The height of a ball above the ground t seconds after it is thrown is h(t)= 20 + 32t - 16t^2. How long will it take for the ball to hit the ground? (Round your answer to the nearest hundredth of a second.)
just set h=0 and solve for t. It's just a normal quadratic equation. If you factor out a 4, you can work with smaller numbers.
To find the time it takes for the ball to hit the ground, we need to solve for t when h(t) = 0.
Given the equation h(t) = 20 + 32t - 16t^2, we set it equal to zero:
0 = 20 + 32t - 16t^2
Rearranging the equation, we get:
16t^2 - 32t - 20 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so we'll use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In our equation, a = 16, b = -32, and c = -20. Substituting these values into the quadratic formula, we get:
t = (-(-32) ± √((-32)^2 - 4(16)(-20))) / (2(16))
Simplifying further:
t = (32 ± √(1024 + 1280)) / 32
t = (32 ± √(2304)) / 32
t = (32 ± 48) / 32
Now we have two possible values for t:
t1 = (32 + 48) / 32 = 80 / 32 = 2.5
t2 = (32 - 48) / 32 = -16 / 32 = -0.5
Since time cannot be negative in this context, the ball takes approximately 2.5 seconds to hit the ground.
To find the time it will take for the ball to hit the ground, we need to find the value of t when the height, h(t), is equal to zero.
Given the equation for the height of the ball, h(t) = 20 + 32t - 16t^2, we can set it equal to zero and solve for t:
0 = 20 + 32t - 16t^2
Rearranging the equation, we have:
16t^2 - 32t - 20 = 0
Next, we can attempt to solve this quadratic equation using the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 16, b = -32, and c = -20. Substituting these values into the quadratic formula, we have:
t = (-(-32) ± sqrt((-32)^2 - 4 * 16 * (-20))) / (2 * 16)
Simplifying further, we get:
t = (32 ± sqrt(1024 + 1280)) / 32
t = (32 ± sqrt(2304)) / 32
Now, we need to determine the values of t that give real solutions. Taking the positive square root, we have:
t = (32 + sqrt(2304)) / 32
t = (32 + 48) / 32
t = 80 / 32
t = 2.5 seconds
Therefore, it will take approximately 2.5 seconds for the ball to hit the ground.