Answer:
volume ≈ 930 cm³ (nearest cubic cm)
Step-by-step explanation:
The base of the prism which is a pentagon is regular. The apothem which is a line drawn from the center to any side of the pentagon is 4 cm. The height of the prism is 16 cm.
The volume of a prism = Bh
where
B = base area
h = height of the prism
The base of the prism can be divided into 5 congruent triangles.But the base of the triangles is unknown.
The sum of interior angle of a pentagon is 540°. That means each angle is 108° . The triangle divide the angle into 2. Making the 2 base angle as 54° each. The last angle of the triangle at the center is 72°(remember sum of angle in a triangle is 180° which is 54° + 54° + 72° = 180°) .
The angle form the center can be divided by the line of height of the triangles. The angle will then be 36°. Using tangential ratio half of the length of the base of the triangle can be known. Therefore,
tan 36° = opposite/adjacent
tan 36° = opposite/4
opposite = 4 tan 36°
The full base of the triangles = 2(4 tan 36°)
area of each triangle = 1/2 × 2(4 tan 36°) × 4 = 16 tan 36°
Base area of the prism(pentagon) = 5 × 16 tan 36° = 80 tan 36°
Volume of the prism = Bh
where
h = 16 cm
B = 80 tan 36°
volume = 80 tan 36° × 16 = 1280 tan 36° = 1280 × 0.726542528
volume = 929.974435847 cm³
volume ≈ 930 cm³ (nearest cubic cm)
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