Question

Sales of the Century video game console, in thousands of units, have been declining ever since a rival company released its new Prism gaming system. Monthly sales of the Century console m months after the Prism gaming system was introduced, can be represented by the function below.

s(m) = 120 - √(25m^2+144)
To combat declining sales, retailers have been adjusting the price of the Century console. The adjusted unit price, in dollars, of the Century console is represented by the function below, where m represents the number of months since the Prism was introduced.
p(m) = 499-20m
Which function represents the revenue, in thousands of dollars, from a month's sales of the console m months after the Prism was introduced?

Answer 1

The revenue from a month's sales of the Century console, m months after the Prism was introduced, can be represented by the function:

r(m) = p(m) * s(m)

where r(m) represents the revenue in thousands of dollars, p(m) represents the adjusted unit price of the Century console in dollars, and s(m) represents the monthly sales of the Century console in thousands of units.

To find the revenue function, we need to substitute the given functions for p(m) and s(m) into the equation.

The adjusted unit price function is given as:

p(m) = 499 - 20m

And the monthly sales function is given as:

s(m) = 120 - √(25m^2 + 144)

Now, substitute these functions into the revenue equation:

r(m) = (499 - 20m) * (120 - √(25m^2 + 144))

Simplifying the expression further requires expanding and simplifying the multiplication, but the resulting equation may be complex and involve radicals. Therefore, it is best to leave the expression as is.

So, the function that represents the revenue from a month's sales of the Century console m months after the Prism was introduced is:

r(m) = (499 - 20m) * (120 - √(25m^2 + 144))

Please note that the revenue function may not simplify further, but this equation provides a way to calculate the revenue based on the number of months since the Prism was introduced.