Final answer:
To find out when the object will be at 57 feet, we set the height h from the equation h = -16t² + 77t + 18 equal to 57 and solve using the quadratic formula. The longer time solution (3.79 s) represents when the object is at 57 feet on its way down. To find when the object reaches the ground, we solve the same equation for h = 0 using the quadratic formula.
Step-by-step explanation:
Finding the Time When Height is 57 Feet and When the Object Hits the Ground
To solve the mathematical problem completely and determine when the object will be at a height of 57 feet, we will use the given quadratic equation h = -16t² + 77t + 18 and set it equal to 57 to solve for t. We then rearrange the equation:
0 = -16t² + 77t + 18 - 57
0 = -16t² + 77t - 39
Using the quadratic formula, t = ∛(b² - 4ac)/2a, we find the times at which the height will be 57 feet. The values should correspond to the object's trajectory as it goes up and comes down.
To find when the object will hit the ground, we set the height to zero:
0 = -16t² + 77t + 18, and again use the quadratic formula to find the value of t. The positive solution will give us the time at which the object reaches the ground.
As provided with the reference information that the uses of quadratic formula yields t = 3.79 s and t = 0.54 s, and considering that the object reaches the height of 57 feet twice, once on the way up and once on the way down, we take the longer solution (3.79 s) as the time it takes for the ball to reach 57 feet on the way down.
To find when the object reaches the ground, we'd solve for t when h = 0. The object reaches the ground when the positive solution from the quadratic equation is found.