Hello! I'm having trouble with this related rates problem: A 5ft tall person is walking away from a 16ft tall lamppost at a rate of (2/x) ft/sec, where x is the distance from the person to the lamppost. Assume the scenario can be modeled with right triangles. At what rate is the length of the person's shadow changing when the person is 12 ft from the lamppost?
The answer the teacher gave me is 5/66 ft/sec. I'm wondering how she got that answer??
did you make your sketch?
in my picture, I let the length of the person's shadow be y , and the distance of the person form the post as x as suggested.
so by similar triangles,
5/y = 16/(x+y)
16y = 5x+5y
11y = 5x
11 dy/dt = 5 dx/dt
dy/dt = (5/11)(2/x) = 10/(11x)
this is the rate we want when x = 12
rate at which shadow is changing
= dy/dt
= 10/132
= 5/66 ft/s
To solve this related rates problem, you can use the properties of similar triangles and the chain rule from calculus. Here's a step-by-step explanation of how to solve it:
Step 1: Understand the problem and draw a diagram.
In this problem, we have a person walking away from a lamppost, and we need to find the rate at which the length of the person's shadow is changing. Let's draw a diagram to better visualize the situation.
L P S
_|___ ____|____ _|___
\. x \.
\_. \_.
\_ \_
\_ h=5 \_
\_ \_
\_ \_
------|--------------------------|------
lamppost shadow
Here, L represents the lamppost, P represents the person, S represents the person's shadow, and x represents the distance from the person to the lamppost. The height of the person is given as h = 5 ft.
Step 2: Identify the variables involved and the given information.
From the problem, we know that the person's height (h) is 5 ft, and the rate at which the person is moving away from the lamppost is (2/x) ft/sec.
Step 3: Relate the variables using similar triangles.
Looking at the diagram, we have a pair of similar triangles formed by the lamppost, the person, and the shadow. By the property of similar triangles, the ratio of the corresponding sides will be equal.
Let's denote the length of the person's shadow as s. The corresponding sides of the triangles are:
- For the lamppost triangle: height = h and the distance from the person (x).
- For the shadow triangle: height = s and the distance from the person (12 ft).
Using the property of similar triangles, we can write the following ratio:
h/x = s/12
Step 4: Differentiate with respect to time.
To find the rate at which the length of the person's shadow is changing, we need to differentiate the above equation with respect to time.
Differentiating both sides of the equation with respect to time (t) using the chain rule, we get:
(dh/dt)/x - h(dx/dt)/x^2 = ds/dt/12
Step 5: Plug in the given values and solve for ds/dt.
Now we can substitute the given values into the equation. We know the person's height (h) is 5 ft and the distance from the person (x) is 12 ft.
Plugging in these values, we get:
(dh/dt)/12 - 5(dx/dt)/144 = ds/dt/12
Substituting the given rate at which the person moves away from the lamppost (2/x), we can write dx/dt as 2/12.
Now, solving the equation for ds/dt gives us:
ds/dt = (dh/dt - 5(dx/dt)/144) * 12
Step 6: Calculate the values and find the solution.
Here, we are given that the person's height remains constant at 5 ft, so dh/dt is 0. The rate at which the person moves away from the lamppost (dx/dt) is 2/12.
Plugging in these values, we get:
ds/dt = (0 - 5(2/12)/144) * 12
ds/dt = (-5/72) * 12
ds/dt = -5/6
Since we are interested in the magnitude of the rate, we take the absolute value, giving us:
|ds/dt| = 5/6 ft/sec
Therefore, the rate at which the length of the person's shadow is changing when the person is 12 ft from the lamppost is 5/6 ft/sec.
It seems that you mentioned the answer given by your teacher as 5/66 ft/sec. However, based on the steps explained above, the correct answer should be 5/6 ft/sec. Please double-check the problem statement or consult your teacher for clarification.