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(12,8.4)left parenthesis 12 comma 8.4 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 \( y = ax^2 + bx + c \), which is given by:
\[
x = -\frac{b}{2a}
\]
In our equation:
- \( a = -0.05 \)
- \( b = 45 \)
Now, we can calculate the x-coordinate of the vertex:
\[
x = -\frac{45}{2 \times -0.05} = -\frac{45}{-0.1} = 450
\]
Next, we can substitute this back into the original equation to find the y-coordinate:
\[
y = -0.05(450)^2 + 45(450) + 3
\]
Calculating \( (450)^2 \):
\[
450^2 = 202500
\]
Now plug that value into the equation:
\[
y = -0.05(202500) + 20250 + 3
\]
\[
y = -10125 + 20250 + 3
\]
\[
y = 10125 + 3 = 10128
\]
Thus, the vertex of the parabola is located at the point \( (450, 10128) \). However, this value seems to be off based on the options you provided. Let's go through the options given again—none of them align with what we computed.
Let's check for errors: It seems you've presented some coordinate options instead of the computed vertex coordinate. Assuming the parabola was correctly computed, I'd suggest looking at your initial coefficients if these options are provided from a different polynomial.
However, based on common interpretations and manipulations of this equation, none of the provided options match, indicating a possibility of alterations to other parts of the function or our interpretation. Please double-check the given equation or confirm the structure of the problem posed with respect to the appropriate format here for desired results.
\[
x = -\frac{b}{2a}
\]
In our equation:
- \( a = -0.05 \)
- \( b = 45 \)
Now, we can calculate the x-coordinate of the vertex:
\[
x = -\frac{45}{2 \times -0.05} = -\frac{45}{-0.1} = 450
\]
Next, we can substitute this back into the original equation to find the y-coordinate:
\[
y = -0.05(450)^2 + 45(450) + 3
\]
Calculating \( (450)^2 \):
\[
450^2 = 202500
\]
Now plug that value into the equation:
\[
y = -0.05(202500) + 20250 + 3
\]
\[
y = -10125 + 20250 + 3
\]
\[
y = 10125 + 3 = 10128
\]
Thus, the vertex of the parabola is located at the point \( (450, 10128) \). However, this value seems to be off based on the options you provided. Let's go through the options given again—none of them align with what we computed.
Let's check for errors: It seems you've presented some coordinate options instead of the computed vertex coordinate. Assuming the parabola was correctly computed, I'd suggest looking at your initial coefficients if these options are provided from a different polynomial.
However, based on common interpretations and manipulations of this equation, none of the provided options match, indicating a possibility of alterations to other parts of the function or our interpretation. Please double-check the given equation or confirm the structure of the problem posed with respect to the appropriate format here for desired results.