If X is directly proportional to the square of Y and inversely proportional to cube of Z If X=20 - Maths - Direct and Inverse Proportions

Question

If X is directly proportional to the square of Y and inversely
proportional to cube of Z If X=20 when Y=10 and Z=5 What is the
value of Y when X=10 and Z=10?

Shridhar Kaushik · Accepted Answer

Given that:â€“ (i) x is proportional to y2 and x is inversely proportional to z3 $∴ \frac{x}{y^{2}} = k_{1} (1) and x \cdot z^{3} = k_{2} (2)$ Also given, x = 20 when y = 10 and z = 5 As we know that â€“ $\frac{x_{1}}{y_{1}^{2}} = \frac{x_{2}}{y_{2}^{2}}$ Put x1 = 20, y1 = 10, we have $\frac{20}{(10)^{2}} = \frac{x_{2}}{(y_{2})^{2}} \Rightarrow \frac{x_{2}}{(y_{2})^{2}} = \frac{1}{5} (3)$ We have to find y, when x = 10 take x2 = 10 in (3), we getâ€“ $\frac{10}{(y_{2})^{2}} = \frac{1}{5} \Rightarrow y_{2}^{2} = 50 \Rightarrow y_{2} = \sqrt{50} \Rightarrow y_{2} = 5 \sqrt{2}$ (ii) Also, $x_{1} = z_{1}^{3} = x_{2} z_{2}^{3}$ Given x1 = 20 when z1 = 5 $∴ 20 \times (5)^{3} = x_{2} z_{2}^{3} \Rightarrow x_{2} z_{2}^{3} = 20 \times 125 (4)$ We have to find y, when when z = 10 Put z2 = 10 in (4), we get â€“ $x_{2} {(10)}^{3} = 20 \times 125 \Rightarrow x_{2} = \frac{20 \times 125}{1000} \Rightarrow x_{2} = \frac{25}{10} = 2 5$ When x2 = 2 5, then from (3), we get â€“ $\frac{2 5}{(y_{2})^{2}} = \frac{1}{5} \Rightarrow {(y_{2})}^{2} = 12 5 \Rightarrow y_{2} = \sqrt{12 5} \Rightarrow y_{2} = 3 54$

If X is directly proportional to the square of Y and inversely proportional to cube of Z. If X=20 when Y=10 and Z=5. What is the value of Y when X=10 and Z=10?