Question

A joker's cap is in the form of a right circular cone whose base radius is 7 cm and height is 24 cm. Find the area of the sheet required to make 10 such caps.

Answer 1

Hint: In this problem we have to find the area of the sheet required to make 10 caps. We will find the curved surface area of 1 cap and then multiply it with 10. Here the height and radius of the cap is given and we will find the slant height of the cone with the help of Pythagoras theorem and then apply them in the formula of curved surface area and find the required value.

Formula used:
Curved surface area of right circular cone \[ = \pi rl\]
where, $r$ is the radius of base of the cone, $l$ is the slant height of the cone and $\pi$ is a constant.

Complete step by step solution:
Here the cap is in the form of a right circular cone. Therefore we are going to find the curved surface area of the right circular cone.
It is given that the radius of circular cone \[r = 7{\text{ }}cm\]and height of circular cone \[h = 24{\text{ }}cm\]
Let \[l\] be the slant height

Here the line h, r and l forms a right angled triangle, with hypotenuse $l$, $h$ and $r$ the opposite and adjacent sides,
Now we are going to apply Pythagoras theorem which states that “the sum of square of adjacent and opposite side of a triangle is equal to square of the hypotenuse”
Therefore we get, \[l = \sqrt {{r^2} + {h^2}} \]
Let us substitute the value of $h$ and $r$ in the above equation we get,
$\Rightarrow$\[l = \sqrt {{{\left( {24} \right)}^2} + {{\left( 7 \right)}^2}} \]
On the squaring the terms inside the square root we get,
$\Rightarrow$\[l = \sqrt {576 + 49} \]
On further solving we get the value of $l$ as,
$\Rightarrow$\[l = 25{\text{ }}cm\]
Hence we have found $l$, $h$ and $r$, let us substitute it in the curved surface area of the cone.
Curved surface area of right circular cone \[ = \pi rl\]
Let us substitute $l$, $r$ and \[\pi \] in the above equation the we get,
$\Rightarrow$\[ \dfrac{{22}}{7} \times 7 \times 25\]
The curved surface area of right circular cone \[ = 550{\text{ }}c{m^2}\]
Hence the joker’s cap is in the form of a right circular cone. Therefore the area value is equal to the surface area of one joker’s cap.
So,
Curved surface area of 1 joker’s cap \[ = 550{\text{ }}c{m^2}\]
To find the curved surface area of 10 caps we should multiply the curved surface area of one cap with 10.
That is curved surface area of 10 jokers cap \[ = 10 \times 550\]
\[ = 5500{\text{ }}c{m^2}\]

$\therefore$ The sheet required to make 10 jokers cap should have the area = \[5500{\text{ }}c{m^2}\]

Note: We should not add or multiply ratios directly. So we use $x$ as the common multiplier. Without using the common multiply we may get the correct answer. But the use of common multiply makest the solution more clear to understand by students.