# Math Help - Secant Method (Numerical Analysis)

1. ## Secant Method (Numerical Analysis)

Hello,
I'm having trouble with the following past exam question.

A fisherman wants to set the net at a water depth where the temperature is 40 degrees F. By dropping a line with a thermometer attached, he finds that the temperature is 38 degrees at a depth of 12 meters, and 46 at a depth of 5 meters. Use the Secant Method to determine a best estimate for the depth at which the temperature is 40.

Now from my calculations the equation that fits these points is

$f(x) = \frac{7}{8} x - 45.25$

so I know that the depth at which the temp. is 40 is 10.25m

The thing that Im having a problem with is how to solve it using the Secant method.

The secant method is easy enough to apply if I am asked to find the root of an equation with initial guesses $x_0 \ and \ x_1$.

But Im not trying to find the root here so I am lost

Any ideas?

2. That's what you just did!

That is exactly what the "secant" method is. Given an equation, f(x)= 40, and two points, x and y, such that f(x)< 40< f(y), you construct the straight line from (x, f(x)) to (y,f(y)) and solve for the point, z, on that line that gives a value of 40. If f(z) is not 40, it is either above it or below it so you repeat with this new point.

Here, you are not given "f" but you are given the first two points. The only thing you can do is use the secant method to find "z" for the first step. And that's what you have already done.