Stephanie is a very good soccer player. She can kick a soccer ball with an initial velocity of 21.3m/sec. The function that represents the height of the ball with respect to time is h(x) = -4.9x^2 + 21.3x. About how many seconds will it take for the ball to hit the ground.
when it hits the ground, h = 0
0 = -4.9x^2 + 21.3x
4.9x^2 - 21.3x = 0
x(4.9x - 21.3) = 0
x = 0 , which corresponds to the moment she kicks the ball
or
x = 21.3/4.9 = appr 4.4 seconds
To find out how many seconds it will take for the ball to hit the ground, we need to determine the value of x when the height of the ball, h(x), is equal to zero.
Given the function h(x) = -4.9x^2 + 21.3x, we can set it equal to zero:
0 = -4.9x^2 + 21.3x
To solve this equation, we can factor out a common x:
0 = x(-4.9x + 21.3)
Setting each factor equal to zero will give us the possible solutions for x:
x = 0 or -4.9x + 21.3 = 0
Simplifying the second equation:
-4.9x + 21.3 = 0
-4.9x = -21.3
x = -21.3 / -4.9
x ≈ 4.3469
Since time cannot be negative, we disregard the root x = 0. Thus, it will take approximately 4.3469 seconds for the ball to hit the ground.
To find out how many seconds it will take for the ball to hit the ground, we need to determine when the height of the ball, represented by the function h(x) = -4.9x^2 + 21.3x, becomes equal to zero. When the ball hits the ground, its height will be zero.
Setting the function h(x) equal to zero, we have the equation -4.9x^2 + 21.3x = 0.
Now, let's solve the equation to find the values of x when the ball hits the ground.
First, we'll divide the equation by -4.9 to simplify it:
x^2 - (21.3/4.9)x = 0.
Next, we'll factor the equation:
x(x - (21.3/4.9)) = 0.
Now, applying the zero product property, we can set each factor equal to zero:
x = 0 or x - (21.3/4.9) = 0.
Simplifying the second equation, we have:
x = 21.3/4.9.
Therefore, the ball will hit the ground at x = 0 (immediately after it was kicked) and x = 21.3/4.9 seconds.
To determine the approximate number of seconds, we can evaluate the decimal value of 21.3/4.9:
21.3 ÷ 4.9 ≈ 4.355 seconds.
Thus, it will take approximately 4.355 seconds for the ball to hit the ground.