Geometry

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Question 1

Find the volumes of these objects.

(a)

So

volume = 50.265 cm2 × 10 cm = 502.65 cm3.

Thus the volume is 503 cm3 (to the nearest cubic centimetre).

(If you used the approximate value of 3.14 for , you will have got a cross-sectional area of 50.24 cm2 and a volume of 502.4 cm3.)

• (b)

So

volume = 37.5 m2 × 10 m = 375 m3.

Question 2

Two car manufacturers both claim that their models have an engine capacity of 2 litres. The two models have four-cylinder, four-stroke engines.

The table below shows the details of the four cylinders.

Car model Cylinder diameter (bore)/mm Cylinder height (stroke)/mm Number of cylinders
A 86 86 4
B 92 75 4

By working out the total volume of the four cylinders for each model in cm3, find out if the manufacturers’ claims are true.

(Hint: 1 litre = 1000 cm3.)

Car A has four cylinders, each with a radius of 4.3 cm and a height of 8.6 cm. The volume of one cylinder is calculated by using the formula

So, the four cylinders will have

Car B has four cylinders, each with a radius of 4.6 cm and a height of 7.5 cm. From the same formula, the four cylinders will have

Therefore, both engines have a cubic capacity very close to 2000 cm3. They are both said to have two-litre engines. Hence the claims of both manufacturers are true.

Question 3

The guttering pictured here has a semicircular cross-section. Find the volume of water that the guttering will hold when full.

The cross-section of the guttering is a semicircle of radius 0.05 m. So

Then, since the length of the guttering is 12m,

Therefore the guttering will hold about 0.047 m3 of water.

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