The vertex of a quadratic function application

Question

Leo owns a gym and needs to determine the monthly membership price p (in dollars). The monthly revenue for his business R (in thousands of dollars), as a function of p is

R(p)= -p² + 54p - 547

What price (in dollars) should Leo charge for a monthly membership to maximize his revenue?

How much in total monthly revenue should Leo expect at this price?

Answer 1

In this problem, you're trying to find the point in the parabola in which the y-coordinate is at the maximum point it could possibly reach. To do that, you need to first solve for where along that parabola (as you look at it from left to right, or in other words, along the x-axis) the highest point would be. Then, you can use that point to calculate how high the parabola reaches. These two steps map to "first, solve for the x-coordinate. Then, solve for the y-coordinate."

To solve for the x-coordinate, you need to use the vertex formula: x=(-b)/2a. (Note: I'm only including the parentheses to make sure the negative sign is clear. It's not necessary.) This requires us to figure out what these new variables, "a" and "b" are. Whenever you have a quadratic equation in the form that yours is in, it's actually already in the correct format for figuring out a, b, and another variable, c (which we don't need in this formula). You do this by determining which coefficients are in front of the variables that match to the formula below:

y = ax² + bx + c

Notice how your equation is already close to this:

R(p) = -p² + 54p - 547

Thus, you can identify the variables a, b, and c as -1, 54, and -547 (respectively). Then, you can plug the variables into the equation x=(-b)/2a.

x=(-54)/2*(-1)

From here, you can figure out what x equals. That's the x-coordinate that corresponds to the price that would have to be charged for the maximum revenue. Then, to figure out how much revenue he can expect from this, you would just need to plug the x-coordinate into the original equation to get the R(p) value.

Good luck!