(a) Given that P = (\(\frac{rk}{Q} - ms\))\(^{\frac{2}{3}}\) (i) Make Q the subject of the relation; (ii) find, correct to two decimal places, the value of ...

Answer Details

a)

(i) To make Q the subject of the formula, we need to isolate Q on one side of the equation.

P = \(\left(\frac{rk}{Q} - ms\right)^{\frac{2}{3}}\)

Take the reciprocal of both sides:

\(\frac{1}{P}\) = \(\frac{Q}{rk}\) - \(\frac{ms}{rk}\)

Add \(\frac{ms}{rk}\) to both sides:

\(\frac{ms}{rk}\) + \(\frac{1}{P}\) = \(\frac{Q}{rk}\)

Multiply both sides by \(\frac{rk}{ms + Prk}\):

Q = \(\frac{rk}{ms + Prk}\)

(ii) Now that we have the formula for Q, we can substitute the given values and calculate:

Q = \(\frac{(10)(4)}{(0.2) + (3)(15)(4)}\)

Q = \(\frac{40}{182}\)

Q = 0.22 (correct to two decimal places)

Therefore, when P = 3, m = 15, s = 0.2, k = 4, and r = 10, Q is approximately equal to 0.22.

b) To find x : y, we can use algebraic manipulation to isolate one of the variables in terms of the other:

\(\frac{x + 2y}{5}\) = x - 2y

Distribute the 5 on the right side of the equation:

\(\frac{x + 2y}{5}\) = x - \(\frac{10y}{5}\)

Simplify:

\(\frac{x + 2y}{5}\) = x - 2y

x + 2y = 5x - 10y

12y = 4x

y = \(\frac{1}{3}\)x

Therefore, x : y is 3 : 1.