Question 109703
The cost of the truck must be what was paid at the start ($19,500) plus what it costs per day to operate ($6.75) times the number of days:
:
{{{C(t)=6.75t+19500}}}
:
Revenue is just what you make per day ($55), times the number of days:
:
{{{R(t)= 55t}}}
:
Now, profit is equal to revenue minus cost, or as given:
:
{{{P(t)=R(t)-C(t)}}}, or substituting our definitions of R(t) and C(t) we get:
:
{{{P(t)=55t-6.75t-19500}}}
:
Breakeven is the point where the profit function, P(t) is 0.  That means that you can set the expression for P(t) equal to zero and solve for t:
:
{{{55t-6.75t-19500=0}}}
{{{48.25t-19500=0}}}
{{{48.25t=19500}}}
{{{t=404.1451}}}
:
Since the question asks how many days it will take the company to break even, you can presume that the desired answer is an integer, therefore, round the answer up to 405.  You have to round up because at the end of the 404th day, P(404) < 0 (just slightly less), so it isn't until the end of the next day when the customer on the 405th day pays his bill that the P(t) function actually goes positive.