A golf ball is hit from the top of a tee. The quadratic equation
𝑦 = −5𝑥2 + 20𝑥 + 0.05 describes its height, 𝑦, in metres as time, 𝑥, in seconds
passes. Determine how long the ball is in the air t:
Use the quadratic formula.
To determine how long the ball is in the air, we need to find when its height (y) is equal to zero. This is because when the ball hits the ground, its height is zero.
So, we set the quadratic equation equal to zero:
-5x^2 + 20x + 0.05 = 0
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
where a = -5, b = 20, and c = 0.05
x = (-20 ± √(20^2 - 4(-5)(0.05))) / 2(-5)
x = (-20 ± √(400 + 1))/(-10)
x = (-20 ± √401)/(-10)
x = (-20 + √401)/(-10) or x = (-20 - √401)/(-10)
x ≈ 0.1013 or x ≈ 3.8987
Since the ball cannot be in the air for a negative time, we discard the negative solution.
Therefore, the ball is in the air for approximately t = 0.1013 seconds.
To determine how long the golf ball is in the air, we can set the equation 𝑦 = -5𝑥^2 + 20𝑥 + 0.05 equal to zero, since the ball will be on the ground when its height is zero.
So, we have:
-5𝑥^2 + 20𝑥 + 0.05 = 0
We can use the quadratic formula to solve this equation. The quadratic formula is given by:
𝑥 = (-𝑏 ± √(𝑏^2 - 4𝑎𝑐)) / 2𝑎
Where 𝑎, 𝑏, and 𝑐 are the coefficients of the equation.
In this case, 𝑎 = -5, 𝑏 = 20, and 𝑐 = 0.05. Plugging these values into the formula, we get:
𝑥 = (-20 ± √(20^2 - 4(-5)(0.05))) / 2(-5)
Simplifying further, we get:
𝑥 = (-20 ± √(400 + 1)) / -10
𝑥 = (-20 ± √401) / -10
Now we can calculate the two possible solutions for 𝑥.