A projectile is fired vertically upward and can be modeled by the function $h(t)=-16 t^{2}+800 t+175$. During what time interval will the projectile be more than 1000 feet above the ground? Round your answer to the nearest hundredth.
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Solution
Step 1 ：The question is asking for the time interval during which the height of the projectile is more than 1000 feet. This means we need to solve the inequality \(h(t) > 1000\) for \(t\).
Step 2 ：The function \(h(t)\) is a quadratic function, so we can solve this inequality by finding the roots of the equation \(h(t) = 1000\) and then determining which intervals of \(t\) satisfy the inequality.
Step 3 ：The roots of the equation \(h(t) = 1000\) are approximately 1.05 and 48.95. This means that the projectile is at a height of 1000 feet at these two times.
Step 4 ：Since the function \(h(t)\) is a downward-opening parabola (because the coefficient of \(t^2\) is negative), the projectile will be more than 1000 feet above the ground between these two times.
Step 5 ：Final Answer: The projectile will be more than 1000 feet above the ground during the time interval \(\boxed{1.05 < t < 48.95}\).
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Solution
Step 1 ：The question is asking for the time interval during which the height of the projectile is more than 1000 feet. This means we need to solve the inequality \(h(t) > 1000\) for \(t\).
Step 2 ：The function \(h(t)\) is a quadratic function, so we can solve this inequality by finding the roots of the equation \(h(t) = 1000\) and then determining which intervals of \(t\) satisfy the inequality.
Step 3 ：The roots of the equation \(h(t) = 1000\) are approximately 1.05 and 48.95. This means that the projectile is at a height of 1000 feet at these two times.
Step 4 ：Since the function \(h(t)\) is a downward-opening parabola (because the coefficient of \(t^2\) is negative), the projectile will be more than 1000 feet above the ground between these two times.
Step 5 ：Final Answer: The projectile will be more than 1000 feet above the ground during the time interval \(\boxed{1.05 < t < 48.95}\).
