Surface Areas and Volumes ⇒⇒ Exercise 11.1

Question 1

Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. (Assume π = 22/7) Find its curved surface area

Solution :

To find the curved surface area of the cone

Given, Diameter of the base = 10.5 cm

Therefore, The radius is half of the diameter.

$$ { { 1 \over 2} × 10.5 } $$

$$⇒ r = {5.25 }cm $$

Slant height of cone, say l = 10 cm

Curved surface area of the cone = $ { \pi}× r × l $

$$ { 22 \over 7} × 5.25 × l0 $$

$$⇒ 165 cm^2 $$

Therefore, The curved surface area of the cone = $ {165 } cm^2 $

Question 2

Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m. (Assume π = 22/7)

Solution :

To find the curved surface area of the cone

Given, Diameter of the base = 24 m

Therefore, The radius is half of the diameter.

$$ { { 1 \over 2} × 24 } $$

$$⇒ r = {12 }m $$

Slant height of cone, say l = 21 m

Total Surface Area of a Cone = $ { \pi}× r × (l + r) $

$$ { 22 \over 7} × 12 × (21 + 12) $$

$$ { 22 \over 7} × 12 × (33) $$

$$⇒ 1244.57 m^2 $$

Therefore, The total surface area of the cone = $ {1244.57 } m^2 $

Question 3

Curved surface area of a cone is 308 $cm^2 $ and its slant height is 14 cm. Find
(i) radius of the base
(ii) Total surface area of the cone. (Assume π = 22/7)

Solution :

To find the (i) radius of the base

Given, Curved surface area of a cone is 308 $ cm^2 $

The slant height of the cone, l = 14 cm

We know the Curved surface area of the cone = $ { \pi}× r × l $

$$ 308 = { 22 \over 7} × r × 14 $$

$$⇒ 308 = 44 × r $$

$$⇒ r = { 308 \over 44} $$

$$⇒ r = 7 cm $$

Therefore, the radius of the circular end of the cone is 7 cm .

(ii) To find the Total surface area of the cone

The total surface area of cone = The Curved surface area of cone + Area of base (${ \pi}× r^2 $)

The total surface area of cone = $$⇒ 308 + { 22 \over 7} × 7 × 7 $$

$$⇒ 308 + ( 22 × 7) $$

$$⇒ 308 + 154 $$

$$⇒ 462 cm^2 $$

Therefore, Total Surface Area of the given cone = $ {462 cm^2 } $

Question 4

A conical tent is 10 m high and the radius of its base is 24 m. Find
(i) Slant height of the tent.
(ii) Cost of the canvas required to make the tent, if the cost of 1 $m^2 $ canvas is Rs 70. (Assume π=22/7)

Solution :

To find the (i) Slant Height of the Tent

Given,Height of conical tent h = 10 m

Radius of conical tent, r = 24m

Let the slant height of the tent be l.

To find the slant height (l), we use the height (h) and the radius (r) of the conical tent, which form a right-angled triangle. We can apply the Pythagorean theorem.

In the right triangle ABO, we have

AB$ ^2 $ = AO $ ^2 $ + BO$ ^2 $(using Pythagoras’ theorem)

$$ ⇒ { l^2 = h^2 + r^2 } $$

$$ ⇒ { l^2 = (10)^2 + (24)^2 } $$

$$ ⇒ { l^2 = (100 + 576 )} $$

$$ ⇒ l = {\sqrt{676 } } $$

$$⇒ l = 26 m $$

Therefore, The slant height of the tent is 26 m.

(ii) To find the Cost of the Canvas

The amount of canvas needed is equal to the curved surface area (CSA) of the Cone. The formula for CSA is:

$$⇒ { \pi}× r × l $$

$$⇒ {{ 22 \over 7} × 24 × 26 } $$

$$⇒ {{ 13728 \over 7} }m^2 $$

cost of canvas is Rs 70 per square meter.

Therefore, cost of canvas = Area of canvas × Rate of canvas

$$⇒ {{ 13728 \over 7} × 70 } $$

$$⇒ Rs. 1,37,280 $$

Therefore, The total cost of the canvas required to make the tent is Rs 1,37,280.

Question 5

What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base radius 6 m ? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20 cm (Assume π=22/7)

Solution :

To find the (i) Slant Height of the Tent

Given,Height of conical tent h = 8 m

Radius of conical tent, r = 6m

Let the slant height of the tent be l.

To find the slant height (l), we use the height (h) and the radius (r) of the conical tent, which form a right-angled triangle. We can apply the Pythagorean theorem.

In the right triangle ABO, we have

AB$ ^2 $ = AO $ ^2 $ + BO$ ^2 $(using Pythagoras’ theorem)

$$ ⇒ { l^2 = h^2 + r^2 } $$

$$ ⇒ { l^2 = (8)^2 + (6)^2 } $$

$$ ⇒ { l^2 = (64 + 36 )} $$

$$ ⇒ l = {\sqrt{100 } } $$

$$⇒ l = 10 m $$

Therefore, The slant height of the tent is 10 m.

