Calculus question - Residents of a town are connected to the municipal water supply.?

Question

They are billed a fixed amount yearly plus a charge for each cubic foot of water used. A household using 1000 cubic feet was billed $90, while one using 1600 cubic feet was billed $105.

A. What is the charge per cubic foot?
B. Write an equation for the total cost of a resident's water as a function of cubic feet of water used.
C. How many cubic feet of water used would lead to a bill of $130?

Answer 1

OK, let's define us some variables, shall we?
Let {f(x) equal total bill; c equal the fixed service charge; r equal the charge per cubic foot; and x equal cubic feet used}.
Now, we know from the question that the bill [f(x)] is going to equal the base rate (c) plus a charge (r) for every (*) cubic foot used (x). So
f(x) = c + r*x
or, for convention's sake
f(x)=rx+c
Now, doesn't that look familiar? Now, one at a time we can plug in what we know and come up with values for r and c.
90 = 1000r + c
105 = 1600r + c
1000r + c - 90 = 0
1600r + c -105 = 0
1000r + c -90 = 1600r + c -105
15 = 600r
A) $0.025 / cu ft = r

Since we've got (r) now, we can plug it back into our function
f(x) = rx+c
f(x) = (0.025)x+c
But that means we still need to find our flat rate. So we plug a set of numbers back in, and go from there. You could use (1600, 105) but for simple math's sake, I'm using the other set.
90 = (0.025)(1000) + c
90 = 25+c
65 = c
To get the answer to B, again, we plug both our numbers in, giving us.

f(x) = (0.025)x + 65

C.
130 = (0.025)x + 65
65 = (0.025)x
2600 = x

Answer 2

This is basic algebra.

This is the equation of a line.

a = amount of bill
x = cubic feet of water used
f = fixed amount

a = mx + f

90 = 1000m + f
105 = 1600m + f

Subtracting the first equation from the second we get:

15 = 600m
m = 15/600 = 1/40

The cost per cubic foot is $1/40 = 2.5 cents
_________________

Plug back in to solve for f.

90 = 1000m + f
90 = 1000/40 + f
90 = 25 + f
f = 65

The equation is:

a = mx + f
a = (1/40)x + 65
______________

Solve for x.

130 = (1/40)x + 65
65 = x/40
2600 = x

2600 cubic feet would lead to a bill of $130.