Chapter 2.8, Problem 61E

Solution

(a).

To Express: The surface area $S$ of the can as a function of the radius $x$ .

The surface area $S$ of the can is equal to $2\pi x^2+\frac{1000}{x}$ .

Given information:

Design a juice Can Flannery Cannery packs peaches in $0.5-\text{L}$ cylindrical cans.

  

Calculation:

First convert the volume of the can into centimetres.

  $\begin{array}{c}V=0.5\text{L}\\ =0.5\times 1000{\text{cm}}^3\\ =500{\text{cm}}^3\end{array}$

Formula used:

The volume of the cylinder is given by

  $V=\pi r^{2}h$

Substitute $V=500,r=x$ in the above formula and solve for $h$ :

  $\begin{array}{c}500=\pi x^{2}h\\ h=\frac{500}{\pi x^2}\end{array}$

Formula used:

The surface area $S$ of the cylinder is given by

  $S=2\pi r^2+2\pi rh$

Substitute $h=\frac{500}{\pi x^2},r=x$ in the above formula and solve for $S$ :

  $\begin{array}{c}S=2\pi x^2+2\pi x\frac{500}{\pi x^2}\\ =2\pi x^2+\frac{1000}{x}\end{array}$

Therefore, the surface area $S$ of the rectangle as a function of $x$ is $2\pi x^2+\frac{1000}{x}$ .

(b).

To find: The dimensions of the can if the surface is less than $900$ cm².

The radius belongs to $\left(1.12,11.37\right)$ cm and the height belongs to $\left(1.23,126.87\right)$ cm.

Given information:

Design a juice Can Flannery Cannery packs peaches in $0.5-\text{L}$ cylindrical cans.

  

Calculation:

From the part (a),

  $S=2\pi x^2+\frac{1000}{x}$

Since the surface is less than $900$ cm², so it can be expressed as below:

  $2\pi x^2+\frac{1000}{x}<900$

Solve for $x$ :

  $\begin{array}{c}\frac{2\pi x^3+1000}{x}<900\\ 2\pi x^3+1000<900x\\ 2\pi x^3-900x+1000<0\\ \pi x^3-450x+500<0\\ x\in \left(-\infty -12.49\right)\cup \left(1.12,11.37\right)\end{array}$

Since $x$ cannot be negative, so ignore negative values.

Therefore, the values of $x$ are lying in $\left(1.12,11.37\right)$ .

Now, calculate the value of $h$ .

From part (a),

  $h=\frac{500}{\pi x^2}$

Substitute the values of $x$ in the above equation and solve for $h$ :

  $\begin{array}{c}h=\frac{500}{\pi {\left(1.12\right)}^2}\text{or}h=\frac{500}{\pi {\left(11.37\right)}^2}\\ h\approx 126.87\text{or}h\approx 1.23\\ h\in \left(1.23,126.87\right)\end{array}$

Therefore, the radius belongs to $\left(1.12,11.37\right)$ cm and the height belongs to $\left(1.23,126.87\right)$ cm.

(c).

To find: The least possible surface area of the can.

The least possible surface area is equal to $348.7$ cm².

Given information:

Design a juice Can Flannery Cannery packs peaches in $0.5-\text{L}$ cylindrical cans.

  

Calculation:

From the part (a),

  $S=2\pi x^2+\frac{1000}{x}$

For the minimum surface area, take derivative with respect to $x$ and set it equal to zero.

Take derivative with respect to $x$ and set it equal to zero:

  $\begin{array}{c}\frac{dS}{dx}=2\pi \left(2x\right)-\frac{1000}{x^2}\\ 0=4\pi x-\frac{1000}{x^2}\end{array}$

Solve for $x$ :

  $\begin{array}{c}\frac{1000}{x^2}=4\pi x\\ x^3=\frac{250}{\pi }\\ x\approx 4.3\end{array}$

Substitute $x=4.3$ and calculate the least possible surface area below:

  $\begin{array}{c}S=2\pi {\left(4.3\right)}^2+\frac{1000}{4.3}\\ \approx 348.7\end{array}$

Therefore, the least possible surface area is $348.7$ cm².