A man 6 feet tall walks along a walkway which is 30 feet from a the base of a lamp which is 126 feet tall. The man walks at a constant rate of 3 feet per second. How fast is the length of his shadow changing when he is 40 feet along the walkway past
the closest point to the lamp?
My answer always is 3/20 but the right answer is 3/25. What can I do? Help me please!!!!!
To solve this problem, we can use similar triangles. Let's start by drawing a diagram to better understand the situation:
Lamp (126 ft)
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|
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|----------------- Walkway
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| Man (6 ft)
x|
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We need to find out how fast the length of the man's shadow is changing when he is 40 feet along the walkway (represented by x).
Let's call the length of the man's shadow y. From the diagram, we can see that we have two similar triangles - one formed by the man, his shadow, and the lamp, and another formed by the lamp, the man's shadow, and the total shadow.
Using the concept of similar triangles, we can set up the following ratios:
1. For the man's triangle:
(y + 6) / x = 126 / (30 + x)
2. For the total shadow triangle:
y / x = 126 / 30
To find the relationship between y and x, we can solve equation 2 for y:
y = (126 / 30) * x = 21/5 * x
Now, let's substitute this expression for y into equation 1:
(21/5 * x + 6) / x = 126 / (30 + x)
Simplifying the equation:
(21x + 30x + 6x) / 5x = 126 / (30 + x)
(57x) / 5x = 126 / (30 + x)
57 / 5 = 126 / (30 + x)
57(30 + x) = 5 * 126
1710 + 57x = 630
57x = 630 - 1710
57x = -1080
x = -1080 / 57
x ≈ -18.95
Since distance cannot be negative, we discard the negative value and assume that x ≈ 18.95.
So, when the man is 40 feet along the walkway (x = 40), the shadow length (y) is approximately 21/5 * 40 = 168 feet.
To find the rate of change of y with respect to x, we can differentiate equation 2 implicitly with respect to time (t), keeping in mind that the man's rate of walking is constant at 3 ft/s:
d/dt (y) = d/dt(21/5 * x)
dy/dt = (21/5) * dx/dt
Given that dx/dt = 3 ft/s, we can substitute the values:
dy/dt = (21/5) * 3
dy/dt = 63/5
dy/dt = 12.6 ft/s
So, the rate at which the length of the man's shadow is changing when he is 40 feet along the walkway is approximately 12.6 ft/s.
The correct answer is 63/5, which is 12.6, not 3/25. Therefore, it seems like there was an error in your calculations. Double-check your steps and calculations to identify where the mistake occurred.
When the man has walked a distance x, his distance d from the pole is
d^2 = x^2+30^2
If his shadow then has length s,
s/6 = (d+s)/126
or,
d = 20s
So, plugging that in,
400s^2 = x^2+30^2
at x=40, s = 5/2
800s ds/dt = 2x dx/dt
2000 ds/dt = 80*3
ds/dt = 3/25
When you get stuck, or cannot agree with the answer, it'd be nice for you to show your work, huh?