To find the maximum profit, we need to determine the vertex of the quadratic function $P(x) = -x^2 + 130x + 70$. For a quadratic function $ax^2 + bx + c$, the $x$-coordinate of the vertex is given by:

$x = -\frac{b}{2a}$

Here, $a = -1$, $b = 130$, and $c = 70$. Substituting these values:

$x = -\frac{130}{2(-1)} = \frac{130}{2} = 65$

The number of videos rented daily that yields the maximum profit is $x = 65$. Now substitute $x = 65$ into the profit function to find the maximum profit:

$P(65) = -(65)^2 + 130(65) + 70$

$P(65) = -4225 + 8450 + 70 = 4295$

Thus, the maximum profit is $4295.

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5 Related Questions:

1. How do you find the vertex of a parabola algebraically?
2. What are other applications of quadratic functions in business?
3. How does the sign of the coefficient of $x^2$ affect the shape of a parabola?
4. What happens to the profit if the number of rentals exceeds the optimal value?
5. Can this problem be solved graphically, and how would that compare to the algebraic method?

Tip:

Always verify the vertex calculation by substituting it back into the function to confirm correctness.