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)
Answers
GPT-4o mini
The equation of the trajectory of the tennis ball is given by \( y = -0.05x^2 + 45x + 3 \). This is a quadratic equation in the standard form \( y = ax^2 + bx + c \), where \( a = -0.05 \), \( b = 45 \), and \( c = 3 \).
To find the vertex of a parabola represented by this equation, we can use the formula for the x-coordinate of the vertex, which is given by:
\[
x = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = -\frac{45}{2 \times -0.05}
\]
\[
x = -\frac{45}{-0.1} = 450
\]
Now that we have the x-coordinate of the vertex as \( x = 450 \), we need to find the corresponding y-coordinate by substituting \( x \) back into the original equation:
\[
y = -0.05(450^2) + 45(450) + 3
\]
Calculating \( 450^2 \):
\[
450^2 = 202500
\]
Now calculate \( y \):
\[
y = -0.05(202500) + 45(450) + 3
\]
\[
y = -10125 + 20250 + 3
\]
\[
y = 10125 + 3
\]
\[
y = 10128
\]
Thus, the vertex of the parabola is located at the point \( (450, 10128) \).
To find the vertex of a parabola represented by this equation, we can use the formula for the x-coordinate of the vertex, which is given by:
\[
x = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = -\frac{45}{2 \times -0.05}
\]
\[
x = -\frac{45}{-0.1} = 450
\]
Now that we have the x-coordinate of the vertex as \( x = 450 \), we need to find the corresponding y-coordinate by substituting \( x \) back into the original equation:
\[
y = -0.05(450^2) + 45(450) + 3
\]
Calculating \( 450^2 \):
\[
450^2 = 202500
\]
Now calculate \( y \):
\[
y = -0.05(202500) + 45(450) + 3
\]
\[
y = -10125 + 20250 + 3
\]
\[
y = 10125 + 3
\]
\[
y = 10128
\]
Thus, the vertex of the parabola is located at the point \( (450, 10128) \).