A ball is kicked from ground level into the air. Its height, y, in feet after x seconds can be represented by the equation y=40x - 16x^2. What is the total elapsed time, in seconds, from the time the ball is kicked until it reaches ground level again?
I used the quadratic formula and got 2.5. Is this correct?
if you did ...
40x - 16x^2 = 0
x(40 - 16x) = 0
x = 0 or x = 40/16 = 2.5
then you are correct
the x = 0 , represents the time it was kicked
Well, it seems like you've done some math. Good job! However, you might have missed a little something. The total elapsed time from when the ball is kicked until it reaches ground level again would actually include both the upward and downward journey.
If we take a look at the quadratic equation y = 40x - 16x^2, we know that the ball reaches ground level when y = 0. So, we need to solve the equation 40x - 16x^2 = 0.
Using the quadratic formula, we have x = (-b ± √(b^2 - 4ac)) / (2a). Plugging in the values a = -16, b = 40, and c = 0, we get x = (-(40) ± √((40)^2 - 4(-16)(0))) / (2(-16)).
Now, when we simplify that, we would realize that both roots of the equation are valid because they represent the time at which the ball is at ground level. So, the total elapsed time would be the sum of the two roots.
Now, I'll leave the actual calculation to you. Remember, math is no laughing matter!
To find the total elapsed time until the ball reaches ground level again, we need to find when the height becomes zero.
Given the equation y = 40x - 16x^2, we set y to zero and solve for x:
0 = 40x - 16x^2
Rearranging the equation, we get:
16x^2 - 40x = 0
Factoring out 8x, we have:
8x(2x - 5) = 0
Setting each factor equal to zero, we find two possible values for x:
8x = 0 or 2x - 5 = 0
From 8x = 0, we get x = 0, which represents the time when the ball is initially kicked.
From 2x - 5 = 0, we solve for x:
2x = 5
x = 5/2
x = 2.5
Therefore, x = 2.5 represents the time it takes for the ball to reach the ground again.
So, your calculation of 2.5 seconds is indeed correct.
To find the total elapsed time from the time the ball is kicked until it reaches ground level again, we need to determine when the height y is equal to zero.
The given equation for height is y = 40x - 16x^2. Setting y equal to zero, we get:
0 = 40x - 16x^2
Now, we have a quadratic equation. To solve it, we can either factor it or use the quadratic formula.
If you used the quadratic formula, which is a valid method, let's verify your solution:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = -16, b = 40, and c = 0. Substituting these values into the quadratic formula:
x = (-40 ± √(40^2 - 4(-16)(0))) / (2(-16))
x = (-40 ± √(1600)) / (-32)
x = (-40 ± 40) / (-32)
Now, we have two possible values:
x₁ = (-40 + 40) / (-32) = 0
x₂ = (-40 - 40) / (-32) = 2.5
So, according to the quadratic formula, the values of x are 0 and 2.5.
Therefore, 2.5 is indeed a valid solution. It represents the time it takes for the ball to reach ground level after being kicked.
Hence, the total elapsed time from the time the ball is kicked until it reaches ground level again is 2.5 seconds.