Question

A child makes a paper model in the shape of a cylinder surmounted by a right cone. The height of the cylindrical part is 7.5 cm. The base of the model has a diameter of 10 cm and the total height of the model is$19\dfrac{1}{2}{\text{ cm}}$. Find the area of the paper required by the child to make the tent.
$
  {\text{A}}{\text{. 440 c}}{{\text{m}}^2} \\
  {\text{B}}{\text{. 775}}\dfrac{1}{2}{\text{ c}}{{\text{m}}^2} \\
  {\text{C}}{\text{. 880 c}}{{\text{m}}^2} \\
  {\text{D}}{\text{. 1028}}\dfrac{1}{2}{\text{ c}}{{\text{m}}^2} \\
$

Answer 1

Hint: To find the area of paper required by the child to make the tent, we should calculate the total surface area of the cone and the cylinder. We apply the formulae of the surface area of a cylinder and a cone using the data given in the question and add them to obtain the answer.

Complete step-by-step answer:
Given Data,
Height of the cylindrical part h = 7.5 cm
The base of the model has diameter = 10 cm
Height of the model =$19\dfrac{1}{2}{\text{ cm}}$

The paper model made by the child can be segregated into a cylinder and a cone placed on its top. We have been given the height of the cylinder h, the total height of the model is considered to be H, the diameter of the base of the cone, which is also the base of one end of cylinder is taken as d and the curved length of the cone is taken as l.
h = 7.5 cm
H =$19\dfrac{1}{2}{\text{ cm}}$
d = 10 cm, we know diameter d = 2r
Hence the radius of the cone r =$\dfrac{{10}}{2}$= 5 cm
The height of the cone = Total height of the model – height of cylinder = 19.5 – 7.5 = 12 cm.
We make an appropriate figure using all the given data in the question. It looks like the follows:
 

Now we know that the total area of paper required is nothing but the total curved surface area of the cone and the cylinder.

We know the curved surface area of a cylinder is given by the formula S.A = 2πrh
Here the radius of the cylinder is equal to the radius of the cone, i.e. 5 cm.
Height of the cylinder is 7.5 cm
Therefore, the curved surface area of cylinder = 2 × 3.14 × 5 × 7.5 = 235.72 sq cm.

We know the curved surface area of a cone is given by the formula S.A = πrl
Where l is the slant height of the cone. The slant height l of the cone is determined by the formula
${\text{l = }}\sqrt {{{\text{r}}^2} + {{\text{h}}^2}} $, where r and h are the radius and height of the cone respectively.
Here the radius of the cone is 5 cm.
We derived the height of the cone to be 12 cm
Slant height, ${\text{l = }}\sqrt {{{\text{r}}^2} + {{\text{h}}^2}} = \sqrt {{{\text{5}}^2} + {{12}^2}} = 13{\text{ cm}}$
Therefore, the curved surface area of cone = 3.14 × 5 × 13 = 204.28 sq cm.
Hence the total curved surface area of the model is 235.72 + 204.28 = 440 sq cm.
Therefore the area of the paper required by the child to make the tent is 440 sq cm.
Option A is the correct answer.

Note: In order to solve this type of problems the key is to identify that the total area of paper required is the total curved surface area of the cone and cylinder. We should observe the data given in the question carefully because it helps us in finding an unknown variable.
Having adequate knowledge in the formulae of surface areas of geometrical figures like cone and cylinder is essential.
The formula for the slant height of the cone is derived from Pythagora's theorem because the radius and height of the cone are perpendicular to each other and they form a right angle triangle with the slant height.
Area is always in square units. And Sq cm can be expressed as${\text{c}}{{\text{m}}^2}$.