How to form linear equation from two sets of variables

[dnymeyer]

I have a story problem which I cannot for the life of me, figure out how to create an equation out of it.

Suppose a giant sequioia started growing around 1900 and is well established by 1950. Let 1950 be time 0. In 1960, it had an estimated mass of 28 tons. In 2000, it had an estimated mass of 52 tons.

Write a linear model f(t), where f(t) is the mass of the tree after 1950.

There's a bunch of other questions to go with it, but I just can't figure out the equation.

Here's what I know:
Linear equations are straight lines on a graph, with standard equation form y=mx+b
Where m = the slope of the line, b = y intercept.

All I can think to solve this is
when t = 10, mass = 28
and when t = 50, mass = 52

I really have no idea how to put this into an equation. Would I treat t and f(t) as X and Y and find the slope by saying y2-y1 / x2 -x1 to get the slope, and then use either ordered pair (10,28) or (50,52) to plug into y=mx+b with the slope and solve for b?

I don't even know if those are ordered pairs... help!

 

[MarkFL]

Yes, you have two ordered pairs, with time as the independent variable and mass (although we should use weight here since tons are weight not mass) is the dependent variable. Your points will be in the form \(\displaystyle (t,m)\). As you observed, you have the two points:

\(\displaystyle (10,28),\,(50,52)\)

Two points is sufficient to uniquely describe a linear function. First, find the slope \(\displaystyle a\):

\(\displaystyle \displaystyle a=\frac{\Delta m}{\Delta t}=\frac{m_2-m_1}{t_2-t_1}\)

Then, use the point-slope formula to get mass as a linear function of time:

\(\displaystyle m=a\left(t-t_1\right)+m_1\)

What do you find?

 

[mmm4444bot]

… find the slope by saying (y2-y1) / (x2-x1) … then use either ordered pair (10,28) or (50,52) to plug into y=mx+b with the slope and solve for b? …

Yes, solving for b would also work.

Please note the grouping symbols (shown in red). It's important to use grouping symbols around numerators and denominators, when we type algebraic ratios with a keyboard.

Without the grouping symbols, the expression y2 - y1 / x2 - x1 would evalutate as y2 - (y1/x2) - x1 due to Order of Operations.

In the forum guidelines, there's a link to an explanation about how to type math as text, if you're interested. :cool: