A rental company purchases a truck for $14,500. The truck requires an average cost of $6.75 per day in maintenance.

(a) Find a linear function that expresses the total cost C of owning the truck after t days.
C(t) = 1

(b) The truck rents for $55.00 a day. Find a linear function that expresses the revenue R when the truck has been rented for t days.
R(t) = 2

(c) The profit after t days, P(t), is given by the function P(t) = R(t) − C(t). Find the linear function P(t).
P(t) = ___?

(d) Use the function P(t) that you obtained in part (c) to determine how many days it will take the company to break even on the purchase of the truck. Assume that the truck is in use every day. ( round answer to nearest day)

(a) To find a linear function that expresses the total cost C of owning the truck after t days, we need to consider the initial cost of the truck and the maintenance cost per day.

The initial cost of the truck is $14,500. Each day, there is an additional maintenance cost of $6.75. So, the linear function for the total cost C can be expressed as:

C(t) = 14500 + 6.75t

(b) To find a linear function that expresses the revenue R when the truck has been rented for t days, we need to consider the rental price per day.

The rental price for the truck per day is $55.00. So, the linear function for the revenue R can be expressed as:

R(t) = 55t

(c) The profit after t days, P(t), is given by the function P(t) = R(t) - C(t). To find the linear function P(t), we can substitute the expressions for R(t) and C(t) into the equation.

P(t) = R(t) - C(t)
= (55t) - (14500 + 6.75t)
= 55t - 14500 - 6.75t
= 48.25t - 14500

So, the linear function P(t) is given by:
P(t) = 48.25t - 14500

(d) To determine how many days it will take the company to break even on the purchase of the truck, we need to find the value of t when P(t) is equal to zero.

P(t) = 48.25t - 14500
0 = 48.25t - 14500

To solve for t, we can isolate the variable t:

48.25t = 14500
t = 14500 / 48.25
t ≈ 300.518

Since the number of days must be a whole number, we round t to the nearest day:

t ≈ 301

Therefore, it will take the company approximately 301 days to break even on the purchase of the truck, assuming it is in use every day.