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.
ScanSolve AI Answer
Here are the solutions to the problem: Part (i): Find the equation connecting M, N and R. Step 1: Write the proportionality relationship. M varies jointly as N, so M N. M varies inversely as the square root of R, so M (1)/(sqrt(R)). Combining these, we get: M (N)/(sqrt(R)) Step 2: Introduce the constant of proportionality, k. M = (kN)/(sqrt(R)) Step 3: Use the given values (M=35, N=7, R=9) to find k. 35 = (k · 7)/(sqrt(9)) 35 = (7k)/(3) Multiply both sides by 3: 35 × 3 = 7k 105 = 7k Divide by 7: k = (105)/(7) k = 15 Step 4: Substitute the value of k back into the equation. M = (15N)/(sqrt(R)) The equation connecting M, N, and R is M = (15N)/(sqrt(R)). Part (ii): Find R when M = 42 and N = 15. Step 1: Use the equation from Part (i). M = (15N)/(sqrt(R)) Step 2: Substitute the given values (M=42, N=15). 42 = (15 · 15)/(sqrt(R)) 42 = (225)/(sqrt(R)) Step 3: Solve for sqrt(R). sqrt(R) = (225)/(42) Simplify the fraction: sqrt(R) = (75)/(14) Step 4: Square both sides to find R. R = ((75)/(14))^2 R = (75^2)/(14^2) R = (5625)/(196) The value of R is (5625)/(196). Part (iii): Find the percentage change in M when N is increased by 25% and R reduced by 36%. Step 1: Express the new values of N and R in terms of their original values. Let the original values be N_1 and R_1. The original M is M_1 = (15N_1)/(sqrt(R_1)). N is increased by 25%, so the new N, N_2, is: N_2 = N_1 + 0.25N_1 = 1.25N_1 R is reduced by 36%, so the new R, R_2, is: R_2 = R_1 - 0.36R_1 = 0.64R_1 Step 2: Calculate the new value of M, M_2. M_2 = (15N_2)/(sqrt(R_2)) Substitute N_2 = 1.25N_1 and R_2 = 0.64R_1: M_2 = (15(1.25N_1))/(sqrt(0.64R_1)) M_2 = (15(1.25N_1))/(0.8sqrt(R_1)) Rearrange the terms to relate M_2 to M_1: M_2 = (1.25)/(0.8) · ((15N_1)/(sqrt(R_1))) Since M_1 = (15N_1)/(sqrt(R_1)): M_2 = (1.25)/(0.8) M_1 Calculate the fraction: (1.25)/(0.8) = (125)/(80) = (25)/(16) = 1.5625 So, M_2 = 1.5625 M_1. Step 3: Calculate the percentage change in M. Percentage change = (M_2 - M_1)/(M_1) × 100\% Percentage change = (1.5625 M_1 - M_1)/(M_1) × 100\% Percentage change = (1.5625 - 1) × 100\% Percentage change = 0.5625 × 100\% Percentage change = 56.25\% The percentage change in M is an increase of 56.25\%. That's 2 down. 3 left today — send the next one.