Problem 87 - Introductory and Intermediate Algebra for College Students 4th

Question

Suppose you know the cost for building a rectangular deck measuring 8 feet by 10 feet. If you decide to increase the dimensions to 12 feet by 15 feet, by how many times will the cost increase?

Step-by-Step Solution

Answer

The cost of building the rectangular deck will increase by a factor of 2.25.

Step 1: Calculate the original area

The area of a rectangle is calculated by multiplying the length by the width. So, for an 8 feet by 10 feet rectangle, the area would be \(8 \times 10 = 80 \) square feet.

Step 2: Calculate the new area

When the dimensions are increased to 12 feet by 15 feet, the area becomes \(12 \times 15 = 180 \) square feet.

Step 3: Calculate the increase factor

The increase factor is obtained by dividing the new area by the original area. In other words: \(\frac{new \space area}{original \space area} = \frac{180}{80} = 2.25\) . This means that the cost will increase by a factor of 2.25, assuming the cost is directly proportional to the area.

Key Concepts

Rectangular Area Calculation

When calculating the area of a rectangle, it's important to remember this straightforward formula: multiply the length by the width. This method essentially counts how many square units fit into the rectangle. The area expresses how much space there is within the boundary of the rectangle. For the original rectangle with dimensions of 8 feet by 10 feet, the calculation goes like this:

• Use the formula for area: Length × Width
• Substitute the values into the formula: 8 × 10
• Compute the multiplication: 8 × 10 = 80

So, the original deck area is 80 square feet. To find the area for the new dimensions, repeat the process.

• New Dimensions: 12 feet by 15 feet
• Calculate: 12 × 15
• Result: 180 square feet

Remember, the geometric property here is simply counting squares, which are equal in size. Knowing how to calculate rectangular areas is fundamental not only in Algebra, but also in practical situations like construction and design.

Proportional Relationships

Proportional relationships involve comparing two quantities by division, showing how one value relates to another consistently. In this exercise, the relationship between the area of the deck and the cost assumes proportionality. This means as the area of the deck increases, the cost also increases at the same rate.
Using our calculated areas:

• The original area was 80 square feet
• The new area increased to 180 square feet
• The proportional increase is given by the ratio: \(\frac{180}{80}\)
• Simplifying the ratio: 2.25

This means if the cost of building the original deck was some amount, the new deck will cost 2.25 times more. Proportional relationships help in predicting the new outcome based on the scale of change, considering both quantities maintain a constant ratio.

Dimensional Analysis

Dimensional analysis is a process that involves using the units of measurement to help solve problems. By ensuring consistency in units, it allows us to analyze physical quantities by breaking them into parts or dimensions. This method is particularly useful to simplify complex equations and verify that our calculations make sense.
In this exercise, the units involved are feet for length and square feet for area.

• You start with dimensions in feet and compute the area in square feet.
• Always ensure that you multiply dimensions with the same units to maintain dimensional integrity.

The concept of dimensional analysis extends beyond basic area calculations, as it also applies to converting different units (such as converting feet to meters). It is a crucial tool in science and engineering, helping to avoid errors and ensure that solutions are logically consistent.