dog business project for apex

Question

10. What are the break-even points of the profit function (the values of x where profit equals 0)? Use the quadratic formula. (4 points: 2 points for each x-value)

11. According to your model, between what two prices should you price your service? Why would you want to keep your price between those limits? (4 points: 2 points for the break-even values, and 2 points for the explanation)

12. Sketch a graph of your profit function. (4 points: 1 point for each blank and 1 point for graph)

Vertex: ______________

x-intercepts: ______________ and ____________

Answer 1

First things first- your equation doesn't seem to have posted! Wyzant will only let you paste certain types of data into these text boxes, so when you submit anything other than text, it might not show up on our end.

Since we're solving for break-even points using the quadratic formula, we can assume that your equation follows this general format:

f(x) = ax² + bx + c

Any of a, b, or c could be zero or negative. That's okay! For example, we might see that your equation is as follows:

0 = 15x² - 5

This just means that a = 15, b = 0, and c = -5. When we evaluate, we get two break-even x values where the function equals 0.

Now, we use the quadratic formula to solve.

x = -b ± √(b² - 4ac)

2a

You can tell that we're going to get two results for x here because of that ±. Whether we add or subtract the square root of b² - 4ac, we will get a break-even x value.

Let's say a = -2, b = 45, and c = -20.

Plugging these values into the quadratic formula, we get two x values: 22.05 and 0.4536. This means that our price should be somewhere between $0.46 and $22.05. Any lower than $0.46 and we're spending more than we're earning, any higher than $22.05 and we're scaring off customers (thus holding inventory and once again spending more than we're earning).

Your x-intercepts will be these x-values we just solved for, and the vertex will be the point on your curve where f(x) is the highest. To optimize f(x) and find this vertex (where you're making the most money), we need to solve for the x and y coordinates.

x_opt = -b / 2a = -45 / (2 * -2) = 11.25

This is the point at which you should ideally price your service.

For y_opt, we just plug this x value into f(x). -2*(11.25)² + 45*11.25 -20 = 233.125.

This means that if we set our price at $11.25, we will maximize our earnings over the given time period and make $233.13.

Quick final note: I assume here that your x values are price points because of the way the question is phrased, but this can vary in different scenarios. It may be the number of walks over the given time period or the number of customers you're servicing. It really depends on the specific model.