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09 Dec 2021, 03:34
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In the image above, the green bar has a height of 1 and casts a shadow of 2 on the floor and the red bar has a height of 6 and casts a shadow of 4 on the floor which continues to a height of \(x\) on a parallel to the bars wall. What is the value of \(x\)? (Assume that the light source is distant, the floors and walls are perfectly flat, and the floor is perpendicular to the bars and walls)
A. \(1.5\)
B. \(2\)
C. \(2.5\)
D. \(3\)
E. \(4\)
09 Dec 2021, 03:34
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Official Solution:
In the image above, the green bar has a height of 1 and casts a shadow of 2 on the floor and the red bar has a height of 6 and casts a shadow of 4 on the floor which continues to a height of \(x\) on a parallel to the bars wall. What is the value of \(x\)? (Assume that the light source is distant, the floors and walls are perfectly flat, and the floor is perpendicular to the bars and walls)
A. \(1.5\)
B. \(2\)
C. \(2.5\)
D. \(3\)
E. \(4\)
Consider the side view of the figure:
Notice that the rays of light should make the same angle \(α\) at the ground. So, all three angles in two triangles shown are equal to each other, which makes these triangles similar. The ratio of the base to the height of the bottom triangle is \(2:1\), hence ratio of the base to the height of the top triangle must also be \(2:1\). So, the base of the top triangle is \(6*2=12\) (this means that if there were no wall, the red bar would cast a shadow of 12). In this case the base of the little blue triangle at the top is \(12-4=8\).
Now, consider the top triangle. The entire top triangle is similar to its part on the right (little blue triangle) so again since the ratio of the base to the height of entire top triangle is \(2:1\), hence ratio of the base to the height (\(x\)) of the little blue triangle must also be \(2:1\). So, the height (\(x\)) of the little blue triangle \(8:2=4\)
Answer: E
In the image above, the green bar has a height of 1 and casts a shadow of 2 on the floor and the red bar has a height of 6 and casts a shadow of 4 on the floor which continues to a height of \(x\) on a parallel to the bars wall. What is the value of \(x\)? (Assume that the light source is distant, the floors and walls are perfectly flat, and the floor is perpendicular to the bars and walls)
A. \(1.5\)
B. \(2\)
C. \(2.5\)
D. \(3\)
E. \(4\)
Consider the side view of the figure:
Notice that the rays of light should make the same angle \(α\) at the ground. So, all three angles in two triangles shown are equal to each other, which makes these triangles similar. The ratio of the base to the height of the bottom triangle is \(2:1\), hence ratio of the base to the height of the top triangle must also be \(2:1\). So, the base of the top triangle is \(6*2=12\) (this means that if there were no wall, the red bar would cast a shadow of 12). In this case the base of the little blue triangle at the top is \(12-4=8\).
Now, consider the top triangle. The entire top triangle is similar to its part on the right (little blue triangle) so again since the ratio of the base to the height of entire top triangle is \(2:1\), hence ratio of the base to the height (\(x\)) of the little blue triangle must also be \(2:1\). So, the height (\(x\)) of the little blue triangle \(8:2=4\)
Answer: E
