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To find the maximum height reached by the toy rocket before it explodes, we need to find the time at which the rocket reaches its highest point.
The rocket reaches its highest point when its vertical velocity is 0. We can find this time by setting the derivative of the height function equal to 0 and solving for t:
h'(t) = -9.8t + 19.6
0 = -9.8t + 19.6
9.8t = 19.6
t = 2
Now that we know the time at which the rocket reaches its highest point, we can plug this time back into the height function to find the maximum height:
h(2) = 1/2(-9.8)(2)^2 + 19.6(2) + 0.5
h(2) = -19.6 + 39.2 + 0.5
h(2) = 19.1
Therefore, the toy rocket is 19.1 meters above the ground when it explodes.