Let's analyze each of the given statements based on the quadratic equation [tex]\( y = -6x^2 + 100x - 180 \)[/tex], where [tex]\( x \)[/tex] represents the selling price of each soccer ball and [tex]\( y \)[/tex] represents the daily profit.
1. If the store charges [tex]$\$[/tex] 5[tex]$ per soccer ball, it can expect to sell 170 of them.
- To verify this, we substitute \( x = 5 \) into the quadratic equation to calculate the profit:
\[
y = -6(5)^2 + 100(5) - 180
\]
\[
y = -6(25) + 500 - 180
\]
\[
y = -150 + 500 - 180
\]
\[
y = 170
\]
Thus, the profit when the store charges \$[/tex]5 per soccer ball is indeed \[tex]$170.
2. If the store charges $[/tex]\[tex]$ 8$[/tex] per soccer ball, it makes a daily profit of [tex]$\$[/tex] 236[tex]$.
- To verify this, we substitute \( x = 8 \) into the quadratic equation to calculate the profit:
\[
y = -6(8)^2 + 100(8) - 180
\]
\[
y = -6(64) + 800 - 180
\]
\[
y = -384 + 800 - 180
\]
\[
y = 236
\]
Thus, the profit when the store charges \$[/tex]8 per soccer ball is \[tex]$236.
3. If the store sells 12 soccer balls, it makes \$[/tex]156 in profit.
- This statement involves the number of soccer balls sold, but our original quadratic equation relates to the selling price, not the number of soccer balls sold. Assuming a misinterpretation:
[tex]\[
y = profit(12)
\][/tex]
Substituting [tex]\( x = 12 \)[/tex] into the equation:
[tex]\[
y = -6(12)^2 + 100(12) - 180
\][/tex]
[tex]\[
y = -6(144) + 1200 - 180
\][/tex]
[tex]\[
y = -864 + 1200 - 180
\][/tex]
[tex]\[
y = 156
\][/tex]
Thus, if the store's equation is assumed based on pricing and not actual balls sold, it makes \[tex]$156 when considering evaluation for \( x=12 \).
4. The greater the selling price, the greater the profit.
- To verify this, we consider the nature of the quadratic equation \( y = -6x^2 + 100x - 180 \):
The quadratic term \(-6x^2\) indicates that the equation represents a parabola opening downwards. This means there is a maximum profit at a certain price, but not an ever-increasing profit:
Substituting various reasonable prices will show the maximum and decreasing trend. Additionally, the Python output confirms this pattern is not increasing consistently:
\[
greater\_profit\_check = False
\]
In summary, the true statements based on our calculations are:
- If the store charges \$[/tex]5 per soccer ball, it can expect to make a profit of \[tex]$170.
- If the store charges \$[/tex]8 per soccer ball, it makes a daily profit of \[tex]$236.
- If evaluated at selling price $[/tex]12, the resulting profit calculation indicates \$156.
- The statement that greater prices always yield greater profits is false.
