How do I solve this word problem?

I'm having a really hard time with this word problem. Could someone please solve it and walk me through their method? Thank you.

A large pizza franchise delivers large pizzas for $16. The equation y=-3x^2+72x+480 describes the profit, y, based on the increase, x, in the price of their large pizzas. Determine the range of prices, to the nearest cent, for which the franchise could sell each pizza and still make a profit of at least $640.

PS: I have to solve it without graphing, so please don't graph it. Thank you again.
PPS: I asked this question on another forum and didn't really understand the answers I was shown. I think the reason for that was because the first answer had shortcuts in it and the second answer was just completely beyond me. The first answer made more sense to me, but I think I need to see it done without shortcuts for it to sink in. So if anyone could help me with that I think it would make the difference.

mathdonkey said:

A large pizza franchise delivers large pizzas for $16. The equation y=-3x^2+72x+480 describes the profit, y, based on the increase, x, in the price of their large pizzas. Determine the range of prices, to the nearest cent, for which the franchise could sell each pizza and still make a profit of at least $640.

Click to expand...

How can a profit of $640 or more be made on an item costing only $16...? :shock:

Does this exercise mean to say that there is a fixed number of items being sold, but that the price can vary...? Is the profit meant to be for the total number sold, not for "each" item...?

Kindly please reply with clarification. Thank you!

Eliz.

Re:

stapel said:

mathdonkey said:

A large pizza franchise delivers large pizzas for $16. The equation y=-3x^2+72x+480 describes the profit, y, based on the increase, x, in the price of their large pizzas. Determine the range of prices, to the nearest cent, for which the franchise could sell each pizza and still make a profit of at least $640.

Click to expand...

How can a profit of $640 or more be made on an item costing only $16...? :shock:

Does this exercise mean to say that there is a fixed number of items being sold, but that the price can vary...? Is the profit meant to be for the total number sold, not for "each" item...?

Kindly please reply with clarification. Thank you!

Eliz.

Click to expand...

Good question Eliz, I think by profit they mean total profit for all pizzas sold.

Oh and ty for responding so quick!

mathdonkey said:

I think by profit they mean total profit for all pizzas sold.

Click to expand...

What does the graph of the quadratic, y = -3x[sup:1tgr7cdl]2[/sup:1tgr7cdl] + 72x + 480, look like?

If you find the x-values for which y = 640 (that is, where the graph is 640 units above the x-axis), where would be your "interval" of price-increases? :wink:

Eliz.

Re:

stapel said:

mathdonkey said:

I think by profit they mean total profit for all pizzas sold.

Click to expand...

What does the graph of the quadratic, y = -3x[sup:3jjptafo]2[/sup:3jjptafo] + 72x + 480, look like?

If you find the x-values for which y = 640 (that is, where the graph is 640 units above the x-axis), where would be your "interval" of price-increases? :wink:

Eliz.

Click to expand...

Are the x-values the roots of the parabola (the quadratic is a parabola right)?

mathdonkey said:

Are the x-values the roots of the parabola (the quadratic is a parabola right)?

Click to expand...

If you are referring to the roots of the original quadratic, please clarify why you are trying to find the interval over which the profit is non-negative. Aren't you supposed to be finding the x-values that yield y > 640, not y > 0...?

Please reply with a clear listing of your work and reasoning. Thank you!

Eliz.

Re:

stapel said:

mathdonkey said:

Are the x-values the roots of the parabola (the quadratic is a parabola right)?

Click to expand...

If you are referring to the roots of the original quadratic, please clarify why you are trying to find the interval over which the profit is non-negative. Aren't you supposed to be finding the x-values that yield y > 640, not y > 0...?

Please reply with a clear listing of your work and reasoning. Thank you!

Eliz.

Click to expand...

Finding x-values that are > 640 is my goal. But I've never seen the term x-values used before, so it kind of threw me and I thought you were talking about the roots of the parabola. BTW quadratic = parabola, right? Because my textbook doesn't use the term quadratic very often, mainly parabola.

But yes, I need to find a range of price increases (x) that will make the profit (y) > 640. My problem is I don't know how to do that with this equation (or is it called a parabola/quadratic? I'm so confused!). That's why I have no work or reasoning for this, sorry Eliz.

Do you think you can still help me?

y=-3x^2+72x+480 describes the profit

profit of at least $640

Those two statements mean:

\(\displaystyle -3x^2\, +\, 72x\, +\, 480\, \ge\, 640\)

\(\displaystyle -3x^2\, +\, 72x\, -\, 160\, \ge\, 0\)

Can you go from here and find range of 'x' where the above inequality is true. Graph it - to understand the logic.