Question

If ‘a’ varies jointly with ‘b’ and ‘c’, and inversely as the square of ‘d’, how would ‘a’ be affected if ‘b’ is tripled and both ‘c’ and ‘d’ are doubled.

Answer 1

When two or more variables vary jointly, their product is always constant. In this case, if 'a' varies jointly with 'b' and 'c' it means that their product is always constant, so abc = k, where k is a constant value.

When a variable 'a' varies inversely as the square of another variable 'd', it means that a*1/d^2 = k, where k is a constant value.

So, if 'b' is tripled, 'c' is doubled and 'd' is doubled, we can see the effect on 'a' by substituting the new values into the equations.

abc = k => a3b2c = k

a1/d^2 = k => a1/(2d)^2 = k

So the effect on 'a' if 'b' is tripled, 'c' is doubled, and 'd' is doubled is that it will be divided by 4.

a3b2c = a1/(2d)^2 => a = k / (3b2c*(2d)^2) = (k/(12bcd^2))

So a = k/(12bcd^2) = a/4.

Therefore, the value of 'a' is decreased by a factor of 4.