Side wall planes

Find the volume and surface of a cuboid whose side c is 30 cm long and whose body diagonal forms angles of 24°20' and 45°30' with the planes of the side walls.

Correct answer:

S = 5217.801 cm²
V = 24681.8833 cm³

Step-by-step explanation:

c = 30 cm α = 24 + 20 / 60 = \frac{7 3}{3} ≐ 24.3333^{\circ} β = 45 + 30 / 60 = \frac{9 1}{2} = 45.5^{\circ} t_{1} = tan α = tan 24.333333333333 ° = 0.452218 = 0.45222 t_{2} = tan β = tan 45.5 ° = 1.017607 = 1.01761 d^{2} = a^{2} + b^{2} + c^{2} u_{1}^{2} = a^{2} + c^{2} u_{2}^{2} = b^{2} + c^{2} tan α = a : u_{2} = t_{1} t_{1}^{2} \cdot (b^{2} + c^{2}) = a^{2} tan β = b : u_{1} = t_{2} t_{2}^{2} \cdot (a^{2} + c^{2}) = b^{2} t_{1}^{2} \cdot (b^{2} + c^{2}) = a^{2} t_{2}^{2} \cdot (a^{2} + c^{2}) = b^{2} t_{2}^{2} \cdot (t_{1}^{2} \cdot (b^{2} + c^{2}) + c^{2}) = b^{2} t_{2}^{2} \cdot (t_{1}^{2} \cdot (x + c^{2}) + c^{2}) = x 1.017607392972 1^{2} \cdot (0.452217879136 7^{2} \cdot (x + 3 0^{2}) + 3 0^{2}) = x 0.788234 x = 1122.561608 x = \frac{1 1 2 2 . 5 6 1 6 0 7 6 8}{0 . 7 8 8 2 3 4 1 3} = 1424.1474246 x = 1424.147425 b = x = 1424.1474 ≐ 37.7379 cm a = t_{1}^{2} \cdot (b^{2} + c^{2}) = 0.452 2^{2} \cdot (37.737 9^{2} + 3 0^{2}) ≐ 21.8012 cm S = 2 \cdot (a \cdot b + b \cdot c + a \cdot c) = 2 \cdot (21.8012 \cdot 37.7379 + 37.7379 \cdot 30 + 21.8012 \cdot 30) = 5217.801 cm^{2}

V = a \cdot b \cdot c = 21.8012 \cdot 37.7379 \cdot 30 ≐ 24681.8833 cm^{3} Verifying Solution: u_{1} = a^{2} + c^{2} = 21.801 2^{2} + 3 0^{2} ≐ 37.0849 cm u_{2} = b^{2} + c^{2} = 37.737 9^{2} + 3 0^{2} ≐ 48.2094 cm A = \frac{1 8 0 °}{π} \cdot arctan (a / u_{2}) = \frac{1 8 0 °}{π} \cdot arctan (21.8012 / 48.2094) = \frac{7 3}{3} ≐ 24.3333^{\circ} B = \frac{1 8 0 °}{π} \cdot arctan (b / u_{1}) = \frac{1 8 0 °}{π} \cdot arctan (37.7379 / 37.0849) = \frac{9 1}{2} = 45.5^{\circ}

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