To determine the probability that the graph of the function \( y = x^2 + 2(a+4)x - 5a + 64 \) lies above the x-axis for \( a \) in the range \( [-5, 30] \), we need to analyze the condition for the graph to be above the x-axis.
The graph of a quadratic equation \( y = ax^2 + bx + c \) lies above the x-axis when the equation has no real roots or when the discriminant \( b^2 - 4ac < 0 \).
For the given equation, \( a = 1 \), \( b = 2(a+4) = 2a + 8 \), and \( c = -5a + 64 \). The discriminant \( D \) is given by:
\(D = (2a + 8)^2 - 4(1)(-5a + 64)\)
Expanding this, we get:
\(D = (2a + 8)^2 + 20a - 256\)
Calculating \( (2a + 8)^2 \):
\((2a + 8)^2 = 4a^2 + 32a + 64\)
Substituting this into the discriminant calculation:
\(D = 4a^2 + 32a + 64 + 20a - 256\)
Simplifying further:
\(D = 4a^2 + 52a - 192\)
For the graph to lie above the x-axis, the discriminant should be negative:
\(4a^2 + 52a - 192 < 0\)
Solving this inequality will give us the range of \( a \) for which the graph is above the x-axis. First, we solve the equation \( 4a^2 + 52a - 192 = 0 \) using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
Here \( b = 52 \), \( a = 4 \), and \( c = -192 \). Calculation of the discriminant:
\(b^2 - 4ac = 52^2 - 4(4)(-192) = 2704 + 3072 = 5776\)
The roots are:
\(a_1, a_2 = \frac{-52 \pm \sqrt{5776}}{8}\)
Calculating, we get:
\(a_1 = \frac{-52 + 76}{8} = 3\) and \(a_2 = \frac{-52 - 76}{8} = -16\)
The quadratic \( 4a^2 + 52a - 192 \) changes sign at these points. Examining intervals:
- For \( a \in (-\infty, -16) \), \( 4a^2 + 52a - 192 > 0 \)
- For \( a \in (-16, 3) \), \( 4a^2 + 52a - 192 < 0 \)
- For \( a \in (3, \infty) \), \( 4a^2 + 52a - 192 > 0 \)
Therefore, the graph is above the x-axis for \( a \) in the interval \((-16, 3)\). Given \( a \) is restricted to \([-5, 30]\), the valid interval is \([-5, 3)\).
The number of integer values of \( a \) in this interval is 8: \(-5, -4, -3, -2, -1, 0, 1, 2\).
Total possible values of \( a \) in \([-5, 30]\) are 36.
Thus, the probability is:
\(\frac{8}{36} = \frac{2}{9}\)
Hence, the correct answer is \(\frac{2}{9}\).
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