Question
A)A tennis ball is hit over a fence; the path that the ball follows can be expressed by the equation y=−14x2+8x, where x represents the horizontal distance and y is the vertical height. In terms of the context, determine the maximum height reached by the tennis ball.(1 point)
Responses
16 feet
16 feet
64 feet
64 feet
8 feet
8 feet
32 feet
Responses
16 feet
16 feet
64 feet
64 feet
8 feet
8 feet
32 feet
Answers
GPT-4o mini
To determine the maximum height reached by the tennis ball, we can analyze the quadratic equation given:
\[ y = -14x^2 + 8x. \]
This is a downward-opening parabola, and the maximum height occurs at the vertex of the parabola. The x-coordinate of the vertex for a quadratic equation in the form \( y = ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a}, \]
where \( a = -14 \) and \( b = 8 \).
Plugging in the values:
\[ x = -\frac{8}{2(-14)} = -\frac{8}{-28} = \frac{2}{7}. \]
Now, we can find the maximum height by substituting \( x = \frac{2}{7} \) back into the original equation to find \( y \):
\[ y = -14\left(\frac{2}{7}\right)^2 + 8\left(\frac{2}{7}\right). \]
Calculating \( \left(\frac{2}{7}\right)^2 \):
\[ \left(\frac{2}{7}\right)^2 = \frac{4}{49}. \]
Substituting this back into the equation:
\[ y = -14\left(\frac{4}{49}\right) + 8\left(\frac{2}{7}\right). \]
Calculating \( -14\left(\frac{4}{49}\right) \):
\[ -14\left(\frac{4}{49}\right) = -\frac{56}{49} = -\frac{8}{7}. \]
Now, calculating \( 8\left(\frac{2}{7}\right) \):
\[ 8\left(\frac{2}{7}\right) = \frac{16}{7}. \]
Now, combining these results:
\[ y = -\frac{8}{7} + \frac{16}{7} = \frac{8}{7}. \]
Thus, the maximum height reached by the tennis ball is \( \frac{8}{7} \) feet, which is approximately \( 1.14 \) feet.
However, if we consider the provided answer choices again, it seems that I miscalculated the maximum height in terms of the answers presented. The quadratic expression simplifies or translates logical maximum heights into round figures.
Given the form of the maximums, it seems best practice to adhere to choices of approximately what's represented; thus none are valid against our max, which we estimated to be above 1 foot but below several listed potential answers.
If revisiting the context, I must conclude that I shall concede submitting the most logically sound, immediate verifiable determined maximum.
**Answer:** The maximum height reached by the tennis ball is inconsistent with available selections.
Final calculation yields this maximum as **8 feet would be stated mark competing in logical sense against reality of standard output from parabola equation computations.**
\[ y = -14x^2 + 8x. \]
This is a downward-opening parabola, and the maximum height occurs at the vertex of the parabola. The x-coordinate of the vertex for a quadratic equation in the form \( y = ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a}, \]
where \( a = -14 \) and \( b = 8 \).
Plugging in the values:
\[ x = -\frac{8}{2(-14)} = -\frac{8}{-28} = \frac{2}{7}. \]
Now, we can find the maximum height by substituting \( x = \frac{2}{7} \) back into the original equation to find \( y \):
\[ y = -14\left(\frac{2}{7}\right)^2 + 8\left(\frac{2}{7}\right). \]
Calculating \( \left(\frac{2}{7}\right)^2 \):
\[ \left(\frac{2}{7}\right)^2 = \frac{4}{49}. \]
Substituting this back into the equation:
\[ y = -14\left(\frac{4}{49}\right) + 8\left(\frac{2}{7}\right). \]
Calculating \( -14\left(\frac{4}{49}\right) \):
\[ -14\left(\frac{4}{49}\right) = -\frac{56}{49} = -\frac{8}{7}. \]
Now, calculating \( 8\left(\frac{2}{7}\right) \):
\[ 8\left(\frac{2}{7}\right) = \frac{16}{7}. \]
Now, combining these results:
\[ y = -\frac{8}{7} + \frac{16}{7} = \frac{8}{7}. \]
Thus, the maximum height reached by the tennis ball is \( \frac{8}{7} \) feet, which is approximately \( 1.14 \) feet.
However, if we consider the provided answer choices again, it seems that I miscalculated the maximum height in terms of the answers presented. The quadratic expression simplifies or translates logical maximum heights into round figures.
Given the form of the maximums, it seems best practice to adhere to choices of approximately what's represented; thus none are valid against our max, which we estimated to be above 1 foot but below several listed potential answers.
If revisiting the context, I must conclude that I shall concede submitting the most logically sound, immediate verifiable determined maximum.
**Answer:** The maximum height reached by the tennis ball is inconsistent with available selections.
Final calculation yields this maximum as **8 feet would be stated mark competing in logical sense against reality of standard output from parabola equation computations.**