Answer

Verified

53.1k+ views

Hint: Here, we will use the formula of slant height, \[l = \sqrt {{r^2} + {h^2}} \], where \[r\] is the radius of a cone and \[h\] is the height of the cone to find the radius of the base of a cone. Then we will substitute the value of radius in the formula of area of base of cone, \[\pi {r^2}\] to find the required value.

Given that the height \[h\] of a cone is 21 cm and the slant height \[l\] of a cone is 28 cm.

Let the radius of a cone is \[r\].

We know that the slant height is \[l = \sqrt {{r^2} + {h^2}} \], where \[r\] is the radius of a cone and \[h\] is the height of a cone.

Substituting the values of \[r\]and \[l\] in the above formula of \[l\], we get

\[

\Rightarrow 28 = \sqrt {{r^2} + {{21}^2}} \\

\Rightarrow {28^2} = {r^2} + {21^2} \\

\Rightarrow 784 = {r^2} + 441 \\

\]

Subtracting the above equation by 441 on each of the sides, we get

\[

\Rightarrow 784 - 441 = {r^2} + 441 - 441 \\

\Rightarrow 343 = {r^2} \\

\]

Taking the square root on both sides in the above equation, we get

\[

\Rightarrow r = \pm \sqrt {343} \\

\Rightarrow r = \pm 7\sqrt 7 {\text{ cm}} \\

\]

Since the value of the radius can never be negative, so negative value of \[r\] is discarded.

Thus, the radius of the base of a cone is \[7\sqrt 7 \] cm.

We know that the area of the base of a cone is \[\pi {r^2}\], where \[r\]is the radius of the cone.

Substituting the above value of \[r\] in \[\pi {r^2}\] to find the area of the base of cone, we get

\[

{\text{Area of base of a cone}} = \pi {\left( {7\sqrt 7 } \right)^2} \\

= \pi \left( {49 \times 7} \right) \\

\]

Using the value of \[\pi \] in the above equation, we get

\[

\dfrac{{22}}{7}\left( {49 \times 7} \right) = 22 \times 49 \\

= 1078{\text{ c}}{{\text{m}}^2} \\

\]

Thus, the area of the base of a cone is 1078 cm\[^2\].

Note: In this question, students should know the formulae of the slant height of cone, \[l = \sqrt {{r^2} + {h^2}} \], where \[r\] is the radius of a cone and \[h\] is height of cone and the area of the cone, area of the base of a cone is \[\pi {r^2}\], where \[r\]is the radius of the cone properly. Also, we are supposed to write the values properly to avoid any miscalculation.

__Complete step by step answer:__Given that the height \[h\] of a cone is 21 cm and the slant height \[l\] of a cone is 28 cm.

Let the radius of a cone is \[r\].

We know that the slant height is \[l = \sqrt {{r^2} + {h^2}} \], where \[r\] is the radius of a cone and \[h\] is the height of a cone.

Substituting the values of \[r\]and \[l\] in the above formula of \[l\], we get

\[

\Rightarrow 28 = \sqrt {{r^2} + {{21}^2}} \\

\Rightarrow {28^2} = {r^2} + {21^2} \\

\Rightarrow 784 = {r^2} + 441 \\

\]

Subtracting the above equation by 441 on each of the sides, we get

\[

\Rightarrow 784 - 441 = {r^2} + 441 - 441 \\

\Rightarrow 343 = {r^2} \\

\]

Taking the square root on both sides in the above equation, we get

\[

\Rightarrow r = \pm \sqrt {343} \\

\Rightarrow r = \pm 7\sqrt 7 {\text{ cm}} \\

\]

Since the value of the radius can never be negative, so negative value of \[r\] is discarded.

Thus, the radius of the base of a cone is \[7\sqrt 7 \] cm.

We know that the area of the base of a cone is \[\pi {r^2}\], where \[r\]is the radius of the cone.

Substituting the above value of \[r\] in \[\pi {r^2}\] to find the area of the base of cone, we get

\[

{\text{Area of base of a cone}} = \pi {\left( {7\sqrt 7 } \right)^2} \\

= \pi \left( {49 \times 7} \right) \\

\]

Using the value of \[\pi \] in the above equation, we get

\[

\dfrac{{22}}{7}\left( {49 \times 7} \right) = 22 \times 49 \\

= 1078{\text{ c}}{{\text{m}}^2} \\

\]

Thus, the area of the base of a cone is 1078 cm\[^2\].

Note: In this question, students should know the formulae of the slant height of cone, \[l = \sqrt {{r^2} + {h^2}} \], where \[r\] is the radius of a cone and \[h\] is height of cone and the area of the cone, area of the base of a cone is \[\pi {r^2}\], where \[r\]is the radius of the cone properly. Also, we are supposed to write the values properly to avoid any miscalculation.

Recently Updated Pages

In a family each daughter has the same number of brothers class 10 maths JEE_Main

Which is not the correct advantage of parallel combination class 10 physics JEE_Main

If 81 is the discriminant of 2x2 + 5x k 0 then the class 10 maths JEE_Main

What is the value of cos 2Aleft 3 4cos 2A right2 + class 10 maths JEE_Main

If left dfracleft 2sinalpha rightleft 1 + cosalpha class 10 maths JEE_Main

The circumference of the base of a 24 m high conical class 10 maths JEE_Main

Other Pages

Electric field due to uniformly charged sphere class 12 physics JEE_Main

Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main

Vant Hoff factor when benzoic acid is dissolved in class 12 chemistry JEE_Main

Explain the construction and working of a GeigerMuller class 12 physics JEE_Main

If a wire of resistance R is stretched to double of class 12 physics JEE_Main