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Chapter 1: Problem 54

A tank is in the shape of an inverted cone, having height \(h=2.5 \mathrm{~m}\) and base radius \(r=0.75 \mathrm{~m} .\) If water is poured into the tank at a rate of \(15 \mathrm{~L} / \mathrm{s}\), how long will it take to fill the tank?

Short Answer

Expert verified
Answer: The tank will take approximately 117.87 seconds to fill at a rate of 15 L/s.

Step by step solution

01

Find the volume of the cone

To find the volume (V) of the cone, we will use the formula: \(V = \dfrac{1}{3} \pi r^2 h\) Where \(r\) is the base radius, and \(h\) is the height of the cone.
02

Substitute values and calculate the tank volume

Given \(r = 0.75 \mathrm{~m}\) and \(h = 2.5 \mathrm{~m}\), substitute the values into the volume formula: \(V = \dfrac{1}{3} \pi (0.75)^2 (2.5)\) Now, calculate the volume: \(V ≈ 1.768 \mathrm{~m^3}\)
03

Convert the volume to liters

1 cubic meter (\(\mathrm{m^3}\)) is equal to 1000 liters. So to convert \(1.768 \mathrm{~m^3}\) to liters, multiply by 1000: \(1.768 \times 1000 ≈ 1768 \mathrm{~L}\)
04

Find the time to fill the tank

Given the rate of water being poured is \(15 \mathrm{~L/s}\), we can find the time (t) it takes to fill the tank by dividing the total volume by the rate: \(t = \dfrac{\text{total volume}}{\text{rate}}\) Substitute the values and calculate the time: \(t = \dfrac{1768}{15} ≈ 117.87 \mathrm{~s}\)
05

Interpret the result

The tank will take approximately \(117.87\) seconds to fill at a rate of \(15 \mathrm{~L/s}\).

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