Answer:
(i) x = 17 cm
(ii) 360 cm²
(iii) 14400 cm³
(iv) 4000 cm²
Explanation:
You want the slant side length, end area, volume, and total surface area of an isosceles trapezoidal prism with end bases 16 cm and 32 cm and end height 15 cm. The length of the prism is 40 cm.
(i) Slant side
The slant side of the base is the hypotenuse of a right triangle that has legs of 15 cm and (32 -16)/2 = 8 cm. You recognize these numbers as belonging to the {8, 15, 17} Pythagorean triple, so you know the slant side is 17 cm.
If you don't recognize that triple, you can figure the side length as ...
x = √(15² +8²) = √(225 +64) = √289
x = 17 . . . centimeters
(ii) End area
The area of the end trapezoid is given by ...
A = 1/2(b1 +b2)h = 1/2(32 +16)(15) = 360 . . . cm²
The area of the end surface is 360 cm².
(iii) Volume
The volume of the prism is given by ...
V = Bh
where B is the area of the base, and h is the length between bases.
V = (360 cm²)(40 cm) = 14400 cm³
The volume of the prism is 14400 cubic centimeters.
(iv) Surface area
The total surface area is the sum of the areas of the two bases and the lateral area. The lateral area is the product of the perimeter of the base and the length of the prism.
total area = 2 · base area + lateral area
total area = 2 · 360 cm² + (32 + 17 + 16 + 17 cm)(40 cm)
total area = 720 cm² + 3280 cm² = 4000 cm²
The total surface area of the prism is 4000 square centimeters.
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