Calculate the Lateral Surface Area of the Tents

The lateral surface area is the area of the material needed to make the tent, which is the curved surface of the cone. The formula is:

$$⇒ { \pi}× r × l $$

$$⇒ {{ 22 \over 7} × 6 × 10 } $$

$$⇒ { 188.4}m^2 $$

Determine the Required Length of the Tarpaulin

The area of the tarpaulin must equal the tent's lateral surface area. Since the tarpaulin has a width of 3 m, we can find the required length by dividing the area by the width:

$$ Length = { {Area} \over Width} $$

$$⇒ Length = { {188.4} \over 3} $$

$$⇒ Length = { 62.8}m $$

Therefore, length of tarpaulin to cover the given tent = 62.8 m.

Now, Total required length of tarpaulin = Length of tarpaulin + 20 cm Extra for margins and wastage

The extra length of the material required for stitching margins and cutting is 20 cm = 0.2 m.

So the total length of tarpaulin bought is (62.8 + 0.2) m = 63 m.

Question 6

The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of whitewashing its curved surface at the rate of Rs. 210 per 100 $m^2 $

Solution :

Calculate the Radius

Given, Diameter of the base = 14 m

Therefore, The radius is half of the diameter.

$$ { { 1 \over 2} × 14 } $$

$$⇒ r = {7 }m $$

Slant height of cone, say l = 25 m

Calculate the Curved Surface Area

Curved Surface Area = $ { \pi} × r × l $

$$ { 22 \over 7} × 7 × 25 $$

$$ { 22 × 25 } $$

$$⇒ 550 m^2 $$

Therefore, The area to be whitewashed is the curved surface area of the cone = $ {550 } m^2 $

Calculation of Total Cost of White Washing

Cost of white washing 100 $m^2 $ = 210

So, Cost of white washing 1$m^2 $ = $ { 210 \over 100} $

Therefore, cost of white washing of 550 $m^2 $ = $ { 210 \over 100} × 550 $

= Rs. 1155

Question 7

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

Solution :

Calculate the Slant Height of the Cone

Given,Height of conical cap h = 24 cm

Radius of conical cap, r = 7cm

Let the slant height of the Joker’s cap be l.

To find the slant height (l), we use the height (h) and the radius (r) of the conical Joker’s cap, which form a right-angled triangle. We can apply the Pythagorean theorem.

In the right triangle , we have (using Pythagoras’ theorem)

$$ ⇒ { l^2 = h^2 + r^2 } $$

$$ ⇒ { l^2 = (24)^2 + (7)^2 } $$

$$ ⇒ { l^2 = (576 + 49 )} $$

$$ ⇒ l = {\sqrt{625 } } $$

$$⇒ l = 25 cm $$

Therefore, The slant height of the cap is 25 cm.

Find the Area of a Single Cap

A Joker's cap is open at the bottom, so the required sheet area is the curved surface area of the cone. The formula is:

$$⇒ { \pi}× r × l $$

$$⇒ {{ 22 \over 7} × 7 × 25 } $$

$$⇒ { 22 × 25 } $$

$$⇒ { 550}cm^2 $$

Calculate the Total Area for 10 Caps

Total Area = Area of one cap × 10

Total Area = $$ {10 × 550 } $$

$$⇒ 5500 cm^2 $$

Thus, total area of sheet required to make 10 given caps = 5500 $cm^2$ .

Question 8

A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is Rs. 12 per $m^2 $, what will be the cost of painting all these cones? (Use π = 3.14 and take ${\sqrt{(1.04) } } $ =1.02)

Solution :

Calculate the Slant Height of a Cone

Given,Height of Cone h = 1 m

Radius of Cone, r = ${ {Diameter} \over 2} = { {40} \over 2}$ = 20 cm

We need to convert this to meters to match the height and painting cost units.

r = ${ {20} \over 100}$ = 0.20 m

Let the slant height of the tent be l.

To find the slant height (l),

In the right triangle, (using Pythagoras’ theorem)

$$ ⇒ { l^2 = h^2 + r^2 } $$

$$ ⇒ { l^2 = (0.2)^2 + (1)^2 } $$

$$ ⇒ { l^2 = (0.04 + 1 )} $$

$$ ⇒ { l = {\sqrt{1.04 }}} $$

$$ ⇒ l = {1.02} $$

Calculate the Curved Surface Area of One Cone

curved surface area = $$⇒ { \pi}× r × l $$

$$⇒ {3.14 × 0.2 × 1.02 } $$

$$⇒ {0.64056 }m^2 $$

There are 50 cones, so the total area is the area of one cone multiplied by 50

$$ 50 × 0.64056 $$

$$⇒ 32.028 m^2 $$

Calculate the Total Painting Cost

The cost of painting is Rs. 12 per $m^2$

Total Cost = Total Area × Rate

Total Cost = 32.028 × 12 = 384.336

Therefore, the cost of painting all these cones is Rs. 384.34.(approximately)