# Math Help - Function Problem

1. ## Function Problem

A street light is 12 feet above a straight bike path. Olav is bicycling down the path at a rate of 15 MPH. At midnight, Olav is 39 feet from the point on the bike path directly below the street light. (See the picture.) The relationship between the intensity C of light (in candlepower) and the distance d (in feet) from the light source is given by C = (k/d^2), where k is a constant depending on the light source.

a) From22 feet away, the street light has an intensity of 1 candle. What is k?
22^2 = k = 484

b)
Find a function which gives the intensity I of the light shining on Olav as a function of time t, in seconds.

So below is the answer for "b" but I am confused on how to get this, can someone explain to me? Thank you!

2. ## Re: Function Problem

From the result of part a), we know:

$I=\frac{22^2}{d^2}$

Using the Pythagorean theorem, we may state:

$d^2=12^2+x^2$

where $x$ is the distance from the point directly below the lamp that Olav is at at time $t$.

If midnight is set to time $t=0$, then we have:

$\frac{dx}{dt}=15\cdot\frac{\text{mi}}{\text{hr}} \cdot\frac{5280\text{ ft}}{1\text{ mi}} \cdot\frac{1\text{ hr}}{3600\text{ s}}=22\frac{\text{ft}}{\text{s}}$ where $x(0)=-39$

Thus:

$x(t)=22t-39$ and so:

$d^2=12^2+(22t-39)^2=484t^2-1716t+1665$

and thus, we find:

$I(t)=\frac{484}{484t^2-1716t+1665}$

3. ## Re: Function Problem

Thank you so much!