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)
, 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 described by the equation \(y = -0.05x^2 + 45x + 3\), we can use the formula for the x-coordinate of the vertex, which is given by:
\[
x = -\frac{b}{2a}
\]
Here, \(a = -0.05\) and \(b = 45\).
Calculating the x-coordinate of the vertex:
\[
x = -\frac{45}{2 \times -0.05} = -\frac{45}{-0.1} = 450
\]
Now, we substitute \(x = 450\) back into the equation to find the y-coordinate of the vertex:
\[
y = -0.05(450^2) + 45(450) + 3
\]
\[
y = -0.05(202500) + 20250 + 3
\]
\[
y = -10125 + 20250 + 3
\]
\[
y = 10125 + 3 = 10128
\]
So the vertex of the parabola is at the point \( (450, 10128) \).
Given the choices provided, it appears there is no exact match for this calculated vertex. However, the approach to derive the vertex can be followed to clarify the output of the parabola. To express it as one of the solution options, I'd reevaluate or check the available choices ensuring if there was an error in computational values or misinterpretation of the problem conditions.
For the provided response options, though, the vertex is indeed located at none: (12,8.4), (19,0), (0,3), or (8,6.2) based on current calculations.
Thus, further adjustments or clarifying problem parameters may be needed for a successful selection.
\[
x = -\frac{b}{2a}
\]
Here, \(a = -0.05\) and \(b = 45\).
Calculating the x-coordinate of the vertex:
\[
x = -\frac{45}{2 \times -0.05} = -\frac{45}{-0.1} = 450
\]
Now, we substitute \(x = 450\) back into the equation to find the y-coordinate of the vertex:
\[
y = -0.05(450^2) + 45(450) + 3
\]
\[
y = -0.05(202500) + 20250 + 3
\]
\[
y = -10125 + 20250 + 3
\]
\[
y = 10125 + 3 = 10128
\]
So the vertex of the parabola is at the point \( (450, 10128) \).
Given the choices provided, it appears there is no exact match for this calculated vertex. However, the approach to derive the vertex can be followed to clarify the output of the parabola. To express it as one of the solution options, I'd reevaluate or check the available choices ensuring if there was an error in computational values or misinterpretation of the problem conditions.
For the provided response options, though, the vertex is indeed located at none: (12,8.4), (19,0), (0,3), or (8,6.2) based on current calculations.
Thus, further adjustments or clarifying problem parameters may be needed for a successful selection.