Problem 25 - Algebra A Combined Function

Question

The iconic NASDAQ sign in New York's Times Square has a width of 84 feet and an area of 10,080 square feet. Find the height (or length) of the sign.

Step-by-Step Solution

Answer

The height of the sign is 120 feet.

Step 1: Understand the Problem

We are given the width and area of a rectangular sign and need to find its height. The width is 84 feet and the area is 10,080 square feet.

Step 2: Recall the Area Formula for a Rectangle

The area of a rectangle is calculated as the product of its width and height. The formula is:\[ \text{Area} = \text{Width} \times \text{Height} \]

Step 3: Set Up the Equation Using the Known Values

Substitute the given values into the area formula: \[ 10,080 = 84 \times \text{Height} \]

Step 4: Solve for Height

To find the height, divide both sides of the equation by the width (84): \[ \text{Height} = \frac{10,080}{84} \]

Step 5: Calculate the Height

Perform the division: \[ \text{Height} = 120 \]

Step 6: Verify the Solution

Check that multiplying the found height by the width gives the original area: \[ 84 \times 120 = 10,080 \], which confirms our solution is correct.

Key Concepts

Rectangular Area

In geometry, understanding a rectangular area is essential, especially when working with problems related to shapes and measurements. A rectangle is a four-sided polygon where opposite sides are parallel and equal in length.
The area of a rectangle is the measure of the space enclosed within its boundaries. It is calculated in square units, representing the total surface area covered inside the rectangle.

• Rectangular areas are widely used in fields such as engineering, architecture, and everyday activities.
• Common examples include the area of a garden, walls, or signboards.

Understanding the concept of area aids in solving many real-world problems by determining the amount of material needed to cover a space or optimizing space usage.

Width and Height Relationship

The width and height relationship in a rectangle is crucial to unlocking and calculating its area. These two dimensions define the size and shape of the rectangle.
The width refers to the horizontal measurement across the shape. The height, often called the length depending on its orientation, is the vertical distance.

• In the context of the NASDAQ sign, knowing one dimension (like width) and the area allows us to calculate the other dimension (height).
• The relationship can be visualized as a multiplication of these dimensions to supply the entire area of the rectangle.

This relationship helps us form a direct equation to find missing dimensions when given partial information about a shape.

Area Calculation

Calculating the area of a rectangle is straightforward when dimensions are known. The formula for calculating the area is:\[ \text{Area} = \text{Width} \times \text{Height}\]This formula is fundamental in geometry, enabling us to find either the width or the height if the area and one dimension are provided.
Let's put it into perspective:

• Given the NASDAQ sign with an area of 10,080 square feet, and its width of 84 feet,calculating the height becomes a direct application of the formula.
• By rearranging the formula to solve for height, we have:\[ \text{Height} = \frac{\text{Area}}{\text{Width}}\]which simplifies our calculations.

The use of this formula highlights the importance of formulas as tools that simplify complex real-world calculations.

Problem-Solving Steps

Solving geometric problems, like finding the height of a rectangular sign, requires a structured approach.
Using a step-by-step method not only organizes our thought process but also ensures accuracy.
Here’s a quick breakdown based on the NASDAQ sign example:

• **Understand the problem:** Identify what is given and what needs to be found. Here, we know the width and total area, and we need to find the height.
• **Recall formulas:** Refresh the area formula for rectangles \(\text{Area} = \text{Width} \times \text{Height}\).
• **Set up equations:** Substitute known values into the formula to form an equation \(10,080 = 84 \times \text{Height}\).
• **Solve the equation:** Isolate the unknown by dividing, resulting in \(\text{Height} = \frac{10,080}{84}\).
• **Verify:** Ensure the solution is correct by multiplying the derived height by the width to check if it results in the given area.

This methodology is beneficial in solving any geometric problem effectively while building good problem-solving habits.