To analyze the height function of the ball given by \( y = -\frac{1}{14}x^2 + 6x + 5 \), we will answer the three questions step by step.
1. Height when the ball leaves the child's hand:
To find the height when the ball leaves the child's hand, we evaluate \( y \) at \( x = 0 \):
\[ y(0) = -\frac{1}{14}(0)^2 + 6(0) + 5 = 5 \]
The height of the ball when it leaves the child's hand is 5 feet.
2. Maximum height of the ball:
The given quadratic function \( y = -\frac{1}{14}x^2 + 6x + 5 \) is a downward-opening parabola, and its maximum height can be found at the vertex.
The x-coordinate of the vertex can be calculated using the formula \( x = -\frac{b}{2a} \), where \( a = -\frac{1}{14} \) and \( b = 6 \):
\[ x = -\frac{6}{2 \cdot -\frac{1}{14}} = -\frac{6}{-\frac{1}{7}} = 6 \cdot 7 = 42 \]
Now, substitute \( x = 42 \) back into the function to find the maximum height:
\[ y(42) = -\frac{1}{14}(42)^2 + 6(42) + 5 \]
Calculating each term:
- \( -\frac{1}{14}(42)^2 = -\frac{1}{14} \cdot 1764 = -126 \)
- \( 6(42) = 252 \)
- Therefore, \[ y(42) = -126 + 252 + 5 = 126 + 5 = 131 \]
The maximum height of the ball is 131 feet.
3. How far from the child does the ball strike the ground?
The ball strikes the ground when \( y = 0 \). We set the height function equal to zero and solve for \( x \):
\[ -\frac{1}{14}x^2 + 6x + 5 = 0 \]
To eliminate the fraction, multiply through by -14:
\[ x^2 - 84x - 70 = 0 \]
Now, we will use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1 \), \( b = -84 \), and \( c = -70 \):
- Calculate the discriminant:
\[ b^2 - 4ac = (-84)^2 - 4(1)(-70) = 7056 + 280 = 7336 \]
- Apply the quadratic formula:
\[ x = \frac{84 \pm \sqrt{7336}}{2} \]
Calculate \( \sqrt{7336} \):
\[ \sqrt{7336} \approx 85.7 \]
Now plug it into the formula:
\[ x \approx \frac{84 \pm 85.7}{2} \]
Calculate both possible \( x \) values:
- \( x_1 = \frac{169.7}{2} \approx 84.85 \)
- \( x_2 = \frac{-1.7}{2} \) (which we discard as \( x \) cannot be negative)
The ball strikes the ground approximately 84.85 feet from the child.
Summary of Results:
- Height when the ball leaves the child's hand: 5 feet
- Maximum height of the ball: 131 feet
- Distance from the child when the ball strikes the ground: approximately 84.85 feet
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