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Mathematics Theory (a) A boy blew his rubber balloon to a spherical shape. The balloon burst when…

(a) A boy blew his rubber balloon to a spherical shape. The balloon burst when its diameter was 15 cm. Calculate, correct to the nearest whole number, the volume of air in the balloon at the point of bursting. [Take (pi = frac{22}{7})]

(b) A point X is on latitude 28°N and longitude 105°W. Y is another point on the same latitude as X but on longitude 35°E. (i) Calculate, correct to three significant figures, the distance between X and Y along latitude 28°N ; (ii) How far is X from the equator? [Take (pi = frac{22}{7}) and radius of the earth = 6,400km].

Explanation

(a) (d = 15 cm implies r = frac{15}{2} cm)

(V = frac{4pi r^{3}}{3})

(V = frac{4}{3} times frac{22}{7} times (frac{15}{2})^{3})

(V = frac{1125 times 22}{7 times 2})

(V = 1,767.857 cm^{3})

(b)(i)    Longitude difference : 105° + 35° = 140°

Distance along the parallel of latitude = (frac{140}{360} times 2 times frac{22}{7} times 6400 cos 28)

= (frac{140 times 44 times 6400 times 0.8829}{360 times 7})

= (13812.48 km)

(approxeq 13800 km)

(ii) Distance from the equator : (frac{28}{360} times 2 times frac{22}{7} times 6400)

= (3128.89 km)

(approxeq 3130 km)