A golf ball is hit from the top of a tee. The quadratic equation
𝑦 = −5𝑥2 + 20𝑥 + 0.05 describes its height, 𝑦, in meters as time, 𝑥, in seconds
passes. Determine how long the ball is in the air
Use the quadratic formula.
To use the quadratic formula, we first need to put the equation in standard form:
𝑦 = −5𝑥^2 + 20𝑥 + 0.05
𝑦 = -5(𝑥^2 - 4𝑥) + 0.05
𝑦 = -5(𝑥^2 - 4𝑥 + 4) + 20 + 0.05
𝑦 = -5(𝑥 - 2)^2 + 20.05
Now we can see that the vertex of the parabola is at (2, 20.05), which represents the maximum height of the ball.
To find how long the ball is in the air, we need to find the x-intercepts of the parabola. This represents when the ball hits the ground.
𝑦 = −5𝑥^2 + 20𝑥 + 0.05 = 0
Using the quadratic formula:
𝑥 = (−20 ± √(20^2 − 4(−5)(0.05)))/(2(−5))
Simplifying:
𝑥 = (−20 ± √402.05))/(-10)
𝑥 ≈ 0.04 or 𝑥 ≈ 3.96
Since we are only interested in the time that the ball is in the air, we can ignore the solution of 𝑥 ≈ 0.04 (which represents when the ball is initially hit from the tee).
Therefore, the ball is in the air for approximately 3.96 seconds before it hits the ground.
To determine how long the ball is in the air, we need to find the roots of the quadratic equation. The quadratic formula can be used to find the roots of a quadratic equation of the form "ax^2 + bx + c = 0", where 'a', 'b', and 'c' are coefficients.
Here, the equation is 𝑦 = −5𝑥^2 + 20𝑥 + 0.05, so 'a' is -5, 'b' is 20, and 'c' is 0.05.
Using the quadratic formula, the roots of the equation can be found:
𝑥 = (-𝑏 ± √(𝑏^2 - 4𝑎𝑐)) / (2𝑎)
Plugging in the values, we have:
𝑥 = (-20 ± √(20^2 - 4(-5)(0.05))) / (2(-5))
Simplifying further:
𝑥 = (-20 ± √(400 + 1)) / (-10)
𝑥 = (-20 ± √401) / (-10)
Now, we can find the two roots:
𝑥1 = (-20 + √401) / (-10)
𝑥2 = (-20 - √401) / (-10)
Calculating these values using a calculator or by hand will give us the values for 𝑥1 and 𝑥2, which will tell us the time the ball is in the air.