Question

A cylinder has radius $r$ and height $h$

(a) How many times greater is the surface area of a cylinder when both dimensions are multiplied by $2$?, $3$?, $5$?, $10$?

(b) Describe the pattern in part (a). Write an expression for the surface area of the cylinder when both dimensions are multiplied by a number.

Answer 1

Hint:

Step 1: (a) The surface area of a cylinder $=2{\mathrm{\pi r}}^2+2\mathrm{\pi rh}$

If both the dimensions are multiplied by $2$ then, Radius $=2r$, Height $=2h$

Then surface area of cylinder $=2\times \mathrm{\pi }\times 2\mathrm{r}\times 2\mathrm{r}+2\times \mathrm{\pi }\times 2\mathrm{r}\times 2\mathrm{h}$

Then surface area of cylinder $=8\mathrm{\pi }{r}^2+8\mathrm{\pi rh}$

$=8\mathrm{\pi r}(\mathrm{r}+\mathrm{h})$

Step 2: The surface area of a cylinder $=2{\mathrm{\pi r}}^2+2\mathrm{\pi rh}$

If both the dimensions are multiplied by $3$ then, Radius $=3r$, Height $=3h$

Then surface area of cylinder $=2\times \mathrm{\pi }\times 3\mathrm{r}\times 3\mathrm{r}+2\times \mathrm{\pi }\times 3\mathrm{r}\times 3\mathrm{h}$

Then surface area of cylinder $=18\mathrm{\pi }{r}^2+18\mathrm{\pi rh}$

$=18\mathrm{\pi r}(\mathrm{r}+\mathrm{h})$

Step 3: The surface area of a cylinder $=2{\mathrm{\pi r}}^2+2\mathrm{\pi rh}$

If both the dimensions are multiplied by $5$ then, Radius $=5r$, Height $=5h$

Then surface area of cylinder $=2\times \mathrm{\pi }\times 5\mathrm{r}\times 5\mathrm{r}+2\times \mathrm{\pi }\times 5\mathrm{r}\times 5\mathrm{h}$

Then surface area of cylinder $=50\mathrm{\pi }{r}^2+50\mathrm{\pi rh}$

$=50\mathrm{\pi r}(\mathrm{r}+\mathrm{h})$

Step 4: The surface area of a cylinder $=2{\mathrm{\pi r}}^2+2\mathrm{\pi rh}$

If both the dimensions are multiplied by $10$ then, Radius $=10r$, Height $=10h$

Then surface area of cylinder $=2\times \mathrm{\pi }\times 10\mathrm{r}\times 10\mathrm{r}+2\times \mathrm{\pi }\times 10\mathrm{r}\times 10\mathrm{h}$

Then surface area of cylinder $=200\mathrm{\pi }{r}^2+200\mathrm{\pi rh}$

$=200\mathrm{\pi r}(\mathrm{r}+\mathrm{h})$

(c) The surface area of the cylinder when both dimensions are multiplied by a number $n=2\times n^2\times \mathrm{\pi r}(\mathrm{r}+\mathrm{h})$