Michael R. answered • 12/08/22
Teacher of Mathematics with 18 years of Experience
Hi Maya. In this problem the profit is described by the quadratic expression, y = -.001x2 + 3.6x - 500, where x is the number of meals sold and y is dollars. This expression is in the standard form y = ax2 + bx + c.
I'm guessing from the phrasing of the question, that your teacher expects you to be familiar with maximums of quadratic relationships. (Minimums, too.) This also makes me think that you've looked at their graphs, as well. Recall that they're called parabola and they form nice smooth curves which open either upward (like a U) or downward (like an up-side-down U)
Do you recall that the quadratic coefficient, "a" determines the orientation of the graph. If it's positive the graph opens upward, and the vertex is the minimum. On the other hand, when the "a" is negative the curve opens downward and the vertex represents the maximum of the graph,
In this problem, a = -.001, which is negative, so the graph opens downward and therefore has a maximum. We need to find the x and y coordinates of this point. The x will be the number of meals and the y will be the profit.
the is a formula that we can use to find the axis of symmetry of the parabola. (That's a vertical line that splits the graph in half. It also is the x-coordinate of the vertex.) The formula is x = -b/2a.
In this problem the linear coefficient b = 3.6. and x = -3.6/-.001 =1800.
So, the maximum profit will occur when they sell 1800 meals.
Next, we need to find what that profit will be. We do this by plugging 1800 for x into the original expression.
y = -.001(1800)2 + 3.6(1800) - 500. Evaluating this results in $2,740.
Remember, word problem must get a word answer, something like...
The concession stand will make a maximum profit of $2,740 when it sells 1800 meals.
We're done with this problem, but I suspect that there may be another issue which I'd like to address.
Many young people can't make sense of the fact that the profit doesn't keep increasing as you sell more and more stuff. Don't try to read too much into the situation. The problem doesn't state WHY the profit adheres to a quadratic model; it ONLY tells us that IT DOES. Just work with the information provided.
I hope this helps.
