## Surface Areas and Volumes

NCERT Exercise 13.5

Q1: A matchbox measures 4 cm × 2.5 cm × 1.5 cm. What will be the volume of a packet containing 12 such boxes?

Answer: Given, l = 4 cm, b = 2.5 cm, h = 1.5 cm
Volume of 1 matchbox = lbh
= 4 × 2.5 × 1.5 cm3 = 15 cm³
Volume of 12 matchboxes = 15 × 12  = 180 cm³ (Answer)

Q2: A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many litres of water can it hold? (1 m³ = 1000 l)

Answer: Given,  l = 6 m, b = 5 m, h = 4.5 m
Volume of the tank = lbh
= 6 × 5 × 4.5 = 135 m³
= 135 × 1000 litres = 1,35,000 litres (Answer)

Q3: A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres of a liquid?

Answer: Here, l = 10 m, b = 8 m, h = ?
Volume of the vessel = lbh
⇒ 380 = 10 × 8 × h
⇒  h = 380 / (10 × 8) = 4.75 m (Answer)

Q4: Find the cost of digging a cuboidal pit 8 m long, 6 m broad and 3 m deep at the rate of Rs 30 per m³.

Answer: Given, l = 8 m, b = 6 m, h = 3 m
Volume of the pit = lbh
= 8 × 6 × 3  = 144 m³

Cost of digging 1m³ = ₹ 30
∴ Cost of digging 144 m³ = ₹ 30 × 144 = ₹ 4320 (Answer)

Q5: The capacity of a cuboidal tank is 50000 litres of water. Find the breadth of the tank, if its length and depth are respectively 2.5 m and 10 m.

Answer: Given, l = 2.5 m, h = 10 m, b = ?

Capacity of the tank = 50000 lites = 50000/1000 = 50 m³
∵ capacity of the tank = lbh
⇒ 50 = 2.5 × b × 10
⇒ b = 50/25 = 2m

Q6: A village, having a population of 4000, requires 150 litres of water per head per day. It has a tank measuring 20 m × 15 m × 6 m. For how many days will the water of this tank last?

Answer: Given, l = 20 m, b = 15 m, h = 6 m
Population of the village = 4000
Water consumed by 1 person in 1 day = 150 litres
∴ Water consumed by 4000 persons in 1 day = 4000 × 150 litres
= 4000 × 150/ 1000 = 600 m³
∵ capacity(volume) of the tank = lbh
= 20 × 15 × 6 m³
∴ Required number of days = Volume of the tank ÷ Water consumed in 1 day
= 20 × 15 × 6 / 600

Q7: A godown measures 40 m × 25 m × 10 m. Find the maximum number of wooden crates each measuring 1.5 m × 1.25 m × 0.5 m that can be stored in the godown.

Answer: Volume of the godown = 40 × 25 × 10 m³
Volume 1 wooden crate = 1.5 × 1.25 × 0.5 m³
∴ Required number of crates = Volume of the godown / Volume of 1 crate
= (40 × 25 × 10) / (1.5 × 1.25 × 0.5)
= 10666.67
Thus, the maximum number of wooden crates that can be stored in the godown = 10666 (Answer)

Q8: A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.

Answer: Here, a = 12 cm
Volume of the cube = a³ = (12)³ cm³ = 1728 cm³

Now, volume of 1 smaller cube = 1728/8 = 216 cm³

Let side of the new cube be A.
Then A³ = 216
⇒ A = ∛216 = 6
Hence, side of the new cube = 6 cm (Answer)

Total surface area of the bigger cube = 6 a²
= 6 × (12)² cm² = 6 × 12 × 12 cm²

Total surface area of 1 smaller cube = 6A²
= 6 × 6² cm² = 6 × 6 × 6 cm²

Hence, required ratio = (6 × 12 × 12) / (6 × 6 × 6) = 4/1

Q9: A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute ?

Answer: Given, b = 40 m, h = 3 m, l = 2 km = 2000 m
Volume of water flowing through the river in 1 hour
= lbh = 2000 × 40 × 3 m³
∴ Volume of water flowing through the river in 1 minute
= (2000 × 40 × 3) / 60  