A man places a torche on the ground, 10m away from a wall. He sees a shade on the wall and gets a friend to measure the height of the shadow as he walks towards the wall. Here are the results:
Distance from torch to man height of shadow
1 20
2 10
3 6.7
4 5
5 4
6 3.3
7 2.9
8 2.5
find the height of the man
Lets denote x = the height of the man, h = the height of the shadow (the right column of data), and y = the distance from torch to the man (the left column of data), then you can write a proportion: x/y = h/10 or x = y*h/10. Use any pair of given data now to find x: the first => 1*20/10 = 2; the second => 2*10/10 = 2, and so on. Finally, x =2.
To find the height of the man, we can use the concept of similar triangles. In this case, we have two sets of similar triangles: the triangles formed by the man, his shadow, and the wall, and the triangles formed by the torch, the man, and his shadow.
Let's denote the height of the man as "h" and the distance from the man to the wall as "x". We are given the distance from the torch to the man (which is also the base of the shadow) and the corresponding height of the shadow at various distances.
Using the concept of similar triangles, we can set up the following proportion:
h / (h + x) = height of shadow / distance from torch to man
Using the given data, we can calculate the height of the man by substituting the values:
1st measurement: h / (h + 1) = 20 / 10 (height of shadow divided by distance from torch to man)
2nd measurement: h / (h + 2) = 10 / 10
3rd measurement: h / (h + 3) = 6.7 / 10
4th measurement: h / (h + 4) = 5 / 10
5th measurement: h / (h + 5) = 4 / 10
6th measurement: h / (h + 6) = 3.3 / 10
7th measurement: h / (h + 7) = 2.9 / 10
8th measurement: h / (h + 8) = 2.5 / 10
To solve for h, we can use any method of solving systems of equations, such as substitution or elimination. Let's use the method of substitution:
From the first equation, we have h / (h + 1) = 2.
We can solve this equation for h: h = 2h + 2
Simplifying the equation, we get: h = -2
Therefore, the height of the man is negative, which does not make sense in this context. Hence, there might be an error in the data or the calculations. Double-checking the data or calculations is recommended to find the correct height of the man.