Related rate expanding rectangle

[synergix]

Homework Statement

The length of a rectangle is increasing at a rate of 8cm/s and its width is increasing at a rate of 3cm/s when the length is 20 cm and the width is 10cm, how fast is the area of the rectangle increasing?

Homework Equations

V=LW

The Attempt at a Solution

dL/dt=8 dW/dt=3
l=20 w= 10
dA/dt=?

(V=LW)'

dA/dt= dL/dt * dW/dt

before I go any further is this correct both the L and the W become one when derived right?

 

[Dick]

A(t)=L(t)W(t), right? Take a look at the 'product rule'. And why would the derivative of L and W be 1?

 

[synergix]

A(t)=L(t)W(t), right? Take a look at the 'product rule'. And why would the derivative of L and W be 1?

I have really just not been using my brain lately. They would be one if they were being added which they're not. (x + a)'= 2 does it not? if they are both variables.

And you have t in parenthesis to make clear that L and W and A change with time and are therefore not constants right?

so according to the product rule->

dA/dt=L(dW/dt)+W(dW/dt)

 

[Dick]

I have really just not been using my brain lately. They would be one if they were being added which they're not. (x + a)'= 2 does it not? if they are both variables.

And you have t in parenthesis to make clear that L and W and A change with time and are therefore not constants right?

so according to the product rule->

dA/dt=L(dW/dt)+W(dW/dt)

Ok, you've got the product rule. But (x+a)'=x'+a' if by ' you mean d/dt. Whether that's 2 or not depends what x and a are. dx/dx=1 but dx/dt doesn't necessarily equal 1.

 

[synergix]

Ok I was just using the power rule (x^1 + a^1)'= 1*x^0 + 1*a^0= 1+1
am I assuming something here I shouldn't be?

 

[Dick]

d(x(t)^1)/dt=1*x(t)^0*dx/dt, chain rule again.

 

[synergix]

Could you explain quickly why you need to use the chain rule here?

 

[Dick]

Because x(t) is a function of t. d(x^1)/dx is 1*x^0 (though you could still throw the chain rule in and write it as 1*x^0*dx/dx, but dx/dx=1). d(x^1)/dt=1*x^0*dx/dt. dx/dx is always 1. dx/dt is not.

 

[synergix]

Because x(t) is a function of t. d(x^1)/dx is 1*x^0 (though you could still throw the chain rule in and write it as 1*x^0*dx/dx, but dx/dx=1). d(x^1)/dt=1*x*dx/dt. dx/dx is always 1. dx/dt is not.

Ok thank you its starting to come together.