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. 10. (a) Step 1: Define variables and set up equations. Let the length of the rectangular field be L and the width be W. The area of the field is A = L × W = 63 m^2. The fencing is for three sides: one length and two widths, as one longer side is an existing wall. So, L > W. The total fencing used is L + 2W = 25 m. Step 2: Solve the system of equations. From the area equation, L = (63)/(W). Substitute this into the fencing equation: (63)/(W) + 2W = 25 Multiply by W to clear the denominator: 63 + 2W^2 = 25W Rearrange into a quadratic equation: 2W^2 - 25W + 63 = 0 Step 3: Solve the quadratic equation for W. Using the quadratic formula W = -b ± sqrt(b^2 - 4ac)2a: W = -(-25) ± sqrt((-25)^2 - 4(2)(63))2(2) W = 25 ± sqrt(625 - 504)4 W = 25 ± sqrt(121)4 W = (25 ± 11)/(4) Two possible values for W: W_1 = (25 + 11)/(4) = (36)/(4) = 9 m W_2 = (25 - 11)/(4) = (14)/(4) = 3.5 m Step 4: Find the corresponding lengths and choose the correct dimensions. If W = 9 m, then L = (63)/(9) = 7 m. In this case, L < W, which contradicts the condition that the wall is one of the longer sides. If W = 3.5 m, then L = (63)/(3.5) = 18 m. In this case, L > W, which satisfies the condition. The dimensions of the field are 18 m by 3.5 m. The dimensions of the field are 18 m by 3.5 m. 10. (b) Step 1: Draw a diagram and identify the trigonometric relationship. Let H be the height of the building. Let D be the distance from point P to the foot of the building, D = 75 m. The angle of depression from the top of the building to point P is 33.6^. This means the angle of elevation from point P to the top of the building is also 33.6^. We have a right-angled triangle where H is the opposite side to the angle and D is the adjacent side. () = oppositeadjacent (33.6^) = (H)/(D) Step 2: Calculate the height of the building. (33.6^) = (H)/(75) H = 75 × (33.6^)