Final answer:
To find when the object is 128 feet in height, the quadratic equation h=-16t²+125t+5 is solved for t by setting h to 128 and using the quadratic formula. After simplifying and calculating the results, the positive value of t is selected as the valid solution indicating the time at which the object reaches the height of 128 feet.
Step-by-step explanation:
To determine when the height of an object thrown upward is 128 feet using the provided equation h = -16t² + 125t + 5, we need to solve for t when h is set to 128 feet. This transforms our equation into the quadratic equation:
128 = -16t² + 125t + 5.
To solve for t, we must first bring all terms to one side of the equation to set it equal to zero, which gives us:
0 = -16t² + 125t - 123.
Now we can use the quadratic formula where a = -16, b = 125, and c = -123:
t = [-b ± √(b² - 4ac)] / (2a)
Substituting our values in gives:
t = [-125 ± √(125² - 4(-16)(-123))] / (2(-16)).
Calculating this expression yields two values for t. The physical scenario dictates that only the positive value of t which represents the time after the launch will be considered as a valid solution. By calculating and choosing the positive root, we get the time when the object reaches 128 feet.
Always remember that in a real-world scenario objects can take different times to reach the same height during their upward and downward journey, hence they may have two different times corresponding to the desired height.