Question #147379
A monument is a stone structure in the form of a frustum of a regular square pyramid whose height is 50m and whose base edges are 4.5m and 10m respectively. Through the center of the monument is a circular opening 3m in diameter at the top and 6m in diameter at the bottom. Refer to figure 1.

8. What is the slant height of the opening?
9. What is the lateral surface area of the opening?
10. Find the volume of stone in the monument.
1
Expert's answer
2020-11-30T12:30:14-0500


Consider frustrum of a right circular cone

  1. Let r1=r_1= the radius of the upper base, r2=r_2= the radius of the lower base, and L=L= the slant height. Then from the right triangle by the Pythagorean Theorem
L2=(r2r1)2+h2L^2=(r_2-r_1)^2+h^2

L=(r2r1)2+h2L=\sqrt{(r_2-r_1)^2+h^2}

Given

r1=32=1.5(m),r2=62=3(m),h=50 mr_1=\dfrac{3}{2}=1.5(m), r_2=\dfrac{6}{2}=3(m), h=50\ m

L=(31.5)2+502=2502.2550.022(m)L=\sqrt{(3-1.5)^2+50^2}=\sqrt{2502.25}\approx50.022(m)

The slant height of the opening is approximately 50.022 m.


9. The lateral area of the frustum of a right circular cone is


AL=12(2πr2+2πr1)LA_L=\dfrac{1}{2}(2\pi r_2+2\pi r_1)L

AL=12(2π(3)+2π(1.5))2502.25707.176362(m2)A_L=\dfrac{1}{2}(2\pi (3)+2\pi (1.5))\sqrt{2502.25}\approx707.176362(m^2)

the lateral surface area of the opening is approximately 707.176362 m2.


10.

For any Frustum, the volume is


V=13(A1+A2+A1A2)hV=\dfrac{1}{3}(A_1+A_2+\sqrt{A_1A_2})h

The volume of the frustrum of the cone


Vc=13(πr12+πr22+πr12πr22)hV_c=\dfrac{1}{3}(\pi r_1^2+\pi r_2^2+\sqrt{\pi r_1^2\pi r_2^2})h

=π3(r12+r22+r1r2)h=\dfrac{\pi}{3}( r_1^2+ r_2^2+r_1r_2)h

Vc=π3((1.5)2+(3)2+1.5(3))(50)V_c=\dfrac{\pi}{3}((1.5)^2+(3)^2+1.5(3))(50)

824.668071567(m3)\approx824.668071567(m^3)

Vp=13((4.5)2+(10)2+4.5(10))(50)V_p=\dfrac{1}{3}((4.5)^2+(10)^2+4.5(10))(50)

2754.666666667(m3)\approx2754.666666667(m^3)

The volume of the stone is


Vstone=VpVcV_{stone}=V_p-V_c

2754.666666667824.668071567\approx2754.666666667-824.668071567

1930 (m3)\approx1930\ (m^3)

The volume of the stone is 1930 m3.



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