Question

The radius and slant height of a cone are in the ratio
4:7. If its curved surface area is 792 cm. find its height.

I want height not radius and pls tell the process​

Answer 1

GiveN :

• Radius & slant height are in ratio 4 : 7
• Curved Surface Area of cone = 792 cm

Here, we're supposed to find the height as it isn't given.

Diagram :

$\setlength{\unitlength}{1.8cm}\begin{picture}(16,4)\thicklines\put(8,1){\line(1,0){1}}\put(8,1){\line(1,2){1}}\put(10,1){\line(-1,2){1}}\put(9,1){\line(0,2){2}}\put(8.4,0.83){\sf{4x m}} \qbezier(8,1)(8.4,1.3)(9,1.3)\qbezier(9,1.3)(9.7,1.3)(10,1)\qbezier(8,1)(8.4,0.6)(9,0.6)\qbezier(9,0.6)(9.7,0.6)(10,1)\put(9.6,2){\sf l = 7x m}\end{picture}$

Let us assume that the the radius of the cone be 4x and the height be 7x respectively.

In order to calculate the height of the cone one should know the Curved Surface Area's formula (as in the case it is given).

We know that,

$\longrightarrow \sf \: CSA \: of \: cone = \pi rl$

Substituting the values,

$\longrightarrow\sf 792 = \dfrac{22}{7}\times4x\times7x$

$\longrightarrow\sf x^2 = \dfrac{792\times7}{22\times4\times7}$

On simplification, we get :

$\longrightarrow\sf x^2 = 9$

$\longrightarrow\sf x = \sqrt{9}$

$\longrightarrow\sf\large{ \boxed{ \sf {x = 3 \:cm}}}$

Now, As per our Assumption's,

• Radius = 4x = 4 × 3 = 12 cm

• Height = 7x= 7 × 3 = 21 cm

As a result,

$\therefore$ Height of the given cone is 21 cm.

Answer 2

Diagram :

$\setlength{\unitlength}{0.99cm}\begin{picture}(6, 4)\linethickness{0.26mm}\qbezier(5.8,2.0)(5.8,2.3728)(4.9799,2.6364)\qbezier(4.9799,2.6364)(4.1598,2.9)(3.0,2.9)\qbezier(3.0,2.9)(1.8402,2.9)(1.0201,2.6364)\qbezier(1.0201,2.6364)(0.2,2.3728)(0.2,2.0)\qbezier(0.2,2.0)(0.2,1.6272)(1.0201,1.3636)\qbezier(1.0201,1.3636)(1.8402,1.1)(3.0,1.1)\qbezier(3.0,1.1)(4.1598,1.1)(4.9799,1.3636)\qbezier(4.9799,1.3636)(5.8,1.6272)(5.8,2.0)\put(0.2,2){\line(1,0){2.8}}\put(3.2,4){\sf{h}}\put(3,2){\line(0,2){4.5}}\put(0.5,1.7){\sf{Radius = 4x\ cm}}\qbezier(.2,2.05)(.7,3)(3,6.5)\qbezier(5.8,2.05)(5.3,3)(3,6.5)\put( - 2,5){\sf{Slant Height = 7x\ cm}}\put(3,2.02){\circle*{0.15}}\put(2.7,2){\dashbox{0.01}(.3,.3)}\end{picture}$

Answer:

Let the radius of the cone be 4x cm and the Slant Height be 7x cm.

☯ $\underline{\boldsymbol{According\: to \:the\: Question\:now :}}$

$:\implies \sf CSA \: of \: Cone = \pi rl$

$:\implies \sf \pi rl = 792 \: {cm}^{2}$

$:\implies \sf \dfrac{22}{7} \times 4x \times 7x = 792$

$:\implies \sf {x}^{2} = \dfrac{792}{22} \times 4$

$:\implies \sf {x}^{2} = 9 \: cm$

$:\implies \underline{ \boxed{\sf x = 3 \: cm}}$

$\: \: \: \: \qquad\tiny\qquad\underline{\frak{ Therefore, the \: radius \: of \: cone \: is :}}$

$\bullet\:\:\textsf{Radius of cone = 4x = 4(3) = \textbf{12 cm}}$