Exercise 10.1

1. Find the lateral surface area and total surface area of the following right prisms.

Solution:

(i) Cube with side 4 cm

Lateral Surface Area (LSA): 4 × side2side3 = 4 × ( cm)^2 = 4 × cm^2 = cm^2

Total Surface Area (TSA): 6 × side2side3 = 6 × ( cm)^2 = 6 × 16 cm^2 = cm^2

(ii) Cuboid with dimensions 8 cm x 6 cm x 5 cm

Solution:

Lateral Surface Area (LSA): 2h(l + b)2(lb + bh + hl) = 2 × cm × ( cm + cm) = 10 cm × 14 cm = cm2

Total Surface Area (TSA): 2(lb + bh + hl)2h(l + b) = 2( × + × + × ) cm^2 = 2( + + ) cm2 = 2 × cm2 = cm2

2. The total surface area of a cube is 1350 sq.m. Find its volume.

Solution:

Given,

Total Surface Area ((TSA)) of cube: 6 × side2 = m2

side2 = 1350 m2 / = m2

side = √225 m√225 m2 = m

Volume of cube: side3 = 15m3 = m3

3. Find the area of four walls of a room (Assume that there are no doors or windows) if its length 12 m., breadth 10 m. and height 7.5 m.

Solution:

Area of four walls (LSA): 2h(l + b) = 2 × m × ( m + m)

= 15 m × 22 m = m^2

4. The volume of a cuboid is 1200 cm3. The length is 15 cm. and breadth is 10 cm. Find its height.

Solution:

Volume of cuboid: l × b × h2h(l + b) = cm3

cm × cm × h = 1200 cm3

cm2 × h = 1200 cm3

h = 1200 cm3 / 150 cm2 = cm

5. How does the total surface area of a box change if

(i) Each dimension is doubled?

Solution:

Let the original dimensions be l, b, h.

Original TSA = 2(lb + bh + hl)2h(l + b)

New dimensions: 2l, 2b, 2hl2, b2, h2

New TSA = 2( + + ) = × 2(lb + bh + hl)

The TSA becomes times the original.

(ii) Each dimension is tripled?

Solution:

Let the original dimensions be l, b, h.

Original TSA = 2(lb + bh + hl)

New dimensions: 3l, 3b, 3hl3, b3, h3

New TSA = 2(3l × 3b + 3b × 3h + 3h × 3l) = 2( + + l) = × 2(lb + bh + hl)

The TSA becomes times the original.

If each dimension is raised to n times:

TSA becomes n2n3 times the original.

6. The base of a prism is triangular in shape with sides 3 cm., 4 cm. and 5 cm. Find the volume of the prism if its height is 10 cm.

Solution:

The base is a right-angled triangle (32 + 42 = 5272).

Area of base = (1/2) × 3 cm × 4 cm = cm2

Volume of prism: Area of base × height = 6 cm2 × 10 cm = cm3

7. A regular square pyramid is 3 m. height and the perimeter of its base is 16 m. Find the volume of the pyramid.

Solution:

Perimeter of base = m

Side of base = 16m416m2 = m

Area of base = side2 = (4 m4 cm)^2 = m^2

Volume of pyramid: (1/3) × Area of base × height = (1/3) × 16 m^2 × 3 m = m3

8. An Olympic swimming pool is in the shape of a cuboid of dimensions 50 m. long and 25 m. wide. If it is 3 m. deep throughout, how many liters of water does it hold? (1 cu.m=1000 liters)

Solution:

Volume of cuboid: l × b × h = 50 m × 25 m × 3 m = m3

Liters of water: 3750 m3 × 1000 liters/m3 = liters