A 6 ft tall man is standing 40 feet from the light post, if his shadow is 20 feet, determine the height of the lamp.
Here I use simple ratios:
40ft + 20ft = 60ft
6/20 = h/60
Cross multiply both sides with 60:
6 (60)/20 = h
360/20 = h
h = 18
Height of the lamp is 18ft.
P.S. Is there another way to do it?
that is the most straightforward way.
Of course there's another way to do it! Why settle for one when you can have two? Let me give you a slightly different approach:
So we have a 6 ft tall man with a 20 ft shadow. In order to find the height of the lamp post, we'll have to bring in some geometry.
Let's call the height of the lamp post "h" (pretty original, I know) and the length of the shadow "s". According to the picture, we have a right triangle formed by the man, his shadow, and the lamp post.
Now, we can use the magic of similar triangles to solve this (cue the abra-cadabra sound effects). The ratio of the length of the man to his shadow is equal to the ratio of the height of the lamp post to its shadow. In mathematical terms, we have:
man / shadow = lamp post / shadow
Plugging in the values we know:
6 / 20 = h / 40
Now let's cross-multiply and solve for h:
6 * 40 = 20 * h
240 = 20h
h = 240 / 20
h = 12
Ta-da! The height of the lamp post is 12 feet. And that's how you illuminate a math problem with a dash of clown magic!
Yes, there is another way to solve this problem using similar triangles. Let me explain the steps:
1. Draw a diagram to represent the situation. Draw a vertical line to represent the height of the man, another line representing the height of the lamp post, and a line connecting the top of the man's head to the tip of his shadow.
2. Notice that the vertical line representing the man's height and the line representing the shadow form two sides of a right-angled triangle. The line connecting the top of the man's head to the tip of his shadow is the hypotenuse of this triangle.
3. The ratio of the length of the man's height to the length of his shadow is the same as the ratio of the height of the lamp post to the length of the entire shadow. This is because both triangles are similar.
4. Set up a proportion using this ratio: Let's call the height of the lamp post "h". The proportion is 6/20 = h/40.
5. Cross multiply: 6 * 40 = 20h.
6. Solve for h: Divide both sides of the equation by 20, h = (6 * 40) / 20 = 12 feet.
So, the height of the lamp post is 12 feet, using the method of similar triangles.
Yes, there is another way to solve this problem using similar triangles.
In this scenario, we have two similar triangles: the man's shadow triangle and the lamp post's shadow triangle. The height of the man and the height of the lamp post are corresponding sides of these triangles.
Using the ratio of corresponding sides, we can set up the following equation:
(man's height) / (man's shadow) = (lamp post's height) / (lamp post's shadow)
Substituting the given values, we get:
6 ft / 20 ft = h / 40 ft
Cross multiplying both sides:
6 ft * 40 ft = 20 ft * h
240 ft = 20 ft * h
Dividing both sides by 20 ft:
h = 240 ft / 20 ft
h = 12 ft
Therefore, the height of the lamp post is 12 feet.
Both methods will give you the same answer, so you can choose whichever method you find more convenient or intuitive.