In order to estimate the height 'h' of a tall pine tree, a student places a mirror on the ground and stands where she can see the top of the tree. The student is 6 feet tall and stands 3 feet from the mirror which is 11 feet from the base of the tree. Another student also wants to see the top of the tree. The other student is 5.5 feet tall. If the mirror is to remain 3 feet from the student's feet, how far from the base of the tree should the mirror be placed?
The only tricky part in this question is to ignore data which is irrelevant, like the 6 ft girl's initial mirror placement.
The only fact we care about is that we have two similar right-angled triangles, formed by the second student and the mirror, and the tree and the mirror.
Make that sketch.
let the distance of the mirror for the tree for the second experiment be x
then x/11 = 3/5.5
x = 33/5.5 = 6 ft
answer
To estimate the height of the tree, we can use similar triangles.
Let's represent the height of the tree as 'h'.
According to the question, the first student is 6 feet tall and stands 3 feet from the mirror. The distance between the mirror and the tree's base is 11 feet.
We can set up the following ratio of similar triangles:
h/6 = (h + x)/3
Where 'x' represents the distance between the mirror and the second student's feet.
Now, let's solve for 'x':
h/6 = (h + x)/3
Cross-multiplying, we get:
3h = 6(h + x)
Distribute on the right side:
3h = 6h + 6x
Rearranging the equation:
6x = 6h - 3h
6x = 3h
Dividing both sides by 6:
x = (3h/6)
Simplifying:
x = h/2
So, the distance 'x' between the mirror and the second student's feet should be half the height 'h' of the tree.
Therefore, the mirror should be placed 3 feet from the second student's feet, which means it should be 3 feet from the base of the tree in addition to the distance between the first student's feet and the mirror:
Mirror distance from the base of the tree = 11 feet (distance between mirror and tree's base) + 3 feet (distance between the first student's feet and the mirror)
= 14 feet
Therefore, the mirror should be placed 14 feet from the base of the tree.
To solve this problem, we can use similar triangles and the concept of proportionality. Let's label the unknown distance from the base of the tree to the mirror as 'x.'
Firstly, let's consider the triangle formed by the first student, the mirror, and the tree. We have a right triangle where the vertical side is the height of the tree (h), the horizontal side is 'x' (the distance from the mirror to the base of the tree), and the hypotenuse is the sum of the distances from the base of the tree to the mirror and from the mirror to the student (x + 11).
Using these lengths, we can set up the following proportion:
(6 feet) / 3 feet = h / (x + 11)
Now, since we are given the height of the second student (5.5 feet) and the distance from their feet to the mirror (3 feet), we can set up another proportion using similar triangles.
(5.5 feet) / 3 feet = h / x
Now, we can solve for 'x' by setting the two proportions equal to each other and solving for 'x':
(6/3) = (h / (x + 11)) = (5.5/3) = (h / x)
Now, let's use cross-multiplication to solve for 'x':
(6/3) * x = (5.5/3) * (x + 11)
2x = 5.5(x + 11)
Distributing 5.5:
2x = 5.5x + 60.5
Subtracting 5.5x from both sides:
2x - 5.5x = 60.5
Combining like terms:
-3.5x = 60.5
Now, divide both sides by -3.5 to solve for 'x':
x = 60.5 / -3.5
x ≈ -17.29
Since distance cannot be negative, we can disregard the negative value. Therefore, the mirror should be placed approximately 17.29 feet from the base of the tree.