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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Step 1: Describe the views from X and Y. a) The rectangular box has dimensions: Length (L) = 6 m, Width (W) = 7 m, Height (H) = 8 m. • The view from X (front view) shows the face ABEH. This is a rectangle with dimensions L × H, which is 6 m × 8 m. • The view from Y (right side view) shows the face BCFE. This is a rectangle with dimensions W × H, which is 7 m × 8 m. Step 2: Calculate the total surface area. b)(i) The formula for the total surface area (SA) of a rectangular box is SA = 2(LW + LH + WH). SA = 2((6 m)(7 m) + (6 m)(8 m) + (7 m)(8 m)) SA = 2(42 m^2 + 48 m^2 + 56 m^2) SA = 2(146 m^2) SA = 292 m^2 Step 3: Calculate the volume. b)(ii) The formula for the volume (V) of a rectangular box is V = LWH. V = (6 m)(7 m)(8 m) V = (42 m^2)(8 m) V = 336 m^3 Step 4: Calculate the length of the space diagonal. b)(iii) The question asks for "diagonal HA". In the provided diagram, HA is a vertical edge of the box, with length 8 m. However, the term "diagonal" usually refers to a line segment connecting two non-adjacent vertices. Given part (iv) asks for the angle this diagonal makes with the base, which is typically a non-90 degree angle for a space diagonal, we will assume "diagonal HA" is a mislabeling and refers to the space diagonal of the rectangular box (e.g., from A to F, the opposite vertex). The length of the space diagonal (d) is given by d = sqrt(L^2 + W^2 + H^2). Here, L = 6 m, W = 7 m, H = 8 m. d = sqrt((6 m))^2 + (7 m)^2 + (8 m)^2 d = sqrt(36 m)^2 + 49 m^2 + 64 m^2 d = sqrt(149 m)^2 d = sqrt(149) m ≈ 12.21 m Step 5: Calculate the angle the space diagonal makes with the base. b)(iv) We need to find the angle () that the space diagonal (e.g., AF) makes with the base (ABCD). Consider the right-angled triangle formed by the space diagonal (AF), the height of the box (CF = H), and the diagonal of the base (AC). First, calculate the length of the base diagonal (AC). AC = sqrt(L^2 + W^2) AC = sqrt((6 m))^2 + (7 m)^2 AC = sqrt(36 m)^2 + 49 m^2 AC = sqrt(85 m)^2 AC = sqrt(85) m Now, in the right triangle formed by A, C, and F, we have: Opposite side to is CF (height) = 8 m. Adjacent side to is AC (base diagonal) = sqrt(85) m. The tangent of the angle is given by = OppositeAdjacent. = (H)/(AC) = 8 msqrt(85) m = ((8)/(sqrt(85))) ≈ ((8)/(9.2195)) ≈ (0.8677) ≈ 41.0^ What's next? Send 'em! 📸