Find the value of R when P = 0.5, given that P is directly proportional to Q and R is inversely proportional to Q squared.

Steps to Solve

1. Determine the constant for P's proportional relationship Since ( P ) is directly proportional to ( Q ), we can express this as: $$ P = k_1 Q $$ To find ( k_1 ), use the given values ( P = 8 ) and ( Q = 2 ): $$ 8 = k_1 \times 2 $$ Solving for ( k_1 ): $$ k_1 = \frac{8}{2} = 4 $$

2. Find the relationship for different values of Q Now that we have the constant, the equation becomes: $$ P = 4Q $$

3. Substituting P to find Q when P is 0.5 We need to find ( Q ) when ( P = 0.5 ): $$ 0.5 = 4Q $$ Solving for ( Q ): $$ Q = \frac{0.5}{4} = 0.125 $$

4. Determine the constant for R's proportional relationship Next, since ( R ) is inversely proportional to ( Q^2 ), we write: $$ R = \frac{k_2}{Q^2} $$ Using the values ( R = 10 ) and ( Q = 3 ), we can find ( k_2 ): $$ 10 = \frac{k_2}{3^2} $$ This simplifies to: $$ 10 = \frac{k_2}{9} $$ Thus: $$ k_2 = 10 \times 9 = 90 $$

5. Construct the formula for R The equation for ( R ) becomes: $$ R = \frac{90}{Q^2} $$

6. Substituting Q to find R when Q is 0.125 Substituting ( Q = 0.125 ): $$ R = \frac{90}{(0.125)^2} $$ Calculating ( (0.125)^2 ): $$ (0.125)^2 = 0.015625 $$ Now substituting in: $$ R = \frac{90}{0.015625} $$

7. Final calculation for R Now perform the division: $$ R = 90 \div 0.015625 = 5760 $$