Question

the swimming pool shown below is a prism. the cross-section of the prism is a trapezium.

the pool is filled at a rate of \\(3000\text{ ml}\\) per second.

how long will it take to fill the pool?

(hint: \\(1\text{ ml} = 1\text{ cm}^3\\))

Explanation:

🆕 New Concept Discovered: Volume of Prisms with Trapezoidal Cross-Sections
Finding volume by multiplying a trapezoid's area by length.

Step 1: Find the area of the trapezoidal cross-section

The front face of the pool is a trapezium (trapezoid). The parallel vertical sides (bases) are \( a = 170\text{ cm} \) and \( b = 200\text{ cm} \). The perpendicular distance between them (the length of the pool along the top) is \( h = 1200\text{ cm} \).

The area of a trapezium is:

$$ A = \frac{a + b}{2} \times h $$

Substitute the given values:

$$ A = \frac{170 + 200}{2} \times 1200 $$

$$ A = \frac{370}{2} \times 1200 $$

$$ A = 185 \times 1200 $$

$$ A = 222,000\text{ cm}^2 $$

Step 2: Calculate the total volume of the pool

The pool is a prism with a length (width) of \( w = 400\text{ cm} \). The volume \( V \) of a prism is the area of its cross-section multiplied by its length:

$$ V = A \times w $$

$$ V = 222,000\text{ cm}^2 \times 400\text{ cm} $$

$$ V = 88,800,000\text{ cm}^3 $$

Step 3: Convert the volume to milliliters

Using the hint \( 1\text{ ml} = 1\text{ cm}^3 \):

$$ V = 88,800,000\text{ ml} $$

Step 4: Calculate the time required to fill the pool

The pool is filled at a rate of \( 3000\text{ ml} \) per second. Divide the total volume by the rate to find the time in seconds:

$$ \text{Time} = \frac{88,800,000\text{ ml}}{3000\text{ ml/s}} $$

$$ \text{Time} = 29,600\text{ seconds} $$

To make this number easier to understand, we can convert it into hours, minutes, and seconds:

$$ 29,600\text{ seconds} \div 3600\text{ seconds/hour} \approx 8.22\text{ hours} $$

$$ 29,600\text{ seconds} = 8\text{ hours, } 13\text{ minutes, and } 20\text{ seconds} $$

Answer:

29,600 seconds (or 8 hours, 13 minutes, and 20 seconds)