Question

A popular video game retailer develops apps. The total profit, in dollars per day, after x days can be modeled by the function R(x) = 3x3 – 21x2 + 21x + 45. The total cost, in dollars per day, after x days, can be modeled by the function A(x) = x2 – 2x – 3. After how many days does the retailer start making money?

Answer 1

The difference between the cost and revenue functions is the profit function.

The retailer will start making money after 3 days.

Given that:

$R(x) = 3x^3\:-\:21x^2\:+\:21x\:+\:45$

$A(x) = x^2 - 2x - 3$

The retailer will start making money when R(x) > 0 i.e. when the profit is greater than 0.

So, we have:

$3x^3 - 21x^2 + 21x + 45 > 0$

Factor out 3

$3(x^3 - 7x^2 +7x + 15) > 0$

Divide through by 3

$x^3 - 7x^2 +7x + 15 > 0$

Expand

$x^3 - 8x^2 + 15x + x^2 - 8x + 15 > 0$

Further expand

$x^3 - 5x^2 - 3x^2 + 15x + x^2 - 5x - 3x + 15 > 0$

Factorize

$x^2(x - 5) - 3x(x - 5) + x(x - 5) - 3(x -5) > 0$

Factor out the common factors

$(x^2 - 3x)(x - 5) + (x - 3)(x - 5) > 0$

Factor out x

$x(x - 3)(x - 5) + (x - 3)(x - 5) > 0$

Factor out (x - 3)(x - 5)

$(x - 3)(x - 5)(x + 1) > 0$

Split

$x - 3> 0$ or $x - 5> 0$ or $x + 1 > 0$

Solve for x

$x > 3$ or $x > 5$ or $x > -1$

x cannot be negative.So:

$x > 3$ or $x > 5$

Because 3 is less than 5, then the reasonable solution to the set is:

$x > 3$

Hence, the retailer will start making money after 3 days.

Read more about profit, cost and revenue functions at:

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