# Calculus

posted by .

A man 2m tall walks away from a lamppost whose light is 5m above the ground. If he walks at a speed of 1.5m/s, at what rate is his shadow growing when he is 10m from the lamppost?

I tried to draw a diagram, but I don't understand where the 5m is, the height of the lamppost? Is the question asking for the derivative of the speed?

• Calculus -

draw a lamppost, vertical line, and label it 5 m high (we assusme the light is at the top of the lamppost)
draw a horizontal line, the sidewalk? , and draw the man , vertical line, somewhere on the sidwalk.
Join the top of the lamppost to the man's head and continue until you hit the sidewalk.
label the distance between the post and the man as y, label the length of his shadow x
we are given dy/dt as 1.5 m/s, we are to find dx/dt when y = 10 m

I see to right-angled triangles, with a smaller inside a larger.
By similar triangles:
5/(x+y) = 2/x
cross-multiply, ...
5x = 2x + 2y
3x = 2y

differentiate with respect to t
3dx/dt = 2dy/dt
dx/dt = 2(1.5)/3 = 1 m/s

Notice we did not need the 10 m, there was no place to sub it in.

We have shown that the man's shadow is growing at a constant rate of 1 m/s, no matter where he is.

Be careful with this question.
Had it asked "how fast is the man's shadow moving", we would have to add the man's speed to the 1 m/s for a speed of 2.5 m/s.

• Calculus -

My goodness!
I just noticed that I typed "to" instead of "two" in "I see to right-angled triangles ..."

• Calculus -

Thank you so much for clearing it up showing the steps crystal clear. How do you know that 1.5m/s is dy/dt?

• Calculus -

2.5 km/h

• Calculus -

thanks a lot Reiny.

• Calculus -

The y distance is based on how fast the man is walking. In respect to time, dy/dt is 1.5m/s.

## Similar Questions

1. ### Calculus

Shadow Length A man 6 feet tall walks at a rate of 3 ft per second away from a light that is 16 ft above the ground (see figure). When he is 11 ft from the base of the light find the following. (a) The rate the tip of the shadow is …
2. ### precalculus

A man is walking away from a lamppost with a light source h = 6 m above the ground. The man is m = 1.5 m tall. How long is the man's shadow when he is d = 11 m from the lamppost?
3. ### calculus

A 6-foot man walks away from a 16-foot lamppost at a speed of 5 ft/sec. Find exactly the rate at which his shadow is increasing in length.
4. ### Calculus

A street light is hung 18 ft. above street level. A 6-foot tall man standing directly under the light walks away at a rate of 3 ft/sec. How fast is the tip of the man's shadow moving?
5. ### algebra

A man is walking away from a lamppost with a light source h = 6 m above the ground. The man is m = 1.5 m tall. How long is the man's shadow when he is d = 12 m from the lamppost?
6. ### Math Urgent help please

A lamppost casts a shadow of a man who is standing 15 feet away from the lamppost. The shadow is 5 feet long. The angle of elevation from the tip of the shadow to the lamp is 50. To the nearest foot, the lamppost is _____ feet tall. …
7. ### Calculus

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?
8. ### AP Calculus AB

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 …
9. ### Calculus

A man is 2 m tall is walking at a rate of 1 m per second in a straight line away from a 10 m lamppost. How fast is the tip of his shadow moving away from the lamppost?
10. ### math

a man 1.9 meters tall walks away from a light 2.4m above the ground. if his shadow lengthens at a rate of 0.6 m/s, how fast is walking?

More Similar Questions

Post a New Question