Solving Related Rates Problem: Find dz/dt Given Coordinates and Derivative

[FallingMan]

Homework Statement

[PLAIN]http://img638.imageshack.us/img638/2218/questionscreenshot.jpg



The attempt at a solution

Step 1. Draw a triangle. Solve for third side (Pythagorean theorem), which I called z.

z^2 = x^2 + y^2

Coordinates are given (4, 6). We know x = 4, y = 6.

z^2 = 16 + 36
z^2 = 52
z = Sqrt(52)

Step 2. Take derivative of Pythagorean theorem:

d/dx (z^2 = x^2 + y^2)

2*z*dz/dt = 2*x*dx/dt + 2*y*dy/dt

Step 3. We know that dx/dt = 3 units per second.
We know that dy/dt can be derived from the function provided to us:

y = 2*sqrt(2x+1)
dy/dt = 2/sqrt(2x+1)

We know that x = 4
dy/dt = 2/sqrt(9)
dy/dt = 2/3

Step 4: Plug in all known values into derivative of Pythagorean theorem and solve for dz/dt


2*z*dz/dt = 2*x*dx/dt + 2*y*dy/dt
2*(Sqrt(52))*dz/dt = (2*4*3) + (2*6*(2/3))

dz/dt = (24 + 8)/(2*(sqrt(52))
dz/dt = 32/(2*sqrt(52))
dz/dt = 2.21880078490092

I tried both 32/(2*sqrt(52)) and 2.21880078490092 as the answer and they still do not work (I tried all sorts of variations of this answer). What mistake am I making?

Damn webwork...

 

[Mark44]

Homework Statement

[PLAIN]http://img638.imageshack.us/img638/2218/questionscreenshot.jpg



The attempt at a solution

Step 1. Draw a triangle. Solve for third side (Pythagorean theorem), which I called z.

z^2 = x^2 + y^2

Coordinates are given (4, 6). We know x = 4, y = 6.

z^2 = 16 + 36
z^2 = 52
z = Sqrt(52)

Step 2. Take derivative of Pythagorean theorem:

d/dx (z^2 = x^2 + y^2)

2*z*dz/dt = 2*x*dx/dt + 2*y*dy/dt

Step 3. We know that dx/dt = 3 units per second.
We know that dy/dt can be derived from the function provided to us:

y = 2*sqrt(2x+1)
dy/dt = 2/sqrt(2x+1)

The line above is incorrect. It seems that you forgot to use the chain rule.

We know that x = 4
dy/dt = 2/sqrt(9)
dy/dt = 2/3

Step 4: Plug in all known values into derivative of Pythagorean theorem and solve for dz/dt


2*z*dz/dt = 2*x*dx/dt + 2*y*dy/dt
2*(Sqrt(52))*dz/dt = (2*4*3) + (2*6*(2/3))

dz/dt = (24 + 8)/(2*(sqrt(52))
dz/dt = 32/(2*sqrt(52))
dz/dt = 2.21880078490092

I tried both 32/(2*sqrt(52)) and 2.21880078490092 as the answer and they still do not work (I tried all sorts of variations of this answer). What mistake am I making?

Damn webwork...

 

[FallingMan]

y = 2*sqrt(2x+1)
dy/dt = 2/sqrt(2x+1)


How is this wrong? I did use chain rule. Even wolfram alpha says it's right:

[PLAIN]http://img143.imageshack.us/img143/8063/wolframss.jpg

 

[Mark44]

My mistake. It seemed to me that you hadn't used the chain rule.

After working the problem through, I'm getting essentially the same answer as you, but in a slightly different form.

I got d' to be 8/sqrt(52), which is equal to your 32/(2sqrt(52). It might be that the application is looking for the answer in a specific form, and isn't smart enough to recognize different forms that have the same value.

Simplifying my answer, I get 4/sqrt(13), which is the same as 4sqrt(13)/13. Maybe one of these will work.