A quantity P is partly constant and partly varies as the square root of a quantity Q. Given that P = 20 when Q = 4 and that P = 60 when Q = 100, find Q when P = 22.

Fresh day Oliverkoome@gmail254.Com., let's solve. 18. a) A quantity P is partly constant and partly varies as the square root of a quantity Q. Given that P = 20 when Q = 4 and that P = 60 when Q = 100, find Q when P = 22. Step 1: Formulate the equation for P in terms of Q. Let P be partly constant k_1 and partly vary as sqrt(Q) with constant k_2. P = k_1 + k_2sqrt(Q) Step 2: Use the given values to form a system of equations. When P = 20, Q = 4: 20 = k_1 + k_2sqrt(4) 20 = k_1 + 2k_2 (Equation 1) When P = 60, Q = 100: 60 = k_1 + k_2sqrt(100) 60 = k_1 + 10k_2 (Equation 2) Step 3: Solve the system of equations for k_1 and k_2. Subtract Equation 1 from Equation 2: (60 - 20) = (k_1 + 10k_2) - (k_1 + 2k_2) 40 = 8k_2 k_2 = (40)/(8) = 5 Substitute k_2 = 5 into Equation 1: 20 = k_1 + 2(5) 20 = k_1 + 10 k_1 = 20 - 10 = 10 Step 4: Write the complete equation for P. P = 10 + 5sqrt(Q) Step 5: Find Q when P = 22. 22 = 10 + 5sqrt(Q) 22 - 10 = 5sqrt(Q) 12 = 5sqrt(Q) sqrt(Q) = (12)/(5) Square both sides: Q = ((12)/(5))^2 Q = (144)/(25) Q = 5.76 When P = 22, Q is: 5.76 b) Three quantities, T, U and V are such that T varies directly as the square of (10-U) and inversely as the cube root of V. When T = 12, U = 4 and V = 8. i) Determine the equation connecting T, U and V. Step 1: Formulate the variation equation. T varies directly as (10-U)^2 and inversely as [3]V. T = k ((10-U)^2)/([3]V) where k is the constant of proportionality. Step 2: Use the given values to find k. Given T = 12, U = 4, V = 8. 12 = k ((10-4)^2)/([3]8) 12 = k ((6)^2)/(2) 12 = k (36)/(2) 12 = k × 18 k = (12)/(18) = (2)/(3) Step 3: Write the equation connecting T, U, and V. T = (2)/(3) ((10-U)^2)/([3]V) The equation connecting T, U, and V is: T = (2(10-U)^2)/(3[3]V) ii) Find U when T = 5(2)/(5) and V = 15(5)/(8). Step 1: Convert the mixed fractions to improper fractions. T = 5(2)/(5) = (5 × 5 + 2)/(5) = (27)/(5) V = 15(5)/(8) = (15 × 8 + 5)/(8) = (120 + 5)/(8) = (125)/(8) Step 2: Calculate [3]V. 3]V = [3/(8) = [3]125[3]8 = (5)/(2) Step 3: Substitute the values of T and [3]V into the equation from part (b)(i). (27)/(5) = (2)/(3) ((10-U)^2)/(5)2 (27)/(5) = (2)/(3) × (2)/(5) (10-U)^2 (27)/(5) = (4)/(15) (10-U)^2 Step 4: Solve for (10-U)^2. (10-U)^2 = (27)/(5) × (15)/(4) (10-U)^2 = (27 × 3)/(4) (10-U)^2 = (81)/(4) Step 5: Solve for U. Take the square root of both sides: 10-U = ±sqrt((81)/(4)) 10-U = ±(9)/(2) Case 1: 10-U = (9)/(2) U = 10 - (9)/(2) U = (20)/(2) - (9)/(2) U = (11)/(2) = 5.5 Case 2: 10-U = -(9)/(2) U = 10 + (9)/(2) U = (20)/(2) + (9)/(2) U = (29)/(2) = 14.5 The possible values for U are: U = 5.5 or U = 14.5 Send me the next one 📸