Question

A cylindrical can of soup has a height of $7$ centimeters and a lateral surface area of $63\mathrm{\pi }$ square centimeters. The can is redesigned to have a lateral surface area of $45\mathrm{\pi }$ square centimeters without changing the radius of the can. What is the height of the new design? Justify your answer.

Answer 1

Hint: Remember the formulas of the cylinder.

Step :Find the height of the new design.

The lateral surface area of the cylinder is given by: $S=2\mathrm{\pi rh}$

Where $r$ is the radius of the base and $h$ is the height of cylinder.

To find the radius of can, substitute $S=63\mathrm{\pi }\&\mathrm{h}=7$ into the equation:

$$\begin{gathered} 63\mathrm{\pi }=2\mathrm{\pi r}·7 \\ \frac{63\mathrm{\pi }}{14\mathrm{\pi }}=\mathrm{r} \\ \mathrm{r}=4.5 \end{gathered}$$

Thus, the radius of the cylindrical can is $4.5cm$

Since, radius is not changed in new design, therefore use the same value of radius to find the height of new design cylindrical can:

$S=2\mathrm{\pi rh}$

To find the height in new can:

$$\begin{gathered} 45\mathrm{\pi }=2\mathrm{\pi }\times 4.5\mathrm{h} \\ \mathrm{h}=5 \end{gathered}$$

The height of new design cylindrical can is $5cm$.

Hence,the height of new design cylindrical can is $5cm$.