(a) The scale of a map is 1 : 20,000. Calculate the area, in square centimetres, on the map of a forest reserve which covers 85(km^{2}).
(b) A rectangular playing field is 18m wide. It is surrounded by a path 6m wide such that its area is equal to the perimeter of the path. Calculate the length of the field.
(c) The diagram shows a circle centre O. If < POQ = x°, the diameter of the circle is 7 cm and the area of the shaded portion is 27.5(cm^{2}). Find, correct to the nearest degree, the value of x. [Take (pi = frac{22}{7})].
Explanation
(a) 1km = 100,000 cm.
Area = (frac{85 times 100,000 times 100,000}{20,000 times 20,000})
= (85 times 5 times 5)
= (2125 cm^{2})
(b)
Perimeter of path = (2(length + width))
= (2(30 + (12 + a)) = 2(42 + a))
= (84 + 2a)
Area of field = (length times width)
= (18 times a)
= 18a
(therefore 18a = 84 + 2a)
(18a – 2a = 84 implies 16a = 84)
(a = frac{84}{16})
= (5.25 cm)
(c) Area of major sector = (frac{360 – x}{360} times pi r^{2})
(27.5 = frac{360 – x}{360} times frac{22}{7} times frac{7}{2} times frac{7}{2})
(27.5 = frac{360 – x}{360} times frac{77}{2})
(360 – x = frac{27.5 times 360 times 2}{77})
(360 – x = 257.14)
(x = 360° – 257.14° = 102.86°)
(x approxeq 103°) (to the nearest degree).