As a man walks away from a 12 foot lamppost, the tip of his shadow moves twice as fast as he does. What is the man's height?
The phrase shadow moves twice as fast as he does dosen't make a lot of sense. The shadow will move at the same rate as the man. I'm guessing this really means the shadow is twice the distance from the lamp post as the man at any given time.
From the diagram:
Let the distance of the shadow from lamp be Let the distance of the man from the lamp be Let height of man be Therefore: The man is 6 foot tall.
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To find the man's height, we can set up a proportion based on the relationship between the man's height and the length of his shadow. Let's denote the man's height as ( h ) and the length of his shadow as ( s ). Since the tip of his shadow moves twice as fast as he does, the rate at which the tip of his shadow moves relative to him is ( 2:1 ).
Using similar triangles, we can set up the following proportion:
[ \frac{h}{s} = \frac{1}{2} ]
We know that the length of the shadow (( s )) is the sum of the distance from the man to the base of the lamppost and the height of the lamppost itself, which is ( 12 ) feet. So, ( s = 12 + h ).
Substituting ( s ) into the proportion, we get:
[ \frac{h}{12 + h} = \frac{1}{2} ]
Cross multiplying:
[ 2h = 12 + h ]
Solving for ( h ):
[ 2h - h = 12 \implies h = 12 ]
Therefore, the man's height is ( 12 ) feet.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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