MathDB

17

Part of 2003 AMC 10

Problems(2)

Circling the Triangle

Source:

1/2/2009
The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle? <spanclass=′latex−bold′>(A)</span> 32π<spanclass=′latex−bold′>(B)</span> 33π<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 6π<spanclass=′latex−bold′>(E)</span> 3π <span class='latex-bold'>(A)</span>\ \frac{3\sqrt2}{\pi} \qquad <span class='latex-bold'>(B)</span>\ \frac{3\sqrt3}{\pi} \qquad <span class='latex-bold'>(C)</span>\ \sqrt3 \qquad <span class='latex-bold'>(D)</span>\ \frac{6}{\pi} \qquad <span class='latex-bold'>(E)</span>\ \sqrt3\pi
geometryperimetercircumcircletrigonometryperpendicular bisectortrig identitiesLaw of Sines
Ice Cream Cone

Source:

1/4/2009
An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies 75% 75\% of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius? <spanclass=′latex−bold′>(A)</span> 2:1<spanclass=′latex−bold′>(B)</span> 3:1<spanclass=′latex−bold′>(C)</span> 4:1<spanclass=′latex−bold′>(D)</span> 16:3<spanclass=′latex−bold′>(E)</span> 6:1 <span class='latex-bold'>(A)</span>\ 2: 1 \qquad <span class='latex-bold'>(B)</span>\ 3: 1 \qquad <span class='latex-bold'>(C)</span>\ 4: 1 \qquad <span class='latex-bold'>(D)</span>\ 16: 3 \qquad <span class='latex-bold'>(E)</span>\ 6: 1
geometry3D geometrysphereratio