The plans for a diving pool call for a rectanglular prism that has a length of 30 meters, a width of 25 meters, and a depth of 5 meters. If the plans are changed to increase both the length and the width of the pool by 10%, what will be the increase, to the nearest 1%, in the volume of the pool?

Options

A. 0.1
B. 0.17
C. 0.2
D. 0.21

Answer

To solve the problem step by step, we will calculate the initial volume of the pool, the new dimensions after the increase, the new volume, and finally the percentage increase in volume.

### Step 1: Calculate the initial volume of the pool.
The formula for the volume \( V \) of a rectangular prism is given by:
\[ V = \text{length} \times \text{width} \times \text{depth} \]
Given:
- Length = 30 meters
- Width = 25 meters
- Depth = 5 meters

Substituting the values:
\[ V_{\text{old}} = 30 \times 25 \times 5 \]
Calculating:
\[ V_{\text{old}} = 3750 \text{ cubic meters} \]

### Step 2: Calculate the new dimensions after a 10% increase.
To find the new length and width after a 10% increase:
- New Length = Old Length + 10% of Old Length
\[ \text{New Length} = 30 + \left(0.10 \times 30\right) = 30 + 3 = 33 \text{ meters} \]
- New Width = Old Width + 10% of Old Width
\[ \text{New Width} = 25 + \left(0.10 \times 25\right) = 25 + 2.5 = 27.5 \text{ meters} \]

### Step 3: Calculate the new volume of the pool.
Using the new dimensions:
\[ V_{\text{new}} = \text{New Length} \times \text{New Width} \times \text{Depth} \]
Substituting the values:
\[ V_{\text{new}} = 33 \times 27.5 \times 5 \]
Calculating:
\[ V_{\text{new}} = 4537.5 \text{ cubic meters} \]

### Step 4: Calculate the increase in volume.
The increase in volume is given by:
\[ \text{Increase in Volume} = V_{\text{new}} - V_{\text{old}} \]
Substituting the values:
\[ \text{Increase in Volume} = 4537.5 - 3750 = 787.5 \text{ cubic meters} \]

### Step 5: Calculate the percentage increase in volume.
The percentage increase is calculated as:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase in Volume}}{V_{\text{old}}}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left(\frac{787.5}{3750}\right) \times 100 \]
Calculating:
\[ \text{Percentage Increase} = 21\% \]

### Final Answer:
The increase in the volume of the pool, to the nearest 1%, is **0.21**.

---