Question

Q.1. A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius on its circular face. The total height of the toy is 15.5 cm. Find the total surface area of the toy​

Answer 1

Answer:

Step-by-step explanation:

$\bf\underline{Given:-}$

Radius of cone = 3.5 cm

Height of cone = 15.5 - 3.5 = 12 cm

$\bf\underline{To \ Find:-}$

Total Surface Area of Toy

$\bf\underline{Solution:-}$

Slant Height of cone = $\sf \sqrt{r^2+h^2}$

$\sf\implies Slant \ Height=\sqrt{12.25+144}$

$\sf\implies Slant \ Height=\sqrt{156.25}$

$\bf\implies Slant \ Height=12.5 \ cm$

$\sf\implies Curved \ Surface \ Area=\pi rl$

$\sf\implies Curved \ Surface \ Area=\frac{22}{7} \times3.5\times12.5$

$\bf\implies Curved \ Surface \ Area=137.5 \ cm^2$

$\sf\implies Curved \ Surface \ Area \ of \ Hemispherical \ Portion=2\pi r^2$

$\sf\implies Curved \ Surface \ Area  \ of  \ Hemispherical \ Portion=2\times\frac{22}{7}\times(3.5)^2$

$\bf\implies Curved \ Surface \ Area \ of \ Hemispherical \ Portion=77 \ cm^2$

$\bf Total \ Surface \ Area= 137.5 + 77 = 214.5 \ cm^2$

Answer 2

$\bold{\underline{\underline{\huge{\sf{AnsWer:}}}}}$

Total surface area of the toy is 214.5 cm²

$\bold{\underline{\underline{\large{\mathfrak{StEp\:by\:stEp\:explanation:}}}}}$

GiVeN :

• We have a toy in the shape of a cone which is mounted on a hemisphere of same radius like the cone on it's circular face.

To FiNd :

• The Total surface area of the toy

SoLuTiOn :

Hemisphere :

• Radius = 3.5 cm

Cone :

• Radius = 3.5 cm
• Height of the cone,

$\hookrightarrow$ $\sf{Total\:height\:of\:the\:toy\:-\:Radius\:of\:hemisphere}$

Block in the values,

$\hookrightarrow$$\sf{15.5\:-\:3.5}$

$\hookrightarrow$$\sf{12\:cm}$

•°• Height of the cone = 12 cm

The total surface area of the toy will be the sum of the curved surface area of the hemisphere and the curved surface area of the cone.

CuRvEd SuRfAcE aReA oF hEmIsPhErE :

Formula :

$\bold{\large{\boxed{\rm{\purple{CSA\:=\:2\:\pi\:r^2}}}}}$

Block in the values,

$\hookrightarrow$ $\sf{2\:\times\:{\dfrac{22}{7}\:\times\:3.5\:\times\:3.5}}$

$\hookrightarrow$ $\sf{\dfrac{44\:\times\:12.25}{7}}$

$\hookrightarrow$ $\sf{\dfrac{539}{7}}$

$\hookrightarrow$ $\sf{77}$

•°• Curved Surface Area of the hemisphere is 77 cm²

CuRvEd SuRfAcE aReA oF cOnE :

FoRmUlA :

$\bold{\large{\boxed{\rm{\pink{CSA\:=\:\pi\:r\:l}}}}}$

Where,

• r = radius of the cone = 3.5 cm
• l = slant height of the cone

We will first calculate the slant height of the cone.

Formula :

$\bold{\large{\boxed{\rm{\orange{l^2\:=\:r^2\:+\:h^2}}}}}$

Block in the values,

$\hookrightarrow$ $\sf{l^2\:=\:(12)^2\:+\:(3.5)^2}$

$\hookrightarrow$ $\sf{l^2\:=\:144\:+\:12.25}$

$\hookrightarrow$ $\sf{l^2\:=\:156.25}$

$\hookrightarrow$ $\sf{l\:=\:{\sqrt{156.25}}}$

$\hookrightarrow$ $\sf{l\:=\:12.5}$

•°• Slant height = 12.5 cm

Now, we can find the curved surface area of the cone by blocking the given data in the formula for the curved surface area.

$\hookrightarrow$ $\sf{\dfrac{22}{7}\:\times\:3.5\:\times\:12.5}$

$\hookrightarrow$ $\sf{\dfrac{22\:\times\:43.75}{7}}$

$\hookrightarrow$ $\sf{\dfrac{962.5}{7}}$

$\hookrightarrow$ $\sf{137.5}$

•°• Curved surface area of the cone is 137.5 cm²

Add the curved surface area of the hemisphere and the cone to get the total surface area of the toy.

Total surface area of the toy,

$\hookrightarrow$ $\sf{77\:+\:137.5}$

$\hookrightarrow$ $\sf{214.5}$

•°• Total surface area of the toy is 214.5 cm²