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Ex.1 Calculate the circumference and area of a circle of radius 5·6 cm.
Sol. We have :
Circumference of the circle
Area of the circle
Ex.2 The circumference of a circle is 123·2 cm. Calculate :
(i) the radius of the circle in cm,
(ii) the area of the circle, correct to nearest cm^{2}.
Sol. (i) Let the radius of the circle be r cm.
Then, its circumference = (2πr) cm.
Ex.3 The area of a circle is 301·84 cm^{2}. Calculate:
(i) the radius of the circle in cm.
(ii) the circumference of the circle, correct to nearest cm.
Sol. (i) Let the radius of the circle be r cm.
Then, its area = πr^{2} cm^{2} = 301.84
= 61.6 cm.
Circumference of the circle, correct to nearest cm = 62 cm.
Ex.4 The perimeter of a semicircular protractor is 32·4 cm. Calculate :
(i) the radius of the protractor in cm,
(ii) the area of the protractor in cm^{2}.
Sol. (i) Let the radius of the protractor be r cm.
Radius of the protractor = 6.3 cm.
(ii) Area of the protractor
Area of the protractor = 62·37 cm^{2}.
Ex.5 The area enclosed by the circumferences of two concentric circles is 346.5 cm^{2}. If the circumference of the inner circle is 88 cm, calculate the radius of the outer circle.
Sol. Let the radius of inner circle be r cm.
Then, its circumference = (2πr) cm.
Hence, the radius of the outer circle is 17·5 cm.
Ex.6 Two circles touch externally. The sum of their areas is 130π sq. cm and the distance between their centres is 14 cm. Determine the radii of the circles.
Sol. Let the radii of the given circles be R cm and r cm respectively. As the circles touch externally, distance between their centres = (R + r) cm.
Ex.7 Two circles touch internally. The sum of their areas is 116π sq. cm and the distance between their centres is 6 cm. Find the radii of the given circles.
Sol. Let the radii of the given circles be R cm and r cm respectively. As the circles touch internally, distance between their centres = (R – r) cm.
Ex.8 The wheel of a cart is making 5 revolutions per second. If the diameter of the wheel is 84 cm, find its speed in km/hr. Give your answer, correct to nearest km.
Sol. Radius of the wheel = 42 cm.
Circumference of the wheel =
Distance moved by the wheel in 1 revolution = 264 cm.
Distance moved by the wheel in 5 revolutions = (264 × 5) cm = 1320 cm.
Distance moved by the wheel in 1 second = 1320 cm.
Distance moved by the wheel in 1 hour = (1320 × 60 × 60) cm.
Hence, the speed of the cart, correct to nearest km/hr is 48 km/hr.
Ex.9 The diameter of the driving wheel of a bus is 140 cm. How many revolutions must the wheel make in order to keep a speed of 66 km/hr?
Sol. Distance to be covered in 1 min.
Hence, the wheel must make 250 revolutions per minute.
Ex.10 A bucket is raised from a well by means of a rope which is wound round a wheel of diameter 77 cm. Given that the bucket ascends in 1 min. 28 seconds with a uniform speed of 1·1 m / sec, calculate the number of complete revolutions the wheel makes in raising the bucket.
Sol. Time taken by bucket to ascend = 1 min. 28 sec. = 88 sec. Speed = 1.1 m/sec.
Length of the rope = Distance covered by bucket to ascend
= (1·1 × 88) m = (1·1 × 88 × 100) cm = 9680 cm.
Circumference of the wheel
Hence, the wheel makes 40 revolutions to raise the bucket.
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