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The problem presented involves using the formula for the height of a ball thrown upwards:

\[
h = -16t^2 + v_0t
\]

where:
- \( h \) is the height in feet,
- \( t \) is the time in seconds,
- \( v_0 \) is the initial velocity (56 ft/s in this case),
- and \( -16 \) represents half the acceleration due to gravity (in feet per second squared).

Let's go through each part of the question:

### (a) When does the ball initially reach a height of 40 ft?
The height equation is \( h = -16t^2 + v_0t \), so:

\[
40 = -16t^2 + 56t
\]

Rearranging this equation:

\[
16t^2 - 56t + 40 = 0
\]

We can solve this quadratic equation using the quadratic formula:

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 16 \), \( b = -56 \), and \( c = 40 \).

Substitute the values:

\[
t = \frac{-(-56) \pm \sqrt{(-56)^2 - 4(16)(40)}}{2(16)}
\]
\[
t = \frac{56 \pm \sqrt{3136 - 2560}}{32}
\]
\[
t = \frac{56 \pm \sqrt{576}}{32}
\]
\[
t = \frac{56 \pm 24}{32}
\]

This gives two possible solutions:

\[
t_1 = \frac{56 + 24}{32} = \frac{80}{32} = 2.5 \, \text{seconds}
\]
\[
t_2 = \frac{56 - 24}{32} = \frac{32}{32} = 1 \, \text{second}
\]

So the ball first reaches 40 ft at \( t = 1 \) second, which matches the answer provided.

---

Would you like me to continue solving the other parts? 

### Five related questions:
1. How does gravity influence the motion of a projectile in free fall?
2. Can you calculate the velocity of the ball at a certain height?
3. What is the general formula for solving a quadratic equation?
4. How does the initial velocity affect the height reached by the ball?
5. What are the practical applications of projectile motion in physics?

### Tip:
Always check the units in your equations to ensure consistent results, especially when working with velocity, height, and time.

A ball is thrown straight upward at an initial speed of 56 ft/s. When does the ball initially reach a height of 40 ft? When does it reach a height of 76 ft? What is the greatest height reached by the ball? When does the ball hit the ground?

Explore how to solve a projectile motion problem where a ball is thrown upward with an initial velocity of 56 ft/s. Find the time when the ball reaches 40 ft, 76 ft, the maximum height, and when it hits the ground using quadratic equations.

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