# How do you write an equation of a line if it take Andrew 20 minutes to get from a depth of 400 feet to a depth of 50 feet?

Dec 25, 2014

We first have to translate to $x$ and $y$.

We can take time as the $x$ (in minutes) and depth as $y$ (in feet)

If we say Andrew started at time zero at a depth of 400 feet, this translates in a point $\left(0 , 400\right)$ on a graph, and after 20 minutes he will be at $\left(20 , 50\right)$

A straight line equation is of the form $y = m \cdot x + b$, where m is the slope of the equation. In this example this will be equivalent to the speed in which Andrew rises, actually the rate at which depth dimishes (in feet per minute).

First we determine $m$:
The change is from $400 \to 50$
change in $y = 50 - 400 = - 350$
change in $x = 20 - 0 = 20$
(Remember: change is always calculated as new minus old)

Then divide the two: $m = - \frac{350}{20} = - 17.5$ (feet per minute)

In the original equation $y = m \cdot x + b$ we can now put in the things we know:
Time 0, depth 400
$400 = - 17.5 \cdot 0 + b \to b = 400$

So the total equation becomes: $y = - 17.5 \cdot x + 400$

Just to make sure you can also check if this is valid for the other point we know and see what happens if we fill in for $x = 20$
$y = - 17.5 \cdot 20 + 400 = 50$ and this is correct.

Challenge:
After how many minutes does Andrew reach the surface?
(tip: solve for $y = 0$)