b is the sum of two quantities, one of which varies inversely as the square of a2 , and the other of which varies directly as a. Identify the relationship between a and b if b = 49 when a = 3 or 5.
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Correct answer: D
Explanation
The correct answer is option 4 i.e b = 8a + 225/a2 .
b = k/a2 + ma
where k and m are constants to be determined.
b = 49 when a = 3 and a = 5, two equations and solve for k and m. This will give the relationship between a and b.
when a = 3, b = 49 ⇒ 49 = k/32 + m × 3 = k/9 + 3m
when a = 5, b = 49:
49 = k/52 + m × 5 = k/25 + 5m
From equation 1: k/9 = 49 − 3m ⟹ k = 9(49 − 3m) = 441 − 27m
Substitute k into equation 2: (441 − 27m)/25 + 5m = 49
⇒ 441 − 27m + 125m = 49 × 25
⇒ 441 + 98m = 1225
⇒ 98m = 1225 − 441 = 784
⇒ m = 784/98 = 8
k using m = 8:
k = 441 − 27 × 8 = 441 − 216 = 225
Hence
b = 225/a2 + 8a