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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Here are the solutions to the problems: 4. Determine the expression for the sides of the board. The area of a rectangular wooden board is given by the expression x^2 + 8x + 15. The area of a rectangle is found by multiplying its length and width. To find the expressions for the sides, we need to factor the quadratic expression. Step 1: Factor the quadratic expression x^2 + 8x + 15. We need two numbers that multiply to 15 and add up to 8. These numbers are 3 and 5. x^2 + 8x + 15 = (x+3)(x+5) The expressions for the sides of the board are x+3 and x+5. The expressions for the sides of the board are (x+3) and (x+5). 5. Determine the width of the mat. The area of a rectangular table mat is 216 cm^2. The width of the mat is 6 cm shorter than the length. Step 1: Define variables for length and width. Let the length of the mat be L and the width be W. Step 2: Write down the given information as equations. Area A = L × W = 216 Width is 6 cm shorter than the length: W = L - 6 Step 3: Substitute the expression for W into the area equation. L(L - 6) = 216 Step 4: Expand and rearrange the equation into a standard quadratic form. L^2 - 6L = 216 L^2 - 6L - 216 = 0 Step 5: Solve the quadratic equation for L using the quadratic formula L = -b ± sqrt(b^2 - 4ac)2a. Here, a=1, b=-6, c=-216. L = -(-6) ± sqrt((-6)^2 - 4(1)(-216))2(1) L = 6 ± sqrt(36 + 864)2 L = 6 ± sqrt(900)2 L = (6 ± 30)/(2) We have two possible values for L: L_1 = (6 + 30)/(2) = (36)/(2) = 18 L_2 = (6 - 30)/(2) = (-24)/(2) = -12 Since length cannot be negative, we take L = 18 cm. Step 6: Calculate the width W. W = L - 6 W = 18 - 6 W = 12 cm The width of the mat is 12 cm. 6. Determine their present ages. A mother is five times as old as her daughter. Six years ago, the product of their ages was 68. Step 1: Define variables for their present ages. Let the daughter's present age be D. Let the mother's present age be M. Step 2: Write down the given information as equations. Mother is five times as old as her daughter: M = 5D Ages six years ago: Daughter's age six years ago: D - 6 Mother's age six years ago: M - 6 Product of their ages six years ago was 68: (D - 6)(M - 6) = 68 Step 3: Substitute M = 5D into the product equation. (D - 6)(5D - 6) = 68 Step 4: Expand and rearrange the equation into a standard quadratic form. 5D^2 - 6D - 30D + 36 = 68 5D^2 - 36D + 36 = 68 5D^2 - 36D + 36 - 68 = 0 5D^2 - 36D - 32 = 0 Step 5: 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) We have 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 6: Calculate the mother's present age M. M = 5D M = 5 × 8 M = 40 years The daughter's present age is 8 years and the mother's present age is 40 years.