Problem 17 - Algebra A Combined Function

Question

Solve.By decreasing each dimension by 1 unit, the area of a rectangle decreased from 40 square feet (on the left) to 28 square feet (on the right). Find the percent decrease in area.

Step-by-Step Solution

Answer

The area of the rectangle decreased by 30%.

Step 1: Understand the initial and final areas

The area of the rectangle initially is 40 square feet, and it decreases to 28 square feet after reducing each dimension by 1 unit. We need to find out what percentage this decrease represents.

Step 2: Calculate the decrease in area

To find the decrease in area, subtract the final area from the initial area: \[ 40 - 28 = 12 \]. So, the area decreased by 12 square feet.

Step 3: Compute the percentage decrease

The percentage decrease can be calculated using the formula: \[ \text{Percentage Decrease} = \left( \frac{\text{Decrease in Area}}{\text{Initial Area}} \right) \times 100 \]. Substitute the known values: \[ \left( \frac{12}{40} \right) \times 100 = 30\% \]. Thus, the area decreased by 30%.

Key Concepts

Rectangle

A rectangle is a four-sided figure with opposite sides that are parallel and equal in length. It's a fundamental shape in geometry with each angle measuring 90 degrees. The two dimensions that define it are its length and width. Understanding rectangles is crucial because they are commonly seen in various applications, from architecture to design.
One interesting property of rectangles is that if either the length or the width changes, the area – which is a measure of the surface inside the rectangle – will also change. Rectangles are robust shapes used frequently because of their symmetry and uniform side lengths. Familiarity with the properties of rectangles allows for effective problem-solving in geometry topics.

Area Calculation

Calculating the area of a rectangle is straightforward and relies on knowing both dimensions of the shape. It is given by the formula:\[\text{Area} = \text{Length} \times \text{Width}\]This formula helps determine how much space is enclosed within a rectangle. Changes in either the length or the width will directly affect the area.
For instance, consider a rectangle with an initial area of 40 square feet. If each dimension is reduced by 1 unit, the new area reduces to 28 square feet. Understanding how changes in dimensions affect area is a common problem in geometry that enhances one's analytical skills.

Percentage Decrease

Percentage decrease is a valuable concept in understanding how much a quantity has diminished compared to its original value. With respect to a rectangle, if its area reduces, the percentage decrease highlights the extent of this reduction.
The formula to calculate percentage decrease is:\[\text{Percentage Decrease} = \left( \frac{\text{Decrease in Value}}{\text{Original Value}} \right) \times 100\]This formula quantifies shifts in value clearly. For example, if the area of a rectangle decreases from 40 square feet to 28 square feet, the decrease is 12 square feet. Applying the formula results in a 30% reduction in area, giving a concrete sense of how much the area has shrunk.

Problem Solving

Effective problem solving in geometry often relies on a systematic approach. In the context of calculating area and changes thereof, the process involves understanding the problem, performing relevant calculations, and interpreting results accurately.
Here’s a general strategy:

• Identify given values and what is being asked. For instance, determine the initial and final areas.
• Perform required calculations, such as finding the initial area using rectangle properties and computing the decrease.
• Use appropriate mathematical formulas, like the percentage decrease formula, to find the desired solution.

Problem-solving isn't just about getting the answer; it's about understanding the "why" and "how" behind calculations, which strengthens mathematical proficiency and confidence.