Chapter 5.4, Problem 52PS

Solution

a.

To calculate: An expression for the efficiency ratio $\frac{S}{V}$ .

The efficiency ratio $\frac{S}{V}$ is $\frac{2\left(r+h\right)}{rh}$ .

Given information:

The surface area for a cylinder: $S=2\pi r^2+2\pi rh$

The volume for a cylinder: $V=\pi r^{2}h$

Concept used:

Algebraically simplifying the equations

Calculation:

The surface area for a cylinder: $S=2\pi r^2+2\pi rh$

The volume for a cylinder: $V=\pi r^{2}h$

Now, For the efficiency ratio $\frac{S}{V}$ ;

  $\begin{array}{l}\frac{S}{V}=\frac{2\pi r^2+2\pi rh}{\pi r^{2}h}\\ \frac{S}{V}=\frac{2\pi r\left(r+h\right)}{\left(\pi r\right)rh}\\ \frac{S}{V}=\frac{2\left(r+h\right)}{rh}\end{array}$

Conclusion:

  $\frac{S}{V}=\frac{2\left(r+h\right)}{rh}$

b.

To calculate: The efficiency ratio for each can list in the table.

The efficiency ratio:

For soup can = $0.7843cm$

For coffee can = $0.3822cm$

For paint can = $0.3412cm$

Given information:

The surface area for a cylinder: $S=2\pi r^2+2\pi rh$

The volume for a cylinder: $V=\pi r^{2}h$

	Soup can	Coffee can	Paint can
Height, $h$	$10.2cm$	$15.9cm$	$19.4cm$
Radius, $r$	$3.4cm$	$7.8cm$	$8.4cm$

Formula used:

The efficiency ratio:

  $\frac{S}{V}=\frac{2\left(r+h\right)}{rh}$

Calculation:

According to given table:

Apply the formula for efficiency ratio:

The efficiency ratio: $\frac{S}{V}=\frac{2\left(r+h\right)}{rh}$ .

	Soup can	Coffee can	Paint can
Height, $h$	$10.2cm$	$15.9cm$	$19.4cm$
Radius, $r$	$3.4cm$	$7.8cm$	$8.4cm$

The efficiency ratio for soup can:

  $\begin{array}{l}\frac{S}{V}=\frac{2\left(r+h\right)}{rh}\\ \frac{S}{V}=\frac{2\left(3.4cm+10.2cm\right)}{\left(3.4cm\right)\left(10.2cm\right)}\Rightarrow 0.7843cm\end{array}$

The efficiency ratio for soup can:

  $\begin{array}{l}\frac{S}{V}=\frac{2\left(r+h\right)}{rh}\\ \frac{S}{V}=\frac{2\left(7.8cm+15.9cm\right)}{\left(7.8cm\right)\left(15.9cm\right)}\Rightarrow 0.3822cm\end{array}$

The efficiency ratio for soup can:

  $\begin{array}{l}\frac{S}{V}=\frac{2\left(r+h\right)}{rh}\\ \frac{S}{V}=\frac{2\left(8.4cm+19.4cm\right)}{\left(8.4cm\right)\left(19.4cm\right)}\Rightarrow 0.3412cm\end{array}$

Conclusion:

The efficiency ratio:

For soup can = $0.7843cm$

For coffee can = $0.3822cm$

For paint can = $0.3412cm$

c.

To calculate: According to efficiency, rank the three cans. Explain your ranking.

According to efficiency, rank the cans: Paint, coffee, soup

The lower the ratio, the more efficient it is, because then it takes less material (surface area) per (volume).

Given information:

The surface area for a cylinder: $S=2\pi r^2+2\pi rh$

The volume for a cylinder: $V=\pi r^{2}h$

Concept used:

The efficiency ratio:

  $\frac{S}{V}=\frac{2\left(r+h\right)}{rh}$

Calculation:

Apply the efficiency ratio: $\frac{S}{V}=\frac{2\left(r+h\right)}{rh}$

The efficiency ratio:

For soup can = $0.7843cm$

For coffee can = $0.3822cm$

For paint can = $0.3412cm$

According to efficiency, rank the cans:

Paint, coffee, soup

The lower the ratio, the more efficient it is, because then it takes less material (surface area) per (volume).

Conclusion: