Volume and Surface Area



Volume and Surface Area

1. The diagram below shows a wooden block. AB is parallel to CD, the length

and height of the block is 20 cm and 8 cm respectively. The length of AC and

BD is the same.

a) Find the volume of the wooden block in cm3.

b) If the total surface area of the block is 1044 cm2, find the length of AC.

[pic]

Solution:

(a) Volume of wooden block [pic]

[pic]

( 1440 cm3

(b) Total surface area ( 1044 cm2

[pic]

[pic]

[pic]

( 540

AC ( 540 ( 40

( 13.5 cm

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2. The floor of a rectangular dining hall measures 15 m by 8 m and the height of the hall is 4 m. A contractor is hired to paint the four walls of the hall.

a) How many litres of paint is required to paint the four walls if one litre

of paint can cover 12 m2?

b) The paint is sold in 5-litre containers. Find the least number of containers of paint to be bought to paint the four walls completely.

Solution:

(a) Area of 4 walls [pic]

( 184 m2

(amount of paint required ( 184 ( 12

( [pic]

b) [pic]( [pic]

Hence, the least number of containers of paint to be bought is 4.

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3. The figure below shows a solid triangular prism. All dimensions in the diagram are in centimetres.

a) Find the total surface area of the prism.

b) If the density of the prism is 2 g/cm3, find the mass of the prism in kg.

[pic]

Solution:

(a) Total surface area [pic]

( 1000 ( 2850

( 3850 cm2

b) Volume of prism [pic]

( 12 500 cm3

(mass of prism ( volume ( density

( 12 500 ( 2

( 25 000 g

( 25 kg

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4. A cylindrical container of radius 40 cm and height 1.2 m weighs 70 kg when empty. Find, to the nearest kg, the mass of the container when it is completely filled with oil if the density of oil is 0.65 g/cm[pic]. (Take ( ( [pic].)

Solution:

Mass of oil ( volume ( density

( ( ( 402 ( 120 ( 0.65

( 392 228.5714 g

( 392.228 571 4 kg

Mass of container when completely filled with oil

( 392.228 571 4 ( 70

( 462 kg (correct to the nearest kg)

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5. The diagram on the left shows the cross sectional view of an antique coin. The diameter of the coin is 4 cm and the dimensions of the square hollow in the middle is 1 cm by 1 cm. The thickness of each coin is 2 mm. 15 such coins are stacked together before they are tied together with strings as shown by the dark lines in the diagram on the right.

Find, in terms of (,

a) the volume of the stacked coins in cm3,

b) the total surface area of the stacked coins in cm2,

c) the minimum length of string needed (assume length of knot is negligible).

[pic]

Solution:

(a) Volume of one coin [pic]

[pic]cm3

( volume of 15 coins [pic]

( 3(4( ( 1)

( (12( ( 3) cm3

(b) Surface area 4 sides of the hollow for 15 coins

( 15 ( (1 ( 1 ( 1 ( 1) [pic]

( 12 cm2

Curved surface area of 15 coins [pic]

[pic] cm2

Area of top and bottom [pic]

[pic]

[pic] cm2

(total surface area ( 12 ( 12( ( 8( ( 2

( 10 ( 20( cm2

c) Minimum length of string needed [pic]

[pic]

[pic]cm

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6. The dimensions of an open rectangular tank is 75 cm by 40 cm by 60 cm high. A solid metal cylinder of volume 37 500 cm3 is placed in the tank and 42 000 cm3 of water is poured into the tank to just submerge the cylinder.

Find

a) the radius of the cylinder,

b) the length of the cylinder, correct to 3 significant figures,

c) the total surface area of the cylinder, correct to 3 significant figures.

(Take ( ( 3.14.)

[pic]

Solution:

(a) Total volume of water and cylinder[pic]

[pic]cm3

Base area of tank [pic]

( 3000 cm2

( diameter of cylinder ( 79 500 ( 3000

( 26.5 cm

Radius of cylinder ( 26.5 ( 2

( 13.25 cm

b) Let length of cylinder be l.

Volume of cylinder ( 37 500 cm3

[pic]

[pic]

l ( 68.0252

( 68.0 cm (correct to 3 sig. fig.)

c) Total surface area of cylinder [pic] [pic]

[pic]

( 6760 cm2 (correct to 3 sig. fig.)

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7. A tank of length 50 cm and width 25 cm is first filled with water to a depth of 15 cm. A solid metal cube is then placed at the centre of the tank and the depth of water rises by x cm. If the metal cube weighs 1.715 kg and its density is 5 g/cm3, find

a) the volume of the metal cube,

b) the rise in the water level in mm,

c) the surface area of the tank in contact with the water correct to 2 decimal places.

[pic]

Solution:

(a) Mass of cube ( 1.715 kg

[pic]g

Density of cube ( 5 g/cm3

(volume of cube ( 1715 ( 5

( 343 cm3

b) Base area of tank [pic]

( 1250 cm2

(rise in water level ( 343 ( 1250

( 0.2744 cm

( 2.744 mm

c) Length of cube [pic] ( 7 cm

( base area ( 72

( 49 cm2

(surface area of tank in contact with the water

( (1250 ( 49) ( (50 ( 25 ( 50 ( 25)(15 ( 0.2744)

( 1201 ( 150(15.2744)

( 3492.16 cm2 (correct to 2 d.p.)

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