The height h (in feet) of an underhand volleyball serve can be modeled by h=−16t2+30t+4, where t is the time in seconds. After how many seconds is the ball 16 feet above the ground? Round your answers to the nearest tenth.
The ball is 16 feet above the ground after about
_______
seconds and after about
__________
seconds.
To find out after how many seconds the ball is 16 feet above the ground, we can set the height equation equal to 16 and solve for t.
Given:
h = -16t^2 + 30t + 4
Setting h = 16:
16 = -16t^2 + 30t + 4
Next, let's move all terms to one side of the equation:
-16t^2 + 30t + 4 - 16 = 0
Simplifying:
-16t^2 + 30t - 12 = 0
To solve this quadratic equation, we can use the quadratic formula, which is given by:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = -16, b = 30, and c = -12.
Plugging in the values:
t = (-(30) ± √((30)^2 - 4(-16)(-12)) / (2(-16))
Simplifying further:
t = (-30 ± √(900 - 768)) / (-32)
t = (-30 ± √132) / (-32)
Taking both positive and negative roots:
t = (-30 + √132) / (-32) and t = (-30 - √132) / (-32)
Using a calculator to evaluate these roots, we get:
t ≈ 0.512 and t ≈ 1.488
Rounded to the nearest tenth:
The ball is 16 feet above the ground after about 0.5 seconds and after about 1.5 seconds.
To find the time when the ball is 16 feet above the ground, we need to solve the equation h = 16.
Plug in 16 for h in the given equation:
16 = -16t^2 + 30t + 4
Rearrange the equation to set it equal to zero:
-16t^2 + 30t + 4 - 16 = 0
-16t^2 + 30t - 12 = 0
Next, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In the quadratic equation -16t^2 + 30t - 12 = 0, a = -16, b = 30, and c = -12.
Plugging these values into the quadratic formula gives:
t = (-30 ± √(30^2 - 4(-16)(-12))) / (2(-16))
Simplifying further:
t = (-30 ± √(900 - 768)) / (-32)
t = (-30 ± √132) / (-32)
Now we have two possible solutions for t:
t1 = (-30 + √132) / (-32)
t2 = (-30 - √132) / (-32)
Evaluating these expressions gives the following approximate values:
t1 ≈ 0.3125 seconds
t2 ≈ 1.1875 seconds
Therefore, the ball is 16 feet above the ground after approximately 0.3 seconds and 1.2 seconds.
2 seconds
Sub h=16 into the equation and re-arrange to get 0 = ... then factor to solve for the two different times that the height is 16 feet.
hint:
0=-16t^2+30t-12