Question

y = -6x² + 100x - 180
x = selling price of each soccer ball
y = daily profit from soccer balls
what are the zeroes of the function? round to the nearest hundredth.
× 0
✓ 2.05
× 8.33
✓ 14.61
complete
what do the zeroes mean in context?
○ if soccer balls are sold for $2.05 each, the store will make a daily profit of $14.61.
○ if soccer balls are sold for $14.61 each, the store will make a daily profit of $2.05.
○ if soccer balls are sold for $2.05 or $14.61 each, the store will maximize their daily profit.
○ if soccer balls are sold for $2.05 or $14.61 each, the store will break even but will not make a profit.
done

Explanation:

First Sub - Question: What are the zeroes of the function?

Step 1: Recall the quadratic formula

For a quadratic function \(y = ax^{2}+bx + c\), the zeroes (roots) are found using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Here, \(a=- 6\), \(b = 100\), and \(c=-180\).

Step 2: Calculate the discriminant

The discriminant \(D=b^{2}-4ac=(100)^{2}-4\times(-6)\times(-180)=10000 - 4320 = 5680\).

Step 3: Find the roots

First root: \(x_1=\frac{-100+\sqrt{5680}}{2\times(-6)}=\frac{-100 + 75.37}{-12}=\frac{-24.63}{-12}\approx2.05\)
Second root: \(x_2=\frac{-100-\sqrt{5680}}{2\times(-6)}=\frac{-100 - 75.37}{-12}=\frac{-175.37}{-12}\approx14.61\)

The zeroes of the profit function \(y=-6x^{2}+100x - 180\) occur when \(y = 0\) (profit is zero). In the context of selling soccer balls, when \(y = 0\), the store is breaking even (no profit and no loss). So when the selling price \(x\) is \(2.05\) or \(14.61\) dollars per soccer ball, the profit \(y\) is zero (break - even point).

Answer:

The zeroes of the function are approximately \(2.05\) and \(14.61\).

Second Sub - Question: What do the zeroes mean in context?