Answer: We can use the given formula to find the height of the projectile at any given time t. In this case, we want to find the time when the height is 85 feet, so we can set H(t) equal to 85 and solve for t:
85 = 16t^2 + 64t + 45
Rearranging and simplifying:
16t^2 + 64t - 40 = 0
Dividing by 16:
t^2 + 4t - 2.5 = 0
Using the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = 1, b = 4, and c = -2.5.
Substituting these values into the formula:
t = (-4 ± sqrt(4^2 - 4(1)(-2.5))) / 2(1)
Simplifying:
t = (-4 ± sqrt(36)) / 2
t = (-4 ± 6) / 2
So, we have two possible solutions:
t = (-4 + 6) / 2 = 1
or
t = (-4 - 6) / 2 = -5
Since time cannot be negative, the only valid solution is t = 1.
Therefore, the projectile will attain a height of 85 feet after 1 second.
Explanation: