The provided equation, y = -4.9t² + 27t + 2.4, represents the height (y) of the water balloon in meters as a function of time (t) in seconds. This equation is a quadratic function, commonly referred to as a parabola when graphed. The coefficients in this equation hold significant meaning:...
- -4.9: Represents half the acceleration due to gravity (approximately 9.8 m/s²). The negative sign indicates that gravity acts downwards, pulling the balloon back towards the ground.
- 27: Represents the initial upward velocity of the balloon.
- 2.4: Represents the initial height of the balloon.
Question 1: Initial Height of the Balloon
The initial height of the balloon is the height at time t = 0. To find this, we simply substitute t = 0 into the equation:
y = -4.9(0)² + 27(0) + 2.4 = 2.4
Therefore, the initial height of the water balloon is 2.4 meters.
Question 2: Time to Reach Maximum Height
The maximum height of the balloon occurs at the vertex of the parabola represented by the equation. The x-coordinate of the vertex represents the time it takes to reach the maximum height. We can find this using the formula:
t = -b / 2a
where a = -4.9 and b = 27 from the equation.
t = -27 / (2 * -4.9) = 2.755 seconds (approximately)
Therefore, the water balloon reaches its maximum height after approximately 2.755 seconds.
Question 3: Time Above 20 Meters
To find the time the balloon spends above 20 meters, we need to solve the inequality:
-4.9t² + 27t + 2.4 > 20
First, we simplify the inequality:
-4.9t² + 27t - 17.6 > 0
Solving this quadratic inequality involves finding the roots (where the expression equals 0) and then determining the intervals where the expression is positive.
Using the quadratic formula, we find the roots:
t = (-b ± √(b² - 4ac)) / 2a
where a = -4.9, b = 27, and c = -17.6.
t ≈ 1.08 seconds or t ≈ 3.34 seconds
These roots divide the time axis into three intervals: t 3.34. We need to determine the interval(s) where the expression is positive.
We can choose a test value from each interval and substitute it into the inequality to see if it holds true. For example, for t
Similarly, for 1.08
Finally, for t > 3.34, we can choose t = 4. The expression becomes -11.2, which is negative. Therefore, the inequality does not hold true for t > 3.34.
Therefore, the balloon is above 20 meters for approximately 2.26 seconds (3.34 - 1.08 = 2.26). This is the time between the balloon reaching a height of 20 meters on its way up and then again on its way down.