1. Cast a metal ball with a radius of 10 into 10 small balls with the same size, and find out the radius of the small balls.

Solution: Let the radius of the ball be r, then 10[(4/3)πr? ]=(4/3)π× 10? , so r? = 100,r= 100^( 1/3)=4.64 1588834

2. The bottom of the prism is a regular hexagon with a side length of 4, and the length of the side perpendicular to the bottom is 5. Find its surface area.

Solution: lateral area S? =6×4×5= 120, and the area of the two bottoms is s? = 2×[( 1/2)×4×4×sin 60 = 8√3；

So the total surface area =S? +S？ = 120+8√3= 133.8564

The bottom of the pyramid is a square with a side length of 6, its height is higher than the center of the bottom and its length is 4. Find its surface area.

Solution: bottom area s? =6×6=36; Lateral area s? =4×[( 1/2)×6×5]=60 (where 5 is the horizontal height);

So surface area = s? +S？ =96

The bottom of the pyramid is a regular triangle with a side length of 4, its height is higher than the center of the bottom and its length is 5. Find its surface area.

Solution: bottom area s? =( 1/2)×4×4×sin 60 = 4√3 = 6.9282；

Lateral area s? =( 1/2)×4×√(79/3)=(2/3)√237= 10.2632; (where √(79/3) is the side height);

So the total surface area S=S? +S？ =4√3+(2/3)√237=6.9282+ 10.2632= 17. 19 14

5. The upper bottom surface of a quadrangular prism is a square with a side length of 6, and each side of the lower bottom surface is a square with a side length of 12. The connecting line between the centers of the two bottoms is perpendicular to the bottom surface and 4 meters high. Find its surface area.

Solution: upper and lower regions s? =36; Bottom area s? = 144; Lateral area s? = 4× [(1/2) (6+12 )× 5] =180 (where 5 is the horizontal height);

So the total area S=S? +S？ +S？ =36+ 144+ 180=360;

6. The radius of the cone bottom surface is 3, and the side development diagram is 3/4 circle. Find its surface area.

Solution: bottom area s? =9π; The central angle θ = 3 π/2 of the side expansion diagram; The arc length of the expanded graph is s = 6 π;

Side bus length l = 6 π/(3 π/2) = 4; Lateral area s? =( 1/2)L？ θ=( 1/2)× 16×(3π/2)= 12π;

So the surface area S=S? +S？ =4+ 12π=4 1.6992.

7. The radii of the upper and lower bottom surfaces of the frustum are r and r respectively, and the side surface area is equal to the sum of the two bottom surfaces. Find its bus length.

Solution: lateral area =π(r+R)L=π(r? +R？ ), so the bus length L=(r? +R？ )/(r+R)