Set up the inequality − 16 t 2 + 64 t + 4 ≥ 52 .
Rearrange the inequality to get t 2 − 4 t + 3 ≤ 0 .
Factor the quadratic expression: ( t − 1 ) ( t − 3 ) ≤ 0 .
Determine the interval where the inequality holds: 1 ≤ t ≤ 3 .
The height of the ball is greater than or equal to 52 feet for 1 ≤ t ≤ 3 .
Explanation
Problem Setup We are given the height of a baseball as a function of time: h ( t ) = − 16 t 2 + 64 t + 4 . We want to find the interval of time t when the height of the ball is greater than or equal to 52 feet. This means we need to solve the inequality = 52"> h ( t ) " >= 52 .
Setting up the Inequality First, let's set up the inequality: − 16 t 2 + 64 t + 4 ≥ 52
Rearranging the Inequality Now, let's rearrange the inequality to get a quadratic expression on one side: − 16 t 2 + 64 t + 4 − 52 ≥ 0 − 16 t 2 + 64 t − 48 ≥ 0
Simplifying the Inequality To simplify, we can divide the entire inequality by -16. Remember that dividing by a negative number reverses the inequality sign: − 16 − 16 t 2 + 64 t − 48 ≤ − 16 0 t 2 − 4 t + 3 ≤ 0
Factoring the Quadratic Now, we factor the quadratic expression: ( t − 1 ) ( t − 3 ) ≤ 0
Analyzing the Sign To find the interval where this inequality holds, we need to determine when the product ( t − 1 ) ( t − 3 ) is negative or zero. We can analyze the sign of each factor:
If t < 1 , then both ( t − 1 ) and ( t − 3 ) are negative, so their product is positive.
If 1 < t < 3 , then ( t − 1 ) is positive and ( t − 3 ) is negative, so their product is negative.
If 3"> t > 3 , then both ( t − 1 ) and ( t − 3 ) are positive, so their product is positive.
Therefore, the inequality ( t − 1 ) ( t − 3 ) ≤ 0 holds when 1 ≤ t ≤ 3 .
Final Answer The height of the ball is greater than or equal to 52 feet for 1 ≤ t ≤ 3 .
Examples
Understanding quadratic inequalities can help in various real-world scenarios. For example, if a company models its profit P ( x ) as a function of the number of units sold x with a quadratic equation, they can determine the range of units they need to sell to achieve a certain profit level. Similarly, in physics, understanding the trajectory of projectiles, like a ball, involves solving quadratic equations and inequalities to determine when the projectile reaches a certain height or distance. This helps in sports, engineering, and other fields where projectile motion is important. For example, if a trebuchet launches a projectile, we can determine the time interval during which the projectile is above a certain height using quadratic inequalities.
The height of the baseball is greater than or equal to 52 feet in the interval from 1 second to 3 seconds. This is derived by solving the quadratic inequality formed from the height function. The answer is option D: 1 ≤ t ≤ 3.
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