Question

A solid cone of radius r and height h is placed over a solid cylinder having the same base radius and height as that of a cone. The total surface area of the combined solid is $\mathrm{\pi r}\sqrt{{\mathrm{r}}^2+{\mathrm{h}}^2+3\mathrm{r}+2\mathrm{h}}.$Write $\text{'}true\text{'}$or $\text{'}false\text{'}$ and justify your answer.

• A. True
• B. False

Answer 1

The correct option is B False

According to the given data

When a solid cone is placed over a solid cylinder of the same base radius, the base of the cone and top of the cylinder will not be covered in total surface area.

The height of the cone and cylinder is the same.

We know that,

The total surface area of cone$=\mathrm{\pi rl}+{\mathrm{\pi r}}^2,$ where $r=$base radius and $l=$slant height

The total surface area of the shape formed $=$ Total surface area of cone $+$ Total Surface area of cylinder$-2$(Area of base)

The total surface area of the cylinder $=2\mathrm{\pi rh}+2{\mathrm{\pi r}}^2\mathrm{h},$ where $r=$ base radius and $h=$ height

$$\begin{gathered} =\mathrm{\pi r}(\mathrm{r}+\mathrm{l})+(2\mathrm{\pi rh}+2{\mathrm{\pi r}}^2)-2{\mathrm{\pi r}}^2 \\ ={\mathrm{\pi r}}^2+\mathrm{\pi rl}+2\mathrm{\pi rh}+2{\mathrm{\pi r}}^2-2{\mathrm{\pi r}}^2 \\ =\mathrm{\pi r}(\mathrm{r}+\mathrm{l}+\mathrm{h}) \\ =\mathrm{\pi r}\sqrt{{\mathrm{r}}^2+{\mathrm{h}}^2+2\mathrm{h}} \end{gathered}$$

The given statement is $false$.