Question

Assertion: The total surface area of a cone whose radius is $\frac{r}{2}$ and slant height $2l$ is $\left(\pi \right)r\left(l+\frac{r}{4}\right)$.

Reason: Total surface area of cone is $\pi r(l+r)$ where $r$ is radius and $l$ is the slant height of the cone.

Which of the following is correct?

• A. If both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
• B. If both Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.
• C. If Assertion is correct but Reason is incorrect.
• D. If Assertion is incorrect but Reason is correct.

Answer 1

The correct option is A

If both Assertion and Reason are correct and Reason is the correct explanation of Assertion.

Explanation for correct option

A:

Assertion:

Step 1: Given data

Radius of original cone$=r$

Slant height of original cone$=l$

Radius of new cone$=\frac{r}{2}$

Slant height of new cone$=2l$

New total surface area$=?$

Step 2: Using the formula

Total surface area of cone$=\mathrm{\pi r}l+{\mathrm{\pi r}}^2$

Step 3: Calculating the new total surface area of cone

TSA of cone$=\mathrm{\pi r}l+{\mathrm{\pi r}}^2$

New TSA of cone,

$$\begin{gathered} =\pi \left(\frac{r}{2}\right)\times 2l+\pi {\left(\frac{r}{2}\right)}^2 \\ =\mathrm{\pi r}\left(\frac{2\mathrm{l}}{2}+\frac{\mathrm{r}}{4}\right) \\ =\mathrm{\pi r}\left(\mathrm{l}+\frac{\mathrm{r}}{4}\right) \end{gathered}$$

Thus, the total surface area of a cone whose radius is $\frac{r}{2}$ and slant height $2l$ is $\left(\pi \right)r\left(l+\frac{r}{4}\right)$.

Assertion is correct.

Reason:

Step 1: Given data

Radius of original cone$=r$

Slant height of original cone$=l$

total surface area$=?$

Step 2: Using the formula

Curved surface area of cone$=\mathrm{\pi r}l$

Area of circle$={\mathrm{\pi r}}^2$

Step 3: Calculating the total surface area of cone

CSA of cone$=\mathrm{\pi r}l$

We know, that base of a cone is a circle.

So, TSA of cone$=$CSA of cone$+$Area of circle

TSA of cone$=\mathrm{\pi rl}+{\mathrm{\pi r}}^2$

Taking $\mathrm{\pi }r$common in the above equation

Therefore,

TSA of cone$=\mathrm{\pi r}(l+r)$

Thus, the total surface area of cone is $\pi r(l+r)$ where $r$ is radius and $l$ is the slant height of the cone.

Reason is correct.

Since, Reason is in relation to the Assertion. Thus, Reason is the correct explanation of Assertion.

Hence, option A is correct.