A rubber ball is thrown upward with an initial speed of 20 m/s. The approximate height of the ball after t seconds is h(t)=20t-4.9t^2. Determine the time period, to the nearest hundredth of a second, during which the ball is higher than 8 m
find where it is equal to8m.
-4.9t^2+20t=8
Find the solutions, and then h>8 between those values.
To determine the time period during which the ball is higher than 8 m, we need to find the values of t for which the height function h(t) is greater than 8.
h(t) = 20t - 4.9t^2
We'll set h(t) greater than 8 and solve for t:
20t - 4.9t^2 > 8
Rearranging, we get:
4.9t^2 - 20t + 8 < 0
To solve this inequality, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 4.9, b = -20, and c = 8. Plugging in these values, we get:
t = (-(-20) ± √((-20)^2 - 4*4.9*8)) / (2*4.9)
Simplifying further, we have:
t = (20 ± √(400 - 156.8)) / 9.8
t ≈ (20 ± √243.2) / 9.8
Now we can evaluate t to find the two time values:
t ≈ (20 + √243.2) / 9.8 ≈ 2.94 seconds
t ≈ (20 - √243.2) / 9.8 ≈ 0.36 seconds
Thus, the time period during which the ball is higher than 8 m is approximately 0.36 seconds to 2.94 seconds.
To find the time period during which the ball is higher than 8m, we need to solve the equation h(t) > 8.
Given: h(t) = 20t - 4.9t^2
Substituting h(t) with 8, we get:
8 = 20t - 4.9t^2
Rearranging the equation, we have:
4.9t^2 - 20t + 8 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = [-b ± √(b^2 - 4ac)] / (2a)
In this equation, a = 4.9, b = -20, and c = 8.
Substituting these values into the formula:
t = [-(-20) ± √((-20)^2 - 4(4.9)(8))] / (2(4.9))
Simplifying this expression further:
t = [20 ± √(400 - 156.8)] / (9.8)
t = [20 ± √(243.2)] / (9.8)
Now, let's calculate the two possible values of t:
t1 = (20 + √243.2) / 9.8 ≈ 2.96 seconds (rounded to the nearest hundredth)
t2 = (20 - √243.2) / 9.8 ≈ 0.21 seconds (rounded to the nearest hundredth)
The time period during which the ball is higher than 8m is between t1 and t2.
Therefore, the time period is approximately 0.21 to 2.96 seconds (rounded to the nearest hundredth).