The function c(x) = 29x + 54.5 represents the cost (in dollars) of cable for x months, including the $54.50 installation fee.

a. how much would you have spent on cable after 6 months?

b. how many months of cable service can you have for $344.50?

a.

b. 10 months.

Step-by-step explanation:

a. Given the function the following function:

 You can calculate the amount of money (in dollars) you would spent on cable after 6 months through this procedure:

- Substitute into the function:

- Evaluating, you get:

b. The steps to calculate the number of months of cable service that you can you have for $344.50, are:

- Rewrite the function in this form:

- Substitute into the function and then solve for "x":

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240