Let's solve the problem step by step for the given quadratic function:

$h(t) = -2t^2 + 8t + 10$ where $h$ represents the height (in meters) and $t$ represents the time (in seconds).

a) After how many seconds does the ball return to the same height from which it was thrown?

The ball starts at $h(0) = 10$ (initial height). To find when the ball returns to this height, set $h(t) = 10$ and solve for $t$:

$-2t^2 + 8t + 10 = 10$

Simplify: $-2t^2 + 8t = 0$

Factorize: $-2t(t - 4) = 0$

Setting each factor equal to zero: $t = 0 \quad \text{or} \quad t = 4$

Thus, the ball returns to the same height after 4 seconds.

b) What height was the ball thrown from?

The ball's initial height corresponds to $t = 0$. Substitute $t = 0$ into $h(t)$:

$h(0) = -2(0)^2 + 8(0) + 10 = 10$

Therefore, the ball was thrown from a height of 10 meters.

c) After how many seconds does the ball hit the ground?

To find when the ball hits the ground, set $h(t) = 0$ and solve for $t$:

$-2t^2 + 8t + 10 = 0$

Divide through by $-2$ to simplify: $t^2 - 4t - 5 = 0$

Factorize: $(t - 5)(t + 1) = 0$

Setting each factor equal to zero: $t = 5 \quad \text{or} \quad t = -1$

Since time cannot be negative, $t = 5$ seconds.

Thus, the ball hits the ground after 5 seconds.

Final Answers:

a) The ball returns to the same height after 4 seconds.
b) The ball was thrown from a height of 10 meters.
c) The ball hits the ground after 5 seconds.

Would you like me to show more detailed steps for the factorization or solving? Let me know!

Related Questions:

1. What is the maximum height the ball reaches, and at what time does it occur?
2. How do you determine the vertex of the quadratic function?
3. Can you calculate the speed of the ball when it hits the ground?
4. How would the graph change if the initial height were 15 meters?
5. What would the equation look like if the ball were thrown with a greater velocity?

Tip:

The vertex of a quadratic function $ax^2 + bx + c$ gives the maximum or minimum value. Use $t = -\frac{b}{2a}$ to find the time at which this occurs!