# Word problem help

• Oct 26th 2012, 05:14 PM
nubshat
Word problem help
The brightness of a lamp is L(s) = 1500s^-2 lumens where s is the distance in metres
from the observer to the lamp. An observer is walking at 1 metre per second along a
straight line that comes within 3 metres of the lamp at its closest point. How fast (in
lumens per second) is the brightness changing when the observer has passed the lamp and
is 5 metres away from it?

• Oct 26th 2012, 05:44 PM
MarkFL
Re: Word problem help
I would orient the coordinate axes such that the lamp is at the origin and the observer moves along the line $\displaystyle y=3$ from left to right. So, put the observer at the point $\displaystyle (x,3)$ and then the distance between the observer and the lamp is:

$\displaystyle s=\sqrt{x^2+3^2}$ and we have:

$\displaystyle L(x)=\frac{1500}{x^2+9}$

Now, we want to find $\displaystyle \frac{dL}{dt}$, so differentiate with respect to time $\displaystyle t$, and use the given:

$\displaystyle \frac{dx}{dt}=1\frac{\text{m}}{\text{s}}$

$\displaystyle s=5\text{ m}$
• Oct 26th 2012, 05:55 PM
chiro
Re: Word problem help
Hey nubshat.

A you familiar with the chain rule? Recall that if you have x(t) = f(g(t)) then dx/dt = dg/dt * f'(g(t)) where ' means the derivative. In this case your f is 1500s^-2 and your g will be how the distance changes as a function of time where s will be the distance from the lamp as a function of t.

Since the closest point is 3m away then intuitively it will look like a triangle where the distance away is the hypotenuse of the triangle where s^2 = 3^2 + (1*t)^2 since the person is moving perpendicular to the line where the shortest distance is (you might want to draw a right angled triangle to see what is going on).

So you have L as a function of s and s as a function of t and you want to find L'(t) when s^2 = 25.