Q 42 - Calculus

Question

Find each mass described in Exercises 41–48.

The mass of a gold walking stick in the shape of a tall, thin cylinder that is 4 feet high and 3 inches wide. Gold has a density of 19.3 grams per cubic centimeter. (Caution:

Watch your units! You will need to use the conversion 2.54 cm = 1 inch.)

Step-by-Step Solution

Answer

 The mass of the gold walking stick is 107.308 grams

Step 1: Given Information

A  gold walking stick in the shape of  cylinder that is 4 feet high and 3 inches wide. Gold has a density of 19.3 grams per cubic centimeter. 

Step 2: Density formula

The density of a substance is the mass per unit volume 

The formula for density is 

$$\begin{gathered} \rho =\frac{M}{V} \\ M is the mass and V is the volume \\ V= {\mathrm{\pi r}}^2\mathrm{h} \\ \text{'}r\text{'} is the radius and h is the height of cylinder \end{gathered}$$

Step 3: Conversion

Convert the given height and radius to centimeter

1 inch = 2.54 cm 

$$\begin{gathered} 3 inches = 2.54* 3= 7.62 inches \\ 1 feet = 12 inches \\ 1 feet = 12(2.54) cm \\ 4 feet = 4*12*2.54=121.92 cm \end{gathered}$$

Step 4: Volume of cylinder

$$\begin{gathered} diameter = 7.62 cm \\ Radius = \frac{diameter}{2}=\frac{7.62}{2}= 3.81 cm \\ Volume ={\mathrm{\pi r}}^2\mathrm{h} \\ \mathrm{Volume} =\mathrm{\pi }{(3.81)}^2(121.92) \\ \mathrm{Volume} = 5560 {\mathrm{cm}}^3 \end{gathered}$$

Step 5: find mass

Density formula is 

$$\begin{gathered} \rho =\frac{M}{V} \\ M is the mass and V is the volume \\ V=5560, \rho =19.3 \\ 19.3=\frac{M}{5560} \\ 19.3\left(5560\right)=M \\ M=107.308 grams \end{gathered}$$