The prism has a cross-sectional area in the shape of a sector. Calculate: a. the radius rr cm, b. the cross-sectional area of the prism, c. the total surface area of the prism, d. the volume of the prism

FizeauV

FizeauV

Answered question

2020-12-24

The prism has a cross-sectional area in the shape of a sector. Calculate: a. the radius rr cm,
b. the cross-sectional area of the prism,
c. the total surface area of the prism,
d. the volume of the prism

Answer & Explanation

irwchh

irwchh

Skilled2020-12-25Added 102 answers

a. To find r, we need to use the given arc length of 20 cm and the angle of 50 that creates the arc. The arc length, s, of a circle is given by the formula s=θ360(2πr) where θ is the angle that creates the arc and r is the radius of the circle. We know the arc length is s=20cm and the angle is θ=50. Therefore:

s=(θ/360)(2πr)

20=(50/360)(2πr)

20=(5π/18)r

20(18/5π)=(5π/18)r(18/5π)

22.9cmr

b. To find the cross-sectional area, we need to find the area of the sector. The area, A, of a sector is given by the formula A=(θ/360)(πr2) where θ is the angle that creates the arc and r is the radius of the circle. We know that θ=50 and r=22.9. The area of the sector is then:

A=(θ/360)(πr2)=50/360[π(22.9)2]228.8cm2

c. The surface area of a prism is S=2B+Ph where B is the area of the base, P is the perimeter of the base, and hh is the height of the prism. The sector is the base so B=228.8cm2. The perimeter of the sector is the arc length plus two times the radius. Since the arc length is 20 cm and the radius is 22.9 cm, then the perimeter is P=20+2(22.9)=65.8cm. The height of the prism is h=8cm. The surface area is then:

S=2B+Ph=2(228.8)+65.8(8)=457.6+526.4=984cm2

d. The volume of a prism is V=Bh where B is the area of the base and hh is the height of the prism. Since the area of the base is B=228.8cm2 and the height of the prism is h=8cm, the volume of the prism is V=228.8(8)=1830.4cm3

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