To determine when 45 impulses occur, we need to set the equation \( y = -x^2 + 2x + 60 \) equal to 45 and solve for \( x \).
So we set up the equation:
\[
- x^2 + 2x + 60 = 45 \]
Next, we'll simplify by moving 45 to the left side:
\[
- x^2 + 2x + 60 - 45 = 0 \]
This simplifies to:
\[
- x^2 + 2x + 15 = 0 \]
To make it easier to work with, we can multiply the entire equation by -1:
\[ x^2 - 2x - 15 = 0 \]
Now we can factor the quadratic equation:
\[ (x - 5)(x + 3) = 0 \]
Setting each factor to zero gives us:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \] \[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]
Since \( x \) typically represents time or a non-negative quantity in this context, we only consider the positive solution:
\[ x = 5 \]
Thus, the number of impulses fired will be 45 when \( x = 5 \).