# Related Rates

## Homework Statement

(Q) A lamp is placed at the point (5,0) and it casts the shadow of an ant onto the y axis. When the ant is at point (1,2), how fast is the ant's shadow moving when the ant's x-coordinate is increasing at the rate of 1/2 units/sec and its y-coordinate is decreasing at 1/5units/sec?

I got -9/16. Is it correct? I just need to confirm whether I'm right or wrong.

## The Attempt at a Solution

HallsofIvy
Homework Helper
That's not at all what I get. Too bad you didn't show your working- I might have been able to point out an error.

Here we go!!

Let the coordinates of the ant be x,y. Let the length of the shadow be s.

Our goal is to find ds/dt. By similar triangles, s/y = 5/(5-x).

So, 5s - sx = 5y
Differentiating,

5(ds/dt) - (s(dx/dt) + x(ds/dt)) = 5(dy/dt)

When (x,y) = (1,2), s = 5/2.

5(ds/dt) - (5/2)(1/2) - (1)(ds/st) = 5(-1/5).

4(ds/dt) = -9/4.

ds/st = -9/16

HallsofIvy
Homework Helper
Let the coordinates of the ant be x,y. Let the length of the shadow be s.

Our goal is to find ds/dt. By similar triangles, s/y = 5/(5-x).

So, 5s - sx = 5y
Differentiating,

5(ds/dt) - (s(dx/dt) + x(ds/dt)) = 5(dy/dt)

When (x,y) = (1,2), s = 5/2.

5(ds/dt) - (5/2)(1/2) - (1)(ds/st) = 5(-1/5).

4(ds/dt) = -9/4.
You "lost a sign"
4(ds/dt)= +5/4- 1= 1/4, not -9/4.

ds/st = -9/16

Dammit!!

All because of this gross gross gross error, I got the answer wrong in the exam. How many marks do you think I'll lose (out of 5?).??

Thanks a ton!!