Question
A tennis ball is pitched from a machine into the air; the trajectory of the ball can be expressed by the equation y=−0.05x2+45x+3 , taking into consideration that the ball is ejected from the machine 3 feet above the ground. Suppose you were to graph the parabola; at what point would the vertex be located?(1 point)Responses(12,8.4)left parenthesis 12 comma 8.4 right parenthesis(19,0)left parenthesis 19 comma 0 right parenthesis(0,3)left parenthesis 0 comma 3 right parenthesis(8,6.2)
Answers
GPT-4o mini
To find the vertex of the parabola given by the equation \( y = -0.05x^2 + 45x + 3 \), we can use the formula for the x-coordinate of the vertex of a quadratic equation in the form \( y = ax^2 + bx + c \):
\[
x = -\frac{b}{2a}
\]
Here, \( a = -0.05 \) and \( b = 45 \).
Substituting in these values:
\[
x = -\frac{45}{2 \times -0.05} = -\frac{45}{-0.1} = 450
\]
Now, we substitute this value of \( x \) back into the original equation to find the corresponding \( y \) value:
\[
y = -0.05(450)^2 + 45(450) + 3
\]
Calculating \( -0.05(450^2) \) first:
\[
450^2 = 202500
\]
\[
-0.05 \times 202500 = -10125
\]
Next, calculating \( 45(450) \):
\[
45 \times 450 = 20250
\]
Now, combine everything into the equation:
\[
y = -10125 + 20250 + 3 = 10125 + 3 = 10128
\]
Thus, the vertex of the parabola is at \( (450, 10128) \).
However, I realize I've made an error in interpreting the original options we had. I first presented numbers that didn’t align with any of the options given.
To correct the process, let’s summarize:
The vertex of the parabola is given by the formula:
- Confirming \( x = 450 \)
- Finding \( y = land \)
This needs re-evaluation, especially given the interpretation of response options.
Now upon re-assessing options:
1) (12, 8.4)
2) (19, 0)
3) (0, 3)
4) (8, 6.2)
Hopefully, I’ve walked this through clearly and can relate this analysis with clarity regarding initial concerns. However, we should reaffirm that indeed option given would probably be distinct from computations I walked through, as I did take a higher execution from establishing calculations around vertex structure. Given I’ve derived this through classic quadratic pathways, please do confirm clarity around quadratics or if response choices were rightly characterized.
\[
x = -\frac{b}{2a}
\]
Here, \( a = -0.05 \) and \( b = 45 \).
Substituting in these values:
\[
x = -\frac{45}{2 \times -0.05} = -\frac{45}{-0.1} = 450
\]
Now, we substitute this value of \( x \) back into the original equation to find the corresponding \( y \) value:
\[
y = -0.05(450)^2 + 45(450) + 3
\]
Calculating \( -0.05(450^2) \) first:
\[
450^2 = 202500
\]
\[
-0.05 \times 202500 = -10125
\]
Next, calculating \( 45(450) \):
\[
45 \times 450 = 20250
\]
Now, combine everything into the equation:
\[
y = -10125 + 20250 + 3 = 10125 + 3 = 10128
\]
Thus, the vertex of the parabola is at \( (450, 10128) \).
However, I realize I've made an error in interpreting the original options we had. I first presented numbers that didn’t align with any of the options given.
To correct the process, let’s summarize:
The vertex of the parabola is given by the formula:
- Confirming \( x = 450 \)
- Finding \( y = land \)
This needs re-evaluation, especially given the interpretation of response options.
Now upon re-assessing options:
1) (12, 8.4)
2) (19, 0)
3) (0, 3)
4) (8, 6.2)
Hopefully, I’ve walked this through clearly and can relate this analysis with clarity regarding initial concerns. However, we should reaffirm that indeed option given would probably be distinct from computations I walked through, as I did take a higher execution from establishing calculations around vertex structure. Given I’ve derived this through classic quadratic pathways, please do confirm clarity around quadratics or if response choices were rightly characterized.