The height of the stone, $s(t)$, is given by the quadratic function:

$s(t) = -16t^2 + 60t + 30$

To determine the time at which the stone reaches its maximum height and to find the maximum height, we can follow these steps:

Step 1: Find the time of maximum height

Since the equation is quadratic, the graph of the function is a parabola. Because the coefficient of $t^2$ is negative ($-16$), the parabola opens downwards, meaning the vertex of the parabola represents the maximum height.

The formula for the time at which a quadratic function $ax^2 + bx + c$ reaches its vertex is given by:

$t_{\text{max}} = \frac{-b}{2a}$

In this case:

• $a = -16$
• $b = 60$

Substitute these values into the formula:

$t_{\text{max}} = \frac{-60}{2(-16)} = \frac{-60}{-32} = 1.875 \, \text{seconds}$

So, the stone reaches its maximum height at $t = 1.875$ seconds.

Step 2: Find the maximum height

Now, substitute $t = 1.875$ back into the height equation $s(t)$ to find the maximum height:

$s(1.875) = -16(1.875)^2 + 60(1.875) + 30$

First, calculate $1.875^2$:

$1.875^2 = 3.515625$

Now substitute back:

$s(1.875) = -16(3.515625) + 60(1.875) + 30$

Simplify each term:

$s(1.875) = -56.25 + 112.5 + 30$

$s(1.875) = 86.25 \, \text{feet}$

Conclusion:

• The stone reaches its maximum height at $t = 1.875$ seconds.
• The maximum height is $86.25$ feet.

Would you like more details on any of these steps?

Here are 5 related questions:

1. What happens to the height of the stone after it reaches its maximum height?
2. How can we calculate when the stone hits the ground?
3. Can you find the velocity of the stone at any given time?
4. How do you find the total time the stone is in the air?
5. What is the significance of the coefficient -16 in the height equation?

Tip: The formula $t_{\text{max}} = \frac{-b}{2a}$ is a quick way to find the maximum or minimum of any quadratic function, depending on the sign of $a$.