Question

A rectangular parking lot 2x feet long and x feet wide is to be enlarged so that the lot will be 2 times as long and 3 times as wide as it is now. The area of the enlarged rectangular lot will be how many times the area of the present lot?

• A. 6
• B. 5
• C. 4
• D. 3
• E. 2

Hint:

When the dimensions of a 2D shape are scaled by factors, the area is scaled by the product of those factors. If length is scaled by a factor of \(k_L\) and width by a factor of \(k_W\), the new area will be \(k_L \times k_W\) times the old area. Here, the factors were 2 and 3, so the area increases by a factor of \(2 \times 3 = 6\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem involves calculating the area of a rectangle and understanding how the area changes when its dimensions are scaled.
Step 2: Key Formula or Approach:
The area of a rectangle is given by the formula \( \text{Area} = \text{length} \times \text{width} \). 1. Calculate the original area of the parking lot. 2. Determine the new dimensions of the enlarged lot. 3. Calculate the new area of the enlarged lot. 4. Find the ratio of the new area to the original area.
Step 3: Detailed Explanation:
1. Original Area: The original dimensions are: Length \(L_1 = 2x\) feet Width \(W_1 = x\) feet The original area (\(A_1\)) is: \[ A_1 = L_1 \times W_1 = (2x)(x) = 2x^2 \] 2. New Dimensions: The new lot will be 2 times as long and 3 times as wide. New Length \(L_2 = 2 \times L_1 = 2 \times (2x) = 4x\) feet New Width \(W_2 = 3 \times W_1 = 3 \times (x) = 3x\) feet 3. New Area: The new area (\(A_2\)) is: \[ A_2 = L_2 \times W_2 = (4x)(3x) = 12x^2 \] 4. Ratio of Areas: We need to find how many times the new area is of the original area. This is the ratio \( \frac{A_2}{A_1} \). \[ \text{Ratio} = \frac{A_2}{A_1} = \frac{12x^2}{2x^2} \] The \(x^2\) terms cancel out. \[ \text{Ratio} = \frac{12}{2} = 6 \] The area of the enlarged lot is 6 times the area of the present lot.
Step 4: Final Answer:
The enlarged area is 6 times the original area.