SURFACE AREA AND VOLUME
EXERCISE 13.5
EXERCISE 13.5
Solution: Measures of matchbox (cuboid) is 4 cm × 2.5 cm × 1.5 cm
⇒ l = 4 cm, b = 2.5 cm and h = 1.5 cm
∴ Volume of matchbox = (l × b) × h
= [4 cm × 2.5 cm] × 1.5 cm3
= 15 cm3
⇒ Volume of 12 boxes = 12 × 15 cm3 = 180 cm3
Solution: Here, Length (l) = 6 m
Breadth (b) = 5 m
Depth (h) = 4.5 m
Capacity = l × b × h = 6 × 5 × 4.5 m3
∵1 m3 can hold 1000 l.
∴135 m3 can hold (135 × 1000 l = 135000 l) of water.
∴ The required amount of water in the tank = 135000 l.
Q 3. A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic meters of liquid?
Solution: Length (l) = 10 m
Breadth (b) = 8 m
Volume (v) = 380 m3
Let height of the cuboid be ‘h’.
Since, volume of a cuboid = 1 × b × h
∴ Volume of the cuboidal vessel = 10 × 8 × h m3 = 80h m3
⇒ 80h = 380
Thus, the required height of the liquid = 4.75 m
Q 4. 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.
Solution: Length (l) = 8 m
Breadth (b) = 6 m
Depth (h) = 3 m
∴ Volume of the cuboidal pit = l × b × h = 8 × 6 × 3 m3 = 144 m3
Since, rate of digging the pit is Rs. 30 per m3.
Cost of digging = Rs. 30 × 144 = Rs. 4320
Q 5. The capacity of 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 in.
Solution: Length of the tank (l) = 2.5 m
Depth of the tank (h) = 10 m
Let breadth of the tank = b m
∴ Volume (capacity) of the tank= l × b × h = 2.5 × b × 10 m3
But the capacity of the tank = 50000 l = 50 m3
∴ 25b = 50 m3
Thus, the depth of the tank = 2 m
Q 6. 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?
Solution: Length of the tank (l) = 20 m
Breadth of the tank (b) = 15 m
Height of the tank (h) = 6 m
Volume of the tank = l × b × h = 20 × 15 × 6 m3 = 1800 m3
Since 1 m3 = 1000 l
∴ Capacity of the tank = 1800 × 1000 l = 1800000 l
Village population = 4000
Since, 150 l of water is required per head per day.
∴ Amount of water is required per day = 150 × 4000 l.
Let the required number of days = x
∴ 4000 × 150 × x = 1800000
Thus, the required number of days is 3.
Q 7. A godown measures 60 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.
Solution: Volume of the godown = 60 × 25 × 10 m3
Volume of a crate = 1.5 × 1.25 × 0.5 m3
Q 8. 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.
Solution: Side of the given cube = 12 cm
∴ Volume of the given cube = (side)3 = (12)3 cm3
Side of the smaller cube:
Let the side of the new (smaller) cube = n
⇒ Volume of smaller cube = n3
⇒ Volume of 8 smaller cubes = 8n3
∴ 8n3 = (12)3 = 12 × 12 × 12
Thus, the required side of the new (smaller) cube is 6 cm.
Ratio between surface areas:
Surface area of the given cube = 6 × (side)2 = 6 × 122 cm2 = 6 × 12 × 12 cm2
Surface area of one smaller cube = 6 × (side)2 = 6 × 62 cm2
= 6 × 6 × 6 cm2
∴ Surface area of 8 smaller cubes = 8 × 6 × 6 × 6 cm2
Thus, the required ratio = 1 : 4
Q 9. A river 3 m deep and 40 m wide is flowing at the rate of 2km per how. How much water will fall into the sea in a minute?
Solution: The water flowing in a river can be considered in the form of a cuboid.
Such that Length (l) = 2 km = 2000 m
Breadth (b) = 40 m
Depth (h) = 3 m
∴ Water volume (volume of the cuboid so formed)
= l × b × h = 2000 × 40 × 3 m3
Now, volume of water fallowing in 1 hr (= 60 minutes)
= 2000 × 40 × 3 m3
Volume of water that will fall in 1 minute = [2000 × 40 × 3] ÷ 60 m3
EXERCISE 13.5