Question
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 vertex formula for a parabola in the standard form \( ax^2 + bx + c \). The x-coordinate of the vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
In this case, \( a = -0.05 \) and \( b = 45 \).
Calculating the x-coordinate of the vertex:
\[
x = -\frac{45}{2(-0.05)} = -\frac{45}{-0.1} = 450
\]
Now that we have the x-coordinate, we can substitute \( x = 450 \) back into the original equation to find the y-coordinate of the vertex:
\[
y = -0.05(450)^2 + 45(450) + 3
\]
Calculating \( y \):
\[
y = -0.05(202500) + 20250 + 3
\]
\[
y = -10125 + 20250 + 3
\]
\[
y = 10125 + 3 = 10128
\]
Thus, the vertex of the parabola is at the point \( (450, 10128) \). However, this value is not provided in the choices listed.
Given that there seem to be contradictions between the values used in the question, the method to find the vertex is still correct. Let's recalculate that:
In case the question is aiming for a practical maximum height and relates this to observed options, we may consider reviewing the required parabola details in another form or deriving essential factors corresponding to the values output presented.
If you previously recorded other values in bounds near the right graph, please scrutinize the resulting peaks using specified vertex nuances to either evaluate computational employ fully or to bring clarity around available vertices nearer anticipated for maximum practical output overall.
As of the provided responses, none directly matches based on calculations due to variance in expected computation. Be sure to review input parameters or calibrate toward expected outputs relative to needed practical height/career graph or adjust coefficients revealing flight path dynamics leading to potential yield points fitting the thematic or physical basis of the interaction fully.
\[
x = -\frac{b}{2a}
\]
In this case, \( a = -0.05 \) and \( b = 45 \).
Calculating the x-coordinate of the vertex:
\[
x = -\frac{45}{2(-0.05)} = -\frac{45}{-0.1} = 450
\]
Now that we have the x-coordinate, we can substitute \( x = 450 \) back into the original equation to find the y-coordinate of the vertex:
\[
y = -0.05(450)^2 + 45(450) + 3
\]
Calculating \( y \):
\[
y = -0.05(202500) + 20250 + 3
\]
\[
y = -10125 + 20250 + 3
\]
\[
y = 10125 + 3 = 10128
\]
Thus, the vertex of the parabola is at the point \( (450, 10128) \). However, this value is not provided in the choices listed.
Given that there seem to be contradictions between the values used in the question, the method to find the vertex is still correct. Let's recalculate that:
In case the question is aiming for a practical maximum height and relates this to observed options, we may consider reviewing the required parabola details in another form or deriving essential factors corresponding to the values output presented.
If you previously recorded other values in bounds near the right graph, please scrutinize the resulting peaks using specified vertex nuances to either evaluate computational employ fully or to bring clarity around available vertices nearer anticipated for maximum practical output overall.
As of the provided responses, none directly matches based on calculations due to variance in expected computation. Be sure to review input parameters or calibrate toward expected outputs relative to needed practical height/career graph or adjust coefficients revealing flight path dynamics leading to potential yield points fitting the thematic or physical basis of the interaction fully.