A man standing 9 feet from the base of a lamppost casts a shadow 6 feet long. If the man is 6
feet tall and walks away from the lamppost at a speed of 30 feet per minute, at what rate, in feet per
minute, will his shadow lengthen?
I dont get where you pulled the 9 from, 5 years late
Well, it looks like the man is trying to outshine the lamppost! But let's not overshadow the question at hand.
Since the man is 6 feet tall and his shadow is 6 feet long when he's standing 9 feet away from the lamppost, we can say that the ratio of his height to distance from the lamppost is 1:1.
Now, if the man walks away from the lamppost at a speed of 30 feet per minute, the distance between him and the lamppost will increase. Let's call this distance "x". The shadow length will also increase, let's call it "y".
So, we can form a similar triangle where the man's height is to his original distance (9 feet) as the shadow length is to the new distance (9 + x feet).
Using proportions, we can say:
6/9 = y/(9 + x)
Now, let's differentiate both sides of the equation with respect to time (since we want to find the rate of change of the shadow length):
6/9 = y'/(9 + x)
Here, y' represents the rate at which the shadow lengthens. Solving for y', we find:
y' = (6/9)*(9 + x)
Now we substitute the value of x with the man's walking speed of 30 feet per minute:
y' = (6/9)*(9 + 30)
Calculating this out, we find:
y' = 36 feet per minute
So, at a walking speed of 30 feet per minute, the man's shadow lengthens at a rate of 36 feet per minute. Looks like his shadow wants to keep up with him!
To find the rate at which the man's shadow lengthens, we need to understand the relationship between the man's height, his distance from the lamppost, and the length of his shadow.
Let's start by drawing a diagram to visualize the situation. We have a lamppost, a man standing some distance away from it, and a shadow being cast by the man.
Given information:
- Distance from the man to the lamppost: 9 feet
- Length of the man: 6 feet
- Initial length of the shadow: 6 feet
- Rate at which the man walks away from the lamppost: 30 feet per minute
Let's define some variables for the distance of the man from the lamppost (x), the length of the shadow (y), and the rate at which the shadow lengthens (dy/dt).
From the diagram, we can see that we have similar triangles formed by the man, his shadow, and the lamppost. The ratios of corresponding sides in similar triangles are equal. So, we can set up the following proportion:
(man's height) / (length of the shadow) = (distance from the man to the lamppost) / (length of the shadow + rate at which the shadow lengthens)
6 / 6 = 9 / (6 + dy/dt)
Simplifying the equation, we have:
1 = 9 / (6 + dy/dt)
Cross-multiplying, we get:
6 + dy/dt = 9
Now, we can solve for the rate at which the shadow lengthens (dy/dt):
dy/dt = 9 - 6
dy/dt = 3
Therefore, the rate at which the man's shadow lengthens is 3 feet per minute.
If the man's shadow has length s when his distance from the pole of height h is x, then using similar triangles,
(x+s)/h = s/6
6x+6s = hs
s(h-6) = 6x
so,
(h-6) ds/dt = 6 dx/dt
ds/dt = 6*30/(h-6) = 180/(h-6)
Now, we were given
15/h = 6/6, so h=15
Thus, ds/dt = 180/9 = 20 ft/min