Question

The semi-vertical angle of a cone is $${45}^{\circ }$$. If the height of the cone is 20.025, then its approximate lateral surface area is
A. $$401\sqrt{2}\pi$$
B. $$400\sqrt{2}\pi$$
C. $$399\sqrt{2}\pi$$
D. None of these

Answer 1

Hint: First of all, find the slant height of the cone and relation between radius and height of the cone by the given semi-vertical angle. Now, the approximate lateral surface area of the cone is given by $$S+\mathrm{\Delta }S$$ where $$S$$ is the LSA of the cone. So, use this concept to reach the solution of the given problem.

Complete step-by-step answer:
Let $$r$$ be the radius, $$h$$ be the height and $$l$$ be the slant height of a cone of semi-vertical angle $${45}^{\circ }$$.

From the figure,
$$\begin{gathered} \Rightarrow \tan {45}^{\circ }={\displaystyle \frac{r}{h}} \\ \Rightarrow 1={\displaystyle \frac{r}{h}} \\ \therefore r=h \end{gathered}$$
We know that $$l=\sqrt{r^2+h^2}$$.
So, slant height of the given cone is
$$\begin{gathered} \Rightarrow l=\sqrt{r^2+h^2} \\ \Rightarrow l=\sqrt{h^2+h^2}=\sqrt{2h^2}\left[\because r=h\right] \\ \therefore l=\sqrt{2}h \end{gathered}$$
Let $$h=20$$ and $$h+\mathrm{\Delta }h=20.0025$$
So, $$\mathrm{\Delta }h=20.025-20=0.025$$
We know that the lateral surface area of the cone with radius $$r$$ and slant height $$l$$ is given by $$S=\pi rl$$.
So, the lateral surface area of the given cone is
$\Rightarrow$ $$S=\pi h\sqrt{2}h=\sqrt{2}\pi h^2\left[\because r=h\text{ and }l=\sqrt{2}h\right]$$
The approximate lateral surface area of the cone is given by $$S+\mathrm{\Delta }S$$.
 Now, consider $$\mathrm{\Delta }S$$
$$\begin{gathered} \Rightarrow \mathrm{\Delta }S={\left({\displaystyle \frac{ds}{dh}}\right)}_{h=20}\mathrm{\Delta }h \\ \Rightarrow \mathrm{\Delta }S={\left[{\displaystyle \frac{d}{dh}}\left(\sqrt{2}\pi h^2\right)\right]}_{h=20}\mathrm{\Delta }h \\ \Rightarrow \mathrm{\Delta }S={\left[2\sqrt{2}\pi h\right]}_{h=20}\mathrm{\Delta }h \\ \Rightarrow \mathrm{\Delta }S=\left[40\sqrt{2}\pi \right]\times 0.025\left[\because \mathrm{\Delta }h=0.025\right] \\ \therefore \mathrm{\Delta }S=\sqrt{2}\pi \end{gathered}$$
And
$$\begin{gathered} \Rightarrow S=\sqrt{2}\pi h^2 \\ \Rightarrow S=\sqrt{2}\pi {\left(20\right)}^2\left[\because h=20\right] \\ \therefore S=400\sqrt{2}\pi \end{gathered}$$
Hence the approximate value of lateral surface area of the cone is
$$S+\mathrm{\Delta }S=400\sqrt{2}\pi +\sqrt{2}\pi =401\sqrt{2}\pi$$
Thus, the correct option is A. $$401\sqrt{2}\pi$$

Note: The semi-vertical angle of the cone is the angle between the height and slant height of the cone. The slant height of the cone of radius $$r$$ and height $$h$$ is given by $$l=\sqrt{r^2+h^2}$$. The lateral surface area of the cone with radius $$r$$ and slant height $$l$$ is given by $$S=\pi rl$$.