The height of a ball thrown upward on the moon with a velocity of 8 meters per second can be modeled by h=-0.8t^2+8t, where h is the height of the ball in meters and t is the time in seconds. At what times will the height of the ball reach 19.2 meters
just solve
-0.8t^2+8t = 19.2
using the quadratic formula
Or, to make things a bit simpler,
8t^2 - 80t + 192 = 0
t^2 - 10t + 24 = 0
which you can easily factor
To find the times at which the height of the ball reaches 19.2 meters, we need to set up the equation h = 19.2 and solve for t.
Given equation: h = -0.8t^2 + 8t
Substitute h with 19.2:
19.2 = -0.8t^2 + 8t
Rearrange and set the equation to 0:
-0.8t^2 + 8t - 19.2 = 0
Now we can solve this quadratic equation by factoring or using the quadratic formula. Let's use the quadratic formula to find the values of t.
The quadratic formula is given by:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -0.8, b = 8, and c = -19.2.
Substituting these values into the quadratic formula:
t = ( -8 ± √(8^2 - 4(-0.8)(-19.2))) / (2(-0.8))
Simplifying this expression:
t = ( -8 ± √(64 - 61.44)) / (-1.6)
t = ( -8 ± √2.56) / (-1.6)
t = ( -8 ± 1.6) / (-1.6)
There are two solutions for t:
1) t = (-8 + 1.6) / (-1.6)
2) t = (-8 - 1.6) / (-1.6)
Simplifying these expressions:
1) t = (-6.4) / (-1.6)
t = 4
2) t = (-9.6) / (-1.6)
t = 6
Therefore, the height of the ball reaches 19.2 meters at t = 4 seconds and t = 6 seconds.
To find the times at which the height of the ball reaches 19.2 meters, we can set up the equation as follows:
h = -0.8t^2 + 8t
Since we want to find the times when the height is 19.2 meters, we can set h equal to 19.2 and solve for t:
19.2 = -0.8t^2 + 8t
Rearranging the equation, we get:
0.8t^2 - 8t + 19.2 = 0
Now we have a quadratic equation in the form ax^2 + bx + c = 0, where a = 0.8, b = -8, and c = 19.2.
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, we get:
t = (-(-8) ± √((-8)^2 - 4 * 0.8 * 19.2)) / (2 * 0.8)
Simplifying further, we have:
t = (8 ± √(64 - 15.36)) / 1.6
t = (8 ± √48.64) / 1.6
t = (8 ± 6.966) / 1.6
Now we can calculate the two possible values for t:
t₁ = (8 + 6.966) / 1.6 ≈ 9.979
t₂ = (8 - 6.966) / 1.6 ≈ 1.079
Therefore, the height of the ball reaches 19.2 meters at approximately t = 1.079 seconds and t = 9.979 seconds.