Chapter 12.2, Problem 42HP

To determine

To give:An example of two cylinders that have the same lateral area and different surface areas. Describe the lateral area and surface area of each.

Answer to Problem 42HP

The radii two cylinders having same lateral area but different surface areas are 4 units and 2 units and their heights are 5 units and 10 units respectively. The lateral area of first cylinder is $40\text{π}$ units, and surface area $72\text{π}$ units and that of second are $40\text{π}$ units and $48\text{π}$ units.

Explanation of Solution

Given data: $y$

The two cylinders that have the same lateral area and different surface areas.

Concept used:

The lateral area of a cylinder of height h and radius r is given by $A=2\text{π}rh$ and surface area $S=2\text{π}rh+2\text{π}{r}^2=L+B$ .

Calculations:

Suppose two cylinders of heights $h_1$ , $h_2$ and radii $r_1$ , $r_2$ have same lateral areas bit different surface areas, i.e.,

  $\text{2π}{r}_1{h}_1=\text{2π}{r}_2{h}_2$

$r_1{h}_1=r_2{h}_2$ .... (1)

Now let we set $r_2=\frac{r_1}{2}$ , and $h_2=2h_1$ , then $r_2{h}_2=\frac{r_1}{2}\cdot 2h_1=r_1\cdot h_1$ .

The surface areas of the cylinders are $S_1=2\text{π}{r}_1{h}_1+2\text{π}{r}_1{}^2$ and $S_2=2\text{π}{r}_2{h}_2+2\text{π}{r}_2{}^2$

  $\Rightarrow S_2=2\text{π}\left(\frac{r_1}{2}\right)\left(2h_1\right)+2\text{π}{\left(\frac{r_1}{2}\right)}^2$

  $\Rightarrow S_2=2\text{π}{r}_1{h}_1+\frac{1}{2}\text{π}{r}_1{}^2\ne S_1$ .

Let height and radius of one cylinder are 4 units and 5 units respectively. The lateral area of this cylinder is $\text{2π}{r}_1{h}_1=\text{2π}\times 4\times 5=40\text{π}$ units, and surface area $S_1=2\text{π}{r}_1{h}_1+2\text{π}{r}_1{}^2\text{}$

  $=40\text{π}+2\text{π}{\left(4\right)}^2=72\text{π}$ units.

The height and radius of other cylinder are 2 units and 10units respectively. The lateral area of this cylinder is $\text{2π}{r}_2{h}_2=\text{2π}\times 2\times 10=40\text{π}$ units, and surface area $S_1=2\text{π}{r}_2{h}_2+2\text{π}{r}_2{}^2\text{}$

  $=2\text{π}\left(2\right)\left(10\right)+2\text{π}{\left(2\right)}^2\text{}$

  $=40\text{π}+2\text{π}{\left(2\right)}^2=48\text{π}$ units.

Conclusion:

The radii two cylinders having same lateral area but different surface areas are 4 units and 2 units and their heights are 5 units and 10 units respectively. The lateral area of first cylinder is $40\text{π}$ units, and surface area $72\text{π}$ units and that of second are $40\text{π}$ units and $48\text{π}$ units.