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
(0,3)
left parenthesis 0 comma 3 right parenthesis
(19,0)
left parenthesis 19 comma 0 right parenthesis
(8,6.2)
left parenthesis 8 comma 6.2 right parenthesis
(12,8.4)
, 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
(0,3)
left parenthesis 0 comma 3 right parenthesis
(19,0)
left parenthesis 19 comma 0 right parenthesis
(8,6.2)
left parenthesis 8 comma 6.2 right parenthesis
(12,8.4)
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 \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients from the standard form of a quadratic equation \( y = ax^2 + bx + c \).
In this equation:
- \( a = -0.05 \)
- \( b = 45 \)
Now, calculate the x-coordinate of the vertex:
\[
x = -\frac{b}{2a} = -\frac{45}{2 \cdot -0.05} = -\frac{45}{-0.1} = 450
\]
Next, substitute \( x = 450 \) back into the equation to find the y-coordinate of the vertex:
\[
y = -0.05(450)^2 + 45(450) + 3
\]
Calculating \( 450^2 \):
\[
450^2 = 202500
\]
Now substitute back into the equation:
\[
y = -0.05(202500) + 45(450) + 3
\]
\[
y = -10125 + 20250 + 3
\]
\[
y = 10125 + 3 = 10128
\]
Thus, the vertex of the parabola is located at \( (450, 10128) \).
In your multiple-choice options, since this calculation does not match any of the given choices, it seems I may have made an error in the measurement context. Could you verify the total calculations or provide any constraints? However, based on standard parabola calculation, the above procedure is accurate.
In this equation:
- \( a = -0.05 \)
- \( b = 45 \)
Now, calculate the x-coordinate of the vertex:
\[
x = -\frac{b}{2a} = -\frac{45}{2 \cdot -0.05} = -\frac{45}{-0.1} = 450
\]
Next, substitute \( x = 450 \) back into the equation to find the y-coordinate of the vertex:
\[
y = -0.05(450)^2 + 45(450) + 3
\]
Calculating \( 450^2 \):
\[
450^2 = 202500
\]
Now substitute back into the equation:
\[
y = -0.05(202500) + 45(450) + 3
\]
\[
y = -10125 + 20250 + 3
\]
\[
y = 10125 + 3 = 10128
\]
Thus, the vertex of the parabola is located at \( (450, 10128) \).
In your multiple-choice options, since this calculation does not match any of the given choices, it seems I may have made an error in the measurement context. Could you verify the total calculations or provide any constraints? However, based on standard parabola calculation, the above procedure is accurate.