We are given the equation $y = -x^2 + 9x - 100$ for a parabola and a horizontal line $y = c$. The problem states that these intersect at exactly one point. To find the value of $c$, we need to understand that for the parabola and the line to intersect at exactly one point, the discriminant of the quadratic equation formed by equating the two expressions must be zero (this indicates a single solution).

Step 1: Set the equations equal to each other

Since $y = c$ and $y = -x^2 + 9x - 100$, we can set:

$c = -x^2 + 9x - 100$

Step 2: Rearrange into a quadratic equation

Rearranging this equation gives:

$0 = -x^2 + 9x - 100 - c$

or

$x^2 - 9x + (100 + c) = 0$

Step 3: Apply the discriminant

For a quadratic equation $ax^2 + bx + c = 0$, the discriminant is $\Delta = b^2 - 4ac$. If the discriminant is zero, there is exactly one solution. In this case:

$\Delta = (-9)^2 - 4(1)(100 + c) = 81 - 4(100 + c)$

Set the discriminant equal to zero for one solution:

$81 - 4(100 + c) = 0$

Step 4: Solve for $c$

$81 = 4(100 + c)$

$81 = 400 + 4c$

$4c = 81 - 400$

$4c = -319$

$c = \frac{-319}{4}$

Thus, the value of $c$ is $\frac{-319}{4}$, which corresponds to option C.

Would you like further clarification or details on this solution?

Related Questions:

1. How would the solution change if the parabola intersected the line at two points?
2. What are the coordinates of the vertex of the parabola $y = -x^2 + 9x - 100$?
3. Can you derive the formula for the axis of symmetry for the parabola in this problem?
4. What is the minimum or maximum value of the function $y = -x^2 + 9x - 100$?
5. How do you determine if a parabola opens upwards or downwards based on its equation?

Tip:

When a quadratic equation is involved and you're asked for the condition of exactly one solution, always think of the discriminant and set it equal to zero!