Suppose you are tossing an apple up to your friend on a third-story balcony. After x seconds the height of the apple is given by h= -16t^2+38.4t+0.96. Your friend catches the apple just as it reaches its highest point. How long does it take for the apple to reach your friend and at what height above the ground does your friend catch it.
To find the time it takes for the apple to reach your friend, we need to determine the highest point of the apple's trajectory.
Step 1: Find the vertex of the parabolic equation h= -16t^2 + 38.4t + 0.96.
The vertex of a parabolic equation in general form y = ax^2 + bx + c can be found using the formula x = -b / (2a).
For our equation h = -16t^2 + 38.4t + 0.96, we have a = -16 and b = 38.4.
Plugging these values into the formula, we have:
t = -38.4 / (2*-16)
t = -38.4 / (-32)
t = 1.2
So the apple reaches its highest point after 1.2 seconds.
Step 2: Find the height of the apple at its highest point.
To do this, substitute the value of t into the equation h = -16t^2 + 38.4t + 0.96:
h = -16(1.2)^2 + 38.4(1.2) + 0.96
h = -16(1.44) + 46.08 + 0.96
h = -23.04 + 46.08 + 0.96
h = 24
Therefore, the apple reaches a height of 24 above the ground at its highest point.
Step 3: The time it takes for the apple to reach your friend is double the time it takes for the apple to reach its highest point.
So, the apple takes 2 * 1.2 = 2.4 seconds to reach your friend.
Step 4: Determine the height at which your friend catches the apple.
To find the height, substitute the time we found in step 3 (t = 2.4) into the equation h = -16t^2 + 38.4t + 0.96:
h = -16(2.4)^2 + 38.4(2.4) + 0.96
h = -16(5.76) + 92.16 + 0.96
h = -92.16 + 92.16 + 0.96
h = 0.96
Therefore, your friend catches the apple at a height of 0.96 above the ground.
To find the time it takes for the apple to reach your friend and the height at which your friend catches it, we need to first determine the highest point the apple reaches.
The height of the apple is given by the equation h = -16t^2 + 38.4t + 0.96. We can find the highest point by finding the vertex of the parabolic equation. The formula for finding the x-coordinate of the vertex of a quadratic equation in the form ax^2 + bx + c is given by x = -b / (2a).
In this case, a = -16 and b = 38.4. Plugging these values into the formula, we get:
x = -38.4 / (2 * -16)
x = -38.4 / -32
x = 1.2
So, the highest point of the apple is reached at t = 1.2 seconds.
To find the height at which your friend catches the apple, we can substitute t = 1.2 into the equation h = -16t^2 + 38.4t + 0.96:
h = -16(1.2)^2 + 38.4(1.2) + 0.96
h = -16(1.44) + 46.08 + 0.96
h = -23.04 + 46.08 + 0.96
h = 24
Therefore, your friend catches the apple at a height of 24 feet above the ground.
To find the time it takes for the apple to reach your friend, we can solve for t when h = 24:
-16t^2 + 38.4t + 0.96 = 24
This equation is a quadratic equation, so we can set it equal to zero:
-16t^2 + 38.4t - 23.04 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -16, b = 38.4, and c = -23.04. Plugging these values into the quadratic formula, we get:
t = (-38.4 ± √(38.4^2 - 4(-16)(-23.04))) / (2 * -16)
Simplifying the equation further, we get:
t = (-38.4 ± √(1478.56 - 1475.52)) / -32
t = (-38.4 ± √3.04) / -32
t = (-38.4 ± 1.74) / -32
Calculating both options:
t1 = (-38.4 + 1.74) / -32
t1 = -36.66 / -32
t1 = 1.15
t2 = (-38.4 - 1.74) / -32
t2 = -40.14 / -32
t2 = 1.26
Therefore, it takes approximately 1.15 seconds or 1.26 seconds for the apple to reach your friend.
you just need to remember that the vertex of the parabola
y = ax^2+bx+c
occurs at x = -b/2a.