To determine when the projectile reaches its maximum height, we need to find the vertex of the quadratic function $h(t) = -16t^2 + 260t$. The vertex of a parabola described by the equation $at^2 + bt + c$ can be found using the formula:

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

In this case, the function is $h(t) = -16t^2 + 260t$, where $a = -16$ and $b = 260$. Plugging these values into the formula, we get:

$t = -\frac{260}{2 \cdot -16}$ $t = -\frac{260}{-32}$ $t = \frac{260}{32}$ $t = 8.125$

So, the projectile reaches its maximum height after 8.125 seconds. However, since none of the options exactly match 8.125 seconds, let's verify that our calculations are correct.

Let's re-evaluate the given options:

• A. $26$ sec
• B. $6$ sec
• C. $8$ sec
• D. $19.5$ sec

Since 8.125 seconds is very close to 8 seconds, the closest and correct answer is:

• C. 8 sec

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. How do you derive the formula for the vertex of a quadratic function?
2. What is the significance of the coefficients in a quadratic function in determining its shape and vertex?
3. How can you find the maximum height of a projectile given its equation?
4. How does the initial velocity of the projectile affect its maximum height?
5. Can the time to reach maximum height be different if the equation is given in a different form?
6. How does gravity affect the motion of a projectile?
7. What are the steps to convert a quadratic function from standard form to vertex form?
8. How do you determine the total time a projectile is in the air?

Tip: Always double-check your calculations, especially when converting fractions to decimals or interpreting values from a formula.