebra A Combined Function

Question

The world's largest sign for Coca-Cola is located in Arica, Chile. The rectangular sign has a length of 400 feet and an area of 52,400 square feet. Find the width of the sign.

Step-by-Step Solution

Answer

The width of the sign is 131 feet.

Step 1: Understanding the Problem

We are given the length and area of a rectangular sign and need to find its width. The formula for the area of a rectangle is given by \( \, \text{Area} = \text{Length} \times \text{Width} \, \).

Step 2: Apply the Formula

We know that the area is 52,400 square feet and the length is 400 feet. Apply these values to the formula: \[ \text{Length} \times \text{Width} = \text{Area} \rightarrow 400 \times \text{Width} = 52,400 \].

Key Concepts

Rectangular Area

When dealing with geometry problems, one of the most basic concepts is the area of a rectangle. The area is essentially a measure of the total space covered by the rectangle. This is especially useful when you want to know how much material you need to cover a surface or the size of a space for a project.

The formula for computing the rectangular area is remarkably simple:

• Area = Length × Width

This equation allows us to calculate the area if we know both the length and the width of the rectangle. In our original exercise, the area and the length are known, giving us the ability to find the missing measurement, which in this case is the width.

Find Width of Rectangle

In geometry problems, especially in everyday life, you often know either the length or the width along with the area and need to find the missing dimension.

• To find the width of a rectangle, rearrange the area formula to solve for width:

Width = Area / Length

This rearrangement stems from simple algebraic manipulation, which allows us to isolate the width on one side of the equation so we can solve for it directly. In our scenario with the Coca-Cola sign, substituting the given numbers:

• Width = 52,400 / 400
• Width = 131

So, the width of the sign is 131 feet.

Algebraic Equations

Algebraic equations are like a puzzle where you use known pieces to find the unknown pieces. In this situation, the algebraic equation helps us find the width when the area and length are known.

By understanding algebraic equations, we can manipulate and rearrange them to find unknowns efficiently. To solve equations, like finding the width of a rectangle, follow these steps:

• Isolate the variable: Use algebraic operations like division or multiplication to get the unknown variable on one side of the equation.
• Substitute known values: Replace the known values in the equation – here length and area.
• Compute the result: After substitutions and isolations, perform the arithmetic to find the solution.

In the rectangle problem, the equation 400 × Width = 52,400 is solved for width, giving us an understanding of how algebraic principles apply to everyday geometry problems.