Answer:
7. P(q) = 60 -0.03(x -80)²
8. P(q) = -0.03x² +4.8x -132
9. q = 35 balloons
10. see attached
Explanation:
You want the profit equation, break-even sales, and a graph for a party supply company that makes a maximum profit of $60 when they sell 80 balloons, and a profit of $33 when they sell 50 balloons.
7. Quadratic equation
If the profit is described by a quadratic equation with a maximum at (quantity, dollars) = (80, 60), and a point on the curve of (50, 33), we can write the equation in two steps.
The vertex-form equation will look like ...
P(q) = 60 -a(q -80)² . . . . . for some scale factor 'a'
We want P(50) = 33, so we have ...
P(50) = 33 = 60 -a(50 -80)²
33 = 60 -900a
a = (60 -33)/900 = 3/100 = 0.03
The profit equation is ...
P(q) = 60 -0.03(q -80)²
8. Standard form
Expanding the equation gives ...
P(q) = 60 -0.03(q² -160q +6400)
P(q) = -0.03q² +4.8q -132
9. Break-even
The profit is zero at ...
P(q) = 0 = 60 -0.03(q -80)²
0.03(q -80)² = 60
(q -80)² = 2000 . . . . . . . . divide by 0.03
q = 80 ±20√5 = 35.3 or 124.7 . . . . . take the square root, add 80
The store will no longer make a profit when sales drops to 35 balloons.
10. Graph
The attachment shows a graph of the profit function.
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Additional comment
You can find where the profit goes to zero using the graph, the quadratic formula, or by solving it using vertex form, as above. The graph is quick and easy; using the vertex form equation works nicely.
Selling 36 balloons will result in a profit of $1.92.
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