Chapter U3.50, Problem 1E

Interpretation Introduction

Interpretation : Two methods to determine the volume when height of water is given are to be explained.

Concept Introduction : Meteorologists keep track of the amount of rainfall by measuring the height of rain water collected in rain gauge.

Answer to Problem 1E

Method 1

  $\begin{array}{l}\text{Volume = basearea×height}\\ {\text{Volume=2×3=6cm}}^{\text{3}}\end{array}$

Method 2 (Usingproportional relationship)

  $\text{Volume of rain=k}\times \text{height of rain}$

Wherek is a proportionality constant.

  ${\text{Volume=2×3=6cm}}^{\text{3}}$

(in case of rain gauge, the proportionality constant is also equal to the area of the base)

Explanation of Solution

Given information:

A cylindrical rain gauge has an area of 2.0 cm² and height of water is 3.0 cm.

Amount of rainfall can be measured using an instrument called rain gauge.

Method 1 (Mathematical calculation)

  $\begin{array}{l}\text{Base area of rain gauge=2}{\text{.0cm}}^2\\ \text{Height of water=3}\text{.0cm}\\ \text{Volume=base area}\times \text{height}\\ \text{Volume=2}\times {\text{3=6cm}}^{\text{3}}\end{array}$

(since the rain gauge is cylindrical)

Method 2 (Usingproportional relationship)

Height and volume for a specific rain gauge are related by a proportionality constant, k .

The relation can be depicted as:

  $\text{Volume of rain=k}\times \text{height of rain}$

Wherek is a proportionality constant.

  ${\text{Volume=2×3=6cm}}^{\text{3}}$

(In case of rain gauge, the proportionality constant is also equal to the area of the base)

Conclusion

Volume and height are proportional for containers with uniform shape and parallel walls.