You can solve for time by solving the equation quadratically.
Your equation is correct; d is actually (x-x0), the 'distance' you travel from start point to end point. Usually it's the number of meters travelled; for your problem it's the number of cigarettes.
You've got positive velocity and negative acceleration, so the shape of this curve (cigarettes on the vertical axis, minutes on the horizontal axis) is an inverted parabola, like the path of a football.
If you wish to know the value of t when you've reached '2 cigarettes', then you solve
(x-x0) = v0t + 0.5at^2
where
x = 2 cig
x0 = 1 cig (x0 is 'x-naught', or 'original x')
v0 = 0.006944 cig/min ('original v')
a = -0.0000017 cig/min/min
t = ?
1 = (0.006944)t - 0.000000085t^2
or
0.000000085t^2 - 0.006944t + 1 = 0
which is an equation of the same form as
2t^2 + 6t - 8 = 0
or
At^2 + Bt + C = 0
and the values of t for which it is true are found by the quadratic formula,
t = -B +/- sqrt(B^2 - 4AC)
----------------------------
2A
where +/- means (plus or minus),
sqrt means 'square root of', and
--------- means 'divided by'.
For my 2t^2 example, the answers are
t = 1 and t = 4.
For the numbers you gave, making sure to remember the minus sign in front of B, I get:
t = 144 minutes
and
t = 8022 minutes which is almost six days
Remember the curve is like a football flying through the air: it starts at one cigarette (or meter) high, arcs up through 2 cigs, and then higher still, peaks out at about 15 cigs high, and starts getting lower and lower, until it passes through the 2 cig mark (again) late in the fifth day
Which is the problem. When I read your question, I assumed that the quantity of cigarettes referred to the number smoked *so far*, a cumulative number. The negative acceleration would refer to a person who was *cutting back*, reducing the rate at which they smoke. This is fine; negative accelerations make sense for smoking. BUT negative velocities do not. I cannot be *undoing* the number of cigs I have smoked at so many cigs per minute; it's nonsense.
Unless you consider a celery stick, or a minute on a treadmill, or whatever, to be equal to one 'negative cigarette'...
anyway, I put the roots at 144 and 8022; that's when the level hits two cigs. If you want to know where it hits zero cigs, redo the problem with d = (x - x0) = (0 - 1) instead of (2 - 1) like I did above; you should get about 8310 min and -141 min. (Yes, a negative number; extend the curve I've described *backwards* past the t = 0 line and see that it hits the horizontal axis back there, too: it's again a nonsense solution, given by the math but ruled out by the physics (or common sense) of the problem.)
I've attached a screen capture of the graph I produced in Excel to help me solve this problem; had I had my TI-83 with me, I could've done it in a tenth of the time it took me this way.
Have I answered your question?
P
