This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Hey, good to see you again. Here are the solutions to your problems. Problem 1: P varies directly as V and inversely as the square root of W. Step 1: Write the initial relationship. P = k (V)/(sqrt(W)) where k is the constant of proportionality. Step 2: Express the new values of V and W. V is decreased by 10%, so V_new = V - 0.10V = 0.90V. W is increased by 21%, so W_new = W + 0.21W = 1.21W. Step 3: Calculate the new value of P (P_new). P_new = k V_newsqrt(W_new) = k (0.90V)/(sqrt(1.21W)) P_new = k (0.90V)/(1.1sqrt(W)) P_new = (0.90)/(1.1) ( k (V)/(sqrt(W)) ) Since P = k (V)/(sqrt(W)), we have: P_new = (0.90)/(1.1) P P_new = (9)/(11) P Step 4: Calculate the percentage change in P. Percentage change = P_new - PP × 100\% Percentage change = ( (9)/(11)P - PP ) × 100\% Percentage change = ( (9)/(11) - 1 ) × 100\% Percentage change = ( (9 - 11)/(11) ) × 100\% Percentage change = ( -(2)/(11) ) × 100\% Percentage change ≈ -18.18\% The percentage change in P is a decrease of approximately 18.18%. Decrease of 18.18\% Problem 2: R varies directly as S and inversely as the square root of T. a) Write an equation connecting R, S and T. Step 1: Write the general variation equation. R = k (S)/(sqrt(T)) where k is the constant of proportionality. b) Find the value of R when S = 360 and T = 1.5. Step 1: Find the constant of proportionality (k) using the given values R = 480, S = 150, and T = 5. 480 = k (150)/(sqrt(5)) k = 480 sqrt(5)150 k = 48 sqrt(5)15 k = 16 sqrt(5)5 Step 2: Write the specific equation for R. R = 16 sqrt(5)5 (S)/(sqrt(T)) Step 3: Substitute the new values S = 360 and T = 1.5 into the equation. R = 16 sqrt(5)5 (360)/(sqrt(1.5)) R = 16 sqrt(5)5 (360)/(sqrt(3)2) R = 16 sqrt(5)5 (360)/(sqrt(3))sqrt(2) R = 16 sqrt(5)5 · 360 sqrt(2)sqrt(3) R = 16 · 360 sqrt(10)5 sqrt(3) R = 16 · 72 sqrt(10)sqrt(3) R = 1152 sqrt(10)sqrt(3) To rationalize the denominator: R = 1152 sqrt(10) · sqrt(3)sqrt(3) · sqrt(3) R = 1152 sqrt(30)3 R = 384 sqrt(30) The value of R is approximately 384 × 5.477 = 2103.17. R = 384 sqrt(30) ≈ 2103.17 c) Find the percentage change in R if S is increased by 5% and T decreased by 20%. Step 1: Express the new values of S and T. S is increased by 5%, so S_new = S + 0.05S = 1.05S. T is decreased by 20%, so T_new = T - 0.20T = 0.80T. Step 2: Calculate the new value of R (R_new). R_new = k S_newsqrt(T_new) = k (1.05S)/(sqrt(0.80T)) R_new = (1.05)/(sqrt(0.80)) ( k (S)/(sqrt(T)) ) Since R = k (S)/(sqrt(T)), we have: R_new = (1.05)/(sqrt(0.80)) R Convert decimals to fractions for precision: 1.05 = (21)/(20) and 0.80 = (4)/(5). sqrt(0.80) = sqrt((4)/(5)) = sqrt(4)sqrt(5) = (2)/(sqrt(5)) R_new = (21)/(20)(2)/(sqrt(5)) R R_new = (21)/(20) · sqrt(5)2 R R_