Question

TSA of a cone that can fit inside a cube of TSA 294 m²​

Answer 1

Given:

Cone inside a cube with Total Surface Area 294 m²​.

To find:

Total Surface Area of cone ?

Solution:

First of all, please refer to the attached image.

It can be observed that:

Height of Cone = Diameter of base of cone = Side of Cube

TSA of Cube = $6 \times side ^2$ = 294 m²​

$\Rightarrow side^2 =49\\\Rightarrow side =7\ m$

Side of cube = 7 m = Height of cone, h = Diameter of cone, 2r

Now, radius of cone = $\frac{7}{2}\ m$

Slant height of cone is given by the formula:

$l =\sqrt{r^{2} +h^{2}}\\\Rightarrow l =\sqrt{(\frac{7}{2})^{2} +7^{2}}\\\Rightarrow l =\sqrt{\frac{49}{4}+49}\\\Rightarrow l =\sqrt{\frac{49\times 5}{4}}\\\Rightarrow l = \dfrac{7}{2}\sqrt5\ m$

Total surface area of a cone is given by the formula:

$TSA = \pi rl+\pi r^2\\\Rightarrow TSA = \dfrac{22}{7} \times \dfrac{7}{2}\times \dfrac{7}{2}\sqrt5+\dfrac{22}{7} \times (\dfrac{7}{2})^2\\\Rightarrow TSA = 11\times \dfrac{7}{2}\sqrt5+11\times \dfrac{7}{2}\\\Rightarrow TSA = \dfrac{77}{2}(\sqrt5+1)\\\Rightarrow TSA = 38.5(2.24+1)\\\Rightarrow TSA = 38.5(3.24)\\\Rightarrow TSA = 124.74\ m^2$

So, TSA of this cone is TSA = 124.74 $m^2$

Answer 2

$Let \: side \: of \: a \:cube = a \: m$

$TSA \: of \: the \: cone = 294 \:m^{2}$

$\implies 6a^{2} = 294$

$\implies a^{2} = \frac{294}{6}$

$\implies a^{2} = 49$

$\blue { a = 7 \: m }$

/* According to the problem given */

A cone fit inside the cube .

$Diameter \: of \: the \: base \: of \: cone (d)$

$= a$

$= 7 \: m$

$base \: radius \: of \: the \:cone (r) = \frac{d}{2}$

$= \frac{7}{2}\:m$

$Height \: of \: the \:cone(h) = a = 7 \:m$

$Slant \: height (l) = \sqrt{r^{2} + h^{2}}$

$= \sqrt{ \big(\frac{7}{2}\Big)^{2} + 7^{2}}$

$=7 \sqrt{ \Big( \frac{1}{4} \Big) + 1 }$

$= 7 \sqrt{ \frac{1+4}{4}}$

$= \frac{7}{2} \sqrt{5}$

$Now, \red{ TSA \: of \: Cone }$

$= \pi r ( r + l )$

$= \frac{22}{7} \times \frac{7}{2} \Big( \frac{7}{2} + \frac{7}{2} \sqrt{5}\Big)$

$= 11\times \frac{7}{2} ( 1+\sqrt{5})$

$= 38.5(1+\sqrt{5})$

$= 38.5( 1 + 2.236)$

$= 38.5 \times 3.236$

$= 124.586 \: m^{2}$

Therefore.,

$\red{ TSA \: of \: Cone } \green { =124.586 \: m^{2}}$

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