Problem 43 - Algebra A Combined Function

Question

By doubling each dimension, the area of a parallelogram increased from 36 square centimeters to 144 square centimeters. Find the percent increase in area.

Step-by-Step Solution

Answer

The percent increase in area is 300%.

Step 1: Find the original area

The problem states that the original area of the parallelogram was 36 square centimeters.

Step 2: Calculate the new area after dimension doubling

When the dimensions of a parallelogram are doubled, each dimension is multiplied by 2. Therefore, the new area becomes the square of twice the original dimension's product. The problem directly gives the new area as 144 square centimeters.

Step 3: Find the change in area

Subtract the original area from the new area to find the increase in area: \( 144 - 36 = 108 \text{ square centimeters} \).

Step 4: Calculate the percent increase

To find the percentage increase, divide the increase in area by the original area, and then multiply by 100 to convert it to a percentage: \[ \frac{108}{36} \times 100 = 300\% \].

Key Concepts

Area Calculation

Calculating the area of a shape is essential in understanding its size and extent. For simple shapes like rectangles and parallelograms, the area is found by multiplying the base by the height. This formula represents the size of the shape in square units, such as square centimeters.
When the dimensions of the shape change, the area also changes accordingly. Doubling the dimensions of a parallelogram means both its base and height are doubled, which significantly affects the area. As shown in the exercise, the area increased as a result from 36 square centimeters to 144 square centimeters.
Understanding how to calculate and adjust the area is crucial in both everyday situations and advanced geometry applications.

Parallelogram Dimensions

The parallelogram, a basic geometrical shape, has distinctive properties which influence how its dimensions impact its area. It is a four-sided polygon with opposite sides that are equal in length, and opposite angles that are also equal.
In a parallelogram, the base can be any of its sides, and the corresponding height is the perpendicular distance from the base to the opposite side. Modifying these dimensions directly influences the overall area.
In the given problem, doubling both the base and the height means each is multiplied by 2. As a result, the entire area changes by a factor of 4, because the area formula, base times height, is applied with doubled dimensions.

Mathematical Problem Solving

Solving mathematical problems involves understanding the question, identifying known values, and applying appropriate formulas or operations to find solutions.
In the exercise, we are asked to find the percent increase in the area of a parallelogram after its dimensions were doubled.

• Step 1 begins by identifying the original area (36 square centimeters).
• Step 2 calculates the increased area as given by the problem (144 square centimeters).
• In Step 3, we determine the change by subtracting the original area from the new area, resulting in a 108 square centimeter increase.
• Finally, Step 4 involves computing the percent increase by dividing the increase (108 square centimeters) by the original area (36 square centimeters) and then multiplying by 100, resulting in a 300% increase.

By breaking down the problem into manageable steps, mathematical problem solving becomes more approachable.