AIME Geometry 3

Question 1

A sphere has radius 3. Find its volume in terms of π.

Accepted Answer

36π

Answer

V=(4/3)πr^3=(4/3)π(27)=36π.

Question 2

In a 30-60-90 triangle, the hypotenuse is 10. Find the length of the shorter leg.

Accepted Answer

5

Answer

In a 30-60-90 triangle, the shorter leg is half the hypotenuse: 10/2=5.

Question 3

What is the area of a trapezoid with parallel bases 6 and 10 and height 4?

Accepted Answer

32

Answer

Area=(b₁+b₂)/2×h=(6+10)/2×4=32.

Question 4

Chord AB and chord CD intersect inside a circle. If AX=3, XB=8, CX=4, find XD.

Accepted Answer

6

Answer

By the intersecting chords theorem, AX·XB=CX·XD: 3×8=4×XD, so XD=6.

Question 5

A cylinder has radius 5 and height 8. What is its total surface area?

Accepted Answer

130π

Answer

Total SA=2πr^2+2πrh=2π(25)+2π(40)=50π+80π=130π.

AIME - American Invitational Mathematics Examination Practice Test