Starting with your equation:

h = (1/2)(-9.8)t² + 24t + 7

Let's simplify that a little:

h = -4.9t² + 24t + 7

You want to know what the maximum height is, and at what time.

Presuming you don't know calculus, we can put this into vertex form, which is:

h = a(t - h)² + k

So the height is "k" and the time of the peak is "h".

Since you are told to estimate, I won't feel bad about using decimal approximations instead of keeping things as fractions.

To start we want the right side to be in the form of (t² + bt), so we will subtract 7 from both sides then divide both sides by -4.9:

h = -4.9t² + 24t + 7

h - 7 = -4.9t² + 24t

(h - 7) / (-4.9) = t² - 4.898t

Now we can complete the square by taking half of t's coefficient, squaring it, then adding it to both sides. So we add 5.997601 to both sides:

(h - 7) / (-4.9) + 5.997601 = t² - 4.898t + 5.997601

Now the right side can be factored:

(h - 7) / (-4.9) + 5.997601 = (t - 2.449)²

Now solve for h again, keeping the squared term intact:

(h - 7) / (-4.9) = (t - 2.449)² - 5.997601

h - 7 = (-4.9)(t - 2.449)² + 29.3882449

h = (-4.9)(t - 2.449)² + 36.3882449

So the peak happens at near the 2.5 second mark and reaches a height of 36.4 m above the ground