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
You're on a roll — here are the solutions to your tasks: a) Help the travellers identify the hidden treasure's location. The given equations are: 1) 5^x+2 + 7^y+1 = 3468 2) 7^y = 5^x - 76 Step 1: Simplify equation (1) using exponent rules. 5^x · 5^2 + 7^y · 7^1 = 3468 25 · 5^x + 7 · 7^y = 3468 Step 2: Introduce substitutions to simplify the system. Let A = 5^x and B = 7^y. The system of equations becomes: 25A + 7B = 3468 (3) B = A - 76 (4) Step 3: Substitute equation (4) into equation (3) and solve for A. 25A + 7(A - 76) = 3468 25A + 7A - 532 = 3468 32A = 3468 + 532 32A = 4000 A = (4000)/(32) A = 125 Step 4: Substitute the value of A back into equation (4) to find B. B = 125 - 76 B = 49 Step 5: Use the substitutions to find x and y. Since A = 5^x: 5^x = 125 5^x = 5^3 x = 3 Since B = 7^y: 7^y = 49 7^y = 7^2 y = 2 The hidden treasure's location is (3, 2). b) Help your uncle to determine the total amount of money needed to be spent on buying trees. Step 1: Determine the dimensions of the rectangular plot. Let the width of the plot be W meters and the length be L meters. Given that the length is 2 meters longer than its width: L = W + 2 Given that the total area of the plot is 35 square meters: L × W = 35 Substitute the expression for L into the area equation: (W + 2)W = 35 W^2 + 2W = 35 W^2 + 2W - 35 = 0 Step 2: Solve the quadratic equation for W. Factor the quadratic equation: (W + 7)(W - 5) = 0 This gives two possible values for W: W = -7 or W = 5. Since width cannot be negative, W = 5 meters. Step 3: Calculate the length L. L = W + 2 = 5 + 2 = 7 meters The dimensions of the plot are 7 meters by 5 meters. Step 4: Calculate the perimeter of the plot. The perimeter P of a rectangle is 2(L + W): P = 2(7 + 5) P = 2(12) P = 24 meters Step 5: Calculate the number of trees needed. Henry plans to use a spacing of 1m from each tree around the perimeter. Number of trees = PerimeterSpacing Number of trees = 24 m1 m/tree Number of trees = 24 trees Step 6: Calculate the total cost of buying the trees. Each tree costs Ugx 2,500. Total Cost = Number of trees × Cost per tree Total Cost = 24 × 2500 Total Cost = 60000 The total amount of money needed to be spent on buying trees is Ugx 60,000. Drop the next question.