Difference between revisions of "2002 AMC 12P Problems/Problem 19"

Revision as of 00:53, 30 December 2023

Problem

In quadrilateral $ABCD$ , $m\angle B = m \angle C = 120^{\circ}, AB=3, BC=4, and CD=5.$ Find the area of $ABCD.$

$\text{(A) }15 \qquad \text{(B) }9 \sqrt{3} \qquad \text{(C) }\frac{45 \sqrt{3}}{4} \qquad \text{(D) }\frac{47 \sqrt{3}}{4} \qquad \text{(E) }15 \sqrt{3}$

Solution

If $\log_{b} 729 = n$ , then $b^n = 729$ . Since $729 = 3^6$ , $b$ must be $3$ to some factor of 6. Thus, there are four (3, 9, 27, 729) possible values of $b \Longrightarrow \boxed{\mathrm{E}}$ .

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 1: / Line 1: @@
 == Problem ==
-How many positive [[integer]]s <math>b</math> have the property that <math>\log_{b} 729</math> is a positive integer?
+In quadrilateral <math>ABCD</math>, <math>m\angle B = m \angle C = 120^{\circ}, AB=3, BC=4, and CD=5.</math> Find the area of <math>ABCD.</math>
-<math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 }  </math>
+<math>
+\text{(A) }15
+\qquad
+\text{(B) }9 \sqrt{3}
+\qquad
+\text{(C) }\frac{45 \sqrt{3}}{4}
+\qquad
+\text{(D) }\frac{47 \sqrt{3}}{4}
+\qquad
+\text{(E) }15 \sqrt{3}
+</math>
 == Solution ==

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Difference between revisions of "2002 AMC 12P Problems/Problem 19"

Revision as of 00:53, 30 December 2023

Problem

Solution

See also