Doug C. answered • 10/28/20

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Sara B.

asked • 10/28/20A student 6 feet tall stands a few feet in front of a 12 feet tall lamppost and starts to walk in a straight path away from the lamppost. When the tip of his shadow is moving 5 feet per second from the base of the lamppost, how fast is he walking away from the base of the lamppost?

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Doug C. answered • 10/28/20

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Daniel B. answered • 10/28/20

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Please draw a picture of the situation:

- A horizontal line represents the ground.

- A vertical line 12 feet tall represents the lamppost.

Let B be the base of the lamppost, and L be its top.

- A vertical line 6 feet tall represents the student.

Let F be the point at his feet, and H be the point at the top of his head.

- Let u(t) be the length of line segment BF, which is a function of time t

as the student walks away from the lamppost.

- Draw a horizontal line from H towards the lamppost.

Let D be the point where it strikes the lamppost.

The distance between D and H is also u(t).

- Draw the line LH connecting the points L and H and extending to the ground.

Let S be the intersection between the ground and the line LH.

The point S is the tip of his shadow.

- Let s(t) be distance between B and S.

It is the distance between the fixed lamppost and the tip of the student's shadow,

and it is also a function of time as the student walks away.

- Let α be the angle between the ground and the line LH (at the point S).

(It also happens to be a function of time, but it is not important here.)

- s(t) = BL . cot(α)

BL = 12 feet is the the lampost's hight.

- The angle between DH and LH is also α.

- u(t) = DL . cot(α)

DL is the difference between the height of the lamppost and the student,

which is also 6 ft.

- By eliminating cot(α) from the equations for u(t) and s(t)

u(t) = s(t) . DL / BL = s(t) . 6ft / 12ft = 0.5 s(t)

- Speed is derivative of distance w.r.t time:

du/dt = 0.5 ds/dt

- That means that the student moves at half the speed of the tip of his shadow.

When the shadow moves at 5 ft/s, the student moves at 2.5 ft/s.

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