Explanation:
To determine when the ball is higher than 10 feet off the ground, we need to find the values of t for which h(t) is greater than 10.Given the equation h(t) = -t^2 + 7t, we can set up the inequality:
t^2 + 7t > 10
Rearranging the inequality, we have:
t^2 + 7t - 10 > 0
To solve this quadratic inequality, we can factorize it or use the quadratic formula. Let's factorize it:
(t - 2)(t + 5) > 0
Now we have two factors: (t - 2) and (t + 5).
To determine when the inequality is true, we consider the signs of each factor.
When (t - 2) > 0 and (t + 5) > 0, both factors are positive. This occurs when t > 2.When (t - 2) < 0 and (t + 5) < 0, both factors are negative. This occurs when t < -5.
Therefore, the ball is higher than 10 feet off the ground when t < -5 or t > 2.In the context of time, it means the ball is higher than 10 feet before it is thrown (t < -5) or after approximately 2 seconds (t > 2).