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Difference between revisions of "2024 AMC 8 Problems/Problem 24"

m (→‎Solution 1)
m (→‎Solution 1)
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==Solution 1==
 
==Solution 1==
 
Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle. The side length of the largest right triangle is <math>12\sqrt{2},</math> which means its area is <math>144.</math> Similarly, the area of the second largest right triangle is <math>64,</math> and the area of the overlap triangle is <math>h^2.</math> Thus,
 
Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle. The side length of the largest right triangle is <math>12\sqrt{2},</math> which means its area is <math>144.</math> Similarly, the area of the second largest right triangle is <math>64,</math> and the area of the overlap triangle is <math>h^2.</math> Thus,
−
<cmath>144+64-h^2=183,</cmath>
+
<cmath>144+64-\frac{h^2}{2}=183,</cmath>
−
which means that <math>\boxed{h=5}.</math>
+
which means that <math>\boxed{h=5\sqrt{2}}.</math>
  
 
~BS2012
 
~BS2012
 +
~WhatdoHumanitariansEat fixed answer because BS2012 forgot to divide by 2

Revision as of 16:07, 25 January 2024

Problem

Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is $8$ feet high while the other peak is $12$ feet high. Each peak forms a $90^\circ$ angle, and the straight sides form a $45^\circ$ angle with the ground. The artwork has an area of $183$ square feet. The sides of the mountain meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h?$

Solution 1

Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle. The side length of the largest right triangle is $12\sqrt{2},$ which means its area is $144.$ Similarly, the area of the second largest right triangle is $64,$ and the area of the overlap triangle is $h^2.$ Thus, \[144+64-\frac{h^2}{2}=183,\] which means that $\boxed{h=5\sqrt{2}}.$

~BS2012 ~WhatdoHumanitariansEat fixed answer because BS2012 forgot to divide by 2

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