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
Hey Janet, good to see you again. Step 1: Define variables for the dimensions of the mat. Let L be the length of the mat and W be the width of the mat. Step 2: Write down the given information as equations. The area of the rectangular mat is 216 cm^2. The formula for the area of a rectangle is A = L × W. L × W = 216 (1) The width of the mat is 6 cm shorter than the length. W = L - 6 (2) Step 3: Substitute equation (2) into equation (1) to form a quadratic equation. L(L - 6) = 216 L^2 - 6L = 216 L^2 - 6L - 216 = 0 Step 4: Solve the quadratic equation for L. We can factor the quadratic equation. We need two numbers that multiply to -216 and add to -6. These numbers are 12 and -18. (L + 12)(L - 18) = 0 This gives two possible values for L: L = -12 or L = 18 Since length cannot be negative, we take L = 18 cm. Step 5: Calculate the width W using the value of L. Substitute L = 18 into equation (2): W = 18 - 6 W = 12 cm The width of the mat is 12 cm. Step 1: Define variables for the present ages. Let M be the mother's present age and D be the daughter's present age. Step 2: Write down the given information as equations. A mother is five times as old as her daughter. M = 5D (1) Six years ago, their ages were M-6 and D-6. The product of their ages was 68. (M - 6)(D - 6) = 68 (2) Step 3: Substitute equation (1) into equation (2) to form a quadratic equation in terms of D. (5D - 6)(D - 6) = 68 Expand the equation: 5D^2 - 30D - 6D + 36 = 68 5D^2 - 36D + 36 - 68 = 0 5D^2 - 36D - 32 = 0 Step 4: Solve the quadratic equation for D using the quadratic formula D = -b ± sqrt(b^2 - 4ac)2a. Here, a = 5, b = -36, c = -32. D = -(-36) ± sqrt((-36)^2 - 4(5)(-32))2(5) D = 36 ± sqrt(1296 + 640)10 D = 36 ± sqrt(1936)10 D = (36 ± 44)/(10) This gives two possible values for D: D_1 = (36 + 44)/(10) = (80)/(10) = 8 D_2 = (36 - 44)/(10) = (-8)/(10) = -0.8 Since age cannot be negative, the daughter's present age is D = 8 years. Step 5: Calculate the mother's present age M using the value of D. Substitute D = 8 into equation (1): M = 5 × 8 M = 40 years The mother's present age is 40 years and the daughter's present age is 8 years. Send me the next one 📸