Problem 96 - Elementary Algebra

Question

Roofing shingles are sold by the "square." A square can cover an area of roof that is \(10 \mathrm{ft}\) long and \(10 \mathrm{ft}\) wide. The area of a roof is \(3450 \mathrm{ft}^{2}\). Find the number of squares of roofing shingles needed to cover this roof. Round up to the nearest whole number.

Step-by-Step Solution

Answer

35

Step 1 - Determine the area a square can cover

A square of roofing shingles covers an area that is 10 feet long and 10 feet wide. To find the area of one square, use the formula for the area of a rectangle: \[ \text{Area} = \text{length} \times \text{width} \] So, \[ \text{Area} = 10 \text{ ft} \times 10 \text{ ft} = 100 \text{ ft}^2 \]

Step 2 - Determine how many squares are needed

The roof has a total area of 3450 square feet. To find out how many squares are needed to cover this area, divide the roof area by the area that one square can cover: \[ \text{Number of squares} = \frac{\text{Roof area}}{\text{Area of one square}} \] So, \[ \text{Number of squares} = \frac{3450 \text{ ft}^2}{100 \text{ ft}^2} = 34.5 \]

Step 3 - Round up to the nearest whole number

Since the number of squares needs to be a whole number and you can't purchase a fraction of a square, round up 34.5 to the nearest whole number: 34.5 rounded up is 35.

Key Concepts

Area of a Rectangle

The area of a rectangle is a fundamental concept in geometry. When you're working with a rectangle, whether it's a simple shape or a piece of a more complex figure, calculating its area is key.
The formula to find the area of a rectangle is straightforward:
Area of a rectangle = length × width
This formula helps you determine how much space is inside a rectangle.
For example, in the exercise, each roofing square is a rectangle that measures 10 feet by 10 feet.
To find the area:

• Length: 10 ft
• Width: 10 ft

So, the calculation is:
Area = 10 ft × 10 ft = 100 ft²
This simple multiplication tells you that each roofing square covers 100 square feet. This concept is useful in many situations, including planning construction projects, designing spaces, and more.

Division in Algebra

Division in algebra is another vital concept that frequently comes into play, especially when dealing with area and spacing calculations. In our exercise, we need to determine how many roofing squares are required to cover a specific area of the roof.
To do this, we divide the total roof area by the area that one roofing square can cover.

Example:

• Total Roof Area: 3450 ft²
• Area of One Roofing Square: 100 ft²

The formula to find how many squares are needed is: Number of squares = Total Roof Area / Area of One Roofing Square
Plugging in the numbers, we get:
Number of squares = 3450 ft² / 100 ft² = 34.5
But since you can't use a fraction of a roofing square, you have to round up to the nearest whole number, which brings us to the next concept.

Rounding Numbers

Rounding numbers is a useful mathematical skill, especially when you deal with quantities that must be whole numbers. In our exercise, the calculation showed that we need 34.5 roofing squares. However, you can't purchase half a roofing square.

To solve this, we round 34.5 up to the nearest whole number.
When you round to the nearest whole number, follow these rules:

• If the digit after the decimal point is 5 or more, round up.
• If the digit is less than 5, round down.

In our exercise, 34.5 rounds up to 35. So, you would need to buy 35 roofing squares to cover the entire roof area.