Answer:
(i) x = 17 cm
(ii) one end: 360 cm²; both ends: 720 cm²
(iii) 14,400 cm³
(iv) 4000 cm²
Explanation:
You want the slant height, base area, volume, and total surface area of a trapezoidal prism with the base isosceles trapezoid having parallel base lengths of 32 and 16, and a height of 15. The distance between bases is 40. All units are cm.
(i) Slant height
If a center rectangle 16 cm wide and 15 cm high is cut from the base trapezoid, the remaining two triangles have a base of 8 cm and a height of 15 cm. The Pythagorean theorem can be used to find the slant height:
x² = a² +b²
x² = 8² +15² = 64 +225 = 289
x = √289 = 17
The slant height, x, is 17 cm.
(ii) Base area
The area of the base trapezoid is given by the formula ...
A = 1/2(b1 +b2)h
A = 1/2(32 +16)(15) = 360 . . . . square cm
The area of one end surface is 360 cm²; the total area of both end surfaces is 720 cm².
(iii) Volume
The volume of the prism is the product of the base area and the length of the prism.
V = Bh
V = (360 cm²)(40 cm) = 14,400 cm³
The volume of the trapezoidal prism is 14,400 cm³.
(iv) Total surface area
The lateral surface area of the prism is the product of the perimeter of the base and the distance between bases.
LA = Ph
LA = (32 +16 +2·17 cm)(40 cm) = 3280 cm²
The total surface area is the sum of the lateral area and the area of the two bases:
SA = LA +2B = (3280 cm²) + 2(360 cm²) = 4000 cm²
The total surface area of the prism is 4000 square centimeters.
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Additional comment
It appears that the top dashed line in the figure is drawn that way in error. It appears to identify a visible edge, so we expect it to be a solid line.