Question

The City Opera House is expanding. Currently the city block containing the opera house is rectangular-shaped with a total volume of 9600 feet. If the expanded Opera House is 2.5 times as long, wide, and deep as the original building, what would the new volume be?

• A. 24,000
• B. 60,000
• C. 72,000
• D. 150,000
• E. 245,000

Hint:

For questions involving scaling of geometric shapes, remember the rules: If linear dimensions are scaled by a factor 'k', the area is scaled by \(k^2\), and the volume is scaled by \(k^3\). This can save a lot of calculation time.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The volume of a rectangular solid (like a building) is calculated by multiplying its length, width, and depth. When all dimensions of a solid are scaled by a certain factor, the new volume is the original volume multiplied by the cube of that factor.
Step 2: Key Formula or Approach:
Let the original dimensions be Length (L), Width (W), and Depth (D).
Original Volume \( V_{original} = L \times W \times D \).
The new dimensions are 2.5 times the original ones: \( L_{new} = 2.5L \), \( W_{new} = 2.5W \), \( D_{new} = 2.5D \).
New Volume \( V_{new} = L_{new} \times W_{new} \times D_{new} \).
Alternatively, if each dimension is scaled by a factor 'k', the new volume is \( k^3 \) times the original volume: \( V_{new} = k^3 \times V_{original} \).
Step 3: Detailed Explanation:
We are given the original volume:
\[ V_{original} = 9600 \text{ cubic feet} \] Each dimension (length, width, and deep/depth) is expanded to be 2.5 times its original size. So, the scaling factor is \( k = 2.5 \).
Using the scaling formula for volume:
\[ V_{new} = (2.5)^3 \times V_{original} \] First, calculate the cube of the scaling factor:
\[ (2.5)^3 = 2.5 \times 2.5 \times 2.5 = 6.25 \times 2.5 = 15.625 \] Now, multiply this by the original volume:
\[ V_{new} = 15.625 \times 9600 \] \[ V_{new} = 150,000 \text{ cubic feet} \] Step 4: Final Answer:
The new volume of the expanded Opera House would be 150,000 cubic feet.