When a soccer ball is kicked with a vertical speed of 25 m/s its height, h metres, after t seconds is given by the equation h=-5t^2+25t
a) how long after it is kicked is the soccer ball at a height of 30 m ?
b) How long is the soccer ball in the air?
a)h=30, solve for t
30=-5t^2+25t
5(t^2-5t+6)=0
5(t-3)(t-2)=0
t=2 (ball going up) or
t=3 (ball going down)
b) h=0, solve for t
0=-5t^2+25t
5t(5-t)=0
t=0 or t=5
Make similar interpretations.
dde
a) To find the time when the soccer ball is at a height of 30 m, we need to solve the equation h = 30.
Given the equation h = -5t^2 + 25t, we substitute h with 30:
30 = -5t^2 + 25t
We then rearrange the equation to have it equal to zero:
-5t^2 + 25t - 30 = 0
This is now a quadratic equation. We can solve it using factoring, completing the square, or using the quadratic formula. Let's use factoring in this example:
-5t^2 + 25t - 30 = 0
Divide the equation by -5 to simplify it:
t^2 - 5t + 6 = 0
Now, we need to factorize the quadratic equation:
(t - 2)(t - 3) = 0
Setting each factor equal to zero, we have:
t - 2 = 0 or t - 3 = 0
Solving for t in both cases:
t = 2 or t = 3
So, the soccer ball is at a height of 30 m after 2 seconds and after 3 seconds.
b) To calculate the time the soccer ball is in the air, we need to find when the height equals zero.
Given the equation h = -5t^2 + 25t, we set h equal to zero:
0 = -5t^2 + 25t
Again, this is a quadratic equation. Let's use factoring to solve it:
0 = t(-5t + 25)
This equation has two solutions:
t = 0 or -5t + 25 = 0
To find the second solution:
-5t + 25 = 0
-5t = -25
Dividing both sides by -5:
t = 5
So, the soccer ball is in the air for 0 seconds (initial kick) and 5 seconds.