A 6 feet tall man walks away from a streetlight that is 15 feet high at a rate of 5 ft/s. Express the length 's' of his shadow as a function of time.
What I have so far:
I set up a triangle and used similar triangles to create a ratio
(6/s) = (s/d+s)
s = (6d + 30/15s)
And I'm stuck...
Please help!
d = 5t, so
6/s = 15/(5t+s)
6(5t+s) = 15s
30t = 9s
s = 30/9 t
To express the length 's' of the man's shadow as a function of time, let's break down the problem step by step.
First, you correctly set up a triangle to visualize the situation. Let's call the length of the shadow 's,' the distance between the man and the streetlight 'd,' and the height of the man 'h.'
Now, let's apply the concept of similar triangles to relate the lengths and heights of the two triangles formed by the man, his shadow, and the streetlight.
Since the height of the man is given as 6 feet and the height of the streetlight is 15 feet, the ratio of the heights can be expressed as:
h / 15 = s / (s + d)
We can then solve this equation for 's':
s = (h * (s + d)) / 15
Substituting the values given:
s = (6 * (s + d)) / 15
Now, let's simplify this equation to express the length 's' of the shadow as a function of time.
s = (6s + 6d) / 15
To eliminate the fraction, we can multiply both sides of the equation by 15 to get:
15s = 6s + 6d
Now, rearrange the equation to solve for 's':
9s = 6d
Finally, divide both sides of the equation by 9 to isolate 's':
s = (6d) / 9
Simplifying this expression gives us the final answer:
s = (2d) / 3
Therefore, the length 's' of the man's shadow in terms of time is given by the function s = (2d) / 3, where 'd' is the distance between the man and the streetlight.