A baseball is thrown upward at an initial speed of 9.5 m/s. The approximate height of the ball, h meters, after t seconds, is given by the equation h= -4.9t+9.5t+1. At the same time, a golf ball is hit upward at an initial speed of 12 m/s. The approximate height of the ball, h meters, after t second, is given by the equation h=-4.9t^2+12t. When are both balls at the same height? Round to the nearest tenth.
At the same height when
-4.9t^2+12t = -4.9t+9.5t+1
2.5t = 1
t = 1/2.5 = .4 seconds
Ah, the battle of the balls! Let's solve this height dilemma, shall we?
First, let's set the two height equations equal to each other:
-4.9t + 9.5t + 1 = -4.9t^2 + 12t
Now, to find out when the balls are at the same height, we need to solve for t. Fun, right?
Rearranging the equation, we get:
-4.9t^2 + (12t - 9.5t) - 1 = 0
Grouping terms, we have:
-4.9t^2 + 2.5t - 1 = 0
Now, to solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, we get:
t = [−2.5 ± √(2.5^2 - 4 × (-4.9) × (-1))] / (2 × (-4.9))
Calculating that gives us two solutions: t ≈ 0.4 and t ≈ 1.8
So, both balls will be at the same height at around 0.4 and 1.8 seconds. Voila!
Remember, laughter is great for your health, so keep smiling!
To find the time when the baseball and golf ball are at the same height, we need to set their height equations equal to each other and solve for t.
The height of the baseball, h1, is given by the equation:
h1 = -4.9t + 9.5t + 1
The height of the golf ball, h2, is given by the equation:
h2 = -4.9t^2 + 12t
Setting h1 equal to h2, we have:
-4.9t + 9.5t + 1 = -4.9t^2 + 12t
Simplifying the equation:
4.9t^2 - 4.9t + 9.5t - 12t + 1 = 0
4.9t^2 + 4.6t + 1 = 0
Using the quadratic formula:
t = [-4.6 ± sqrt((4.6)^2 - 4*4.9*1)] / (2*4.9)
Calculating the value inside the square root:
(4.6)^2 - 4*4.9*1 ≈ 4.6^2 - 4*4.9
≈ 21.16 - 19.6
≈ 1.56
Now, substituting this value back into the quadratic formula:
t = [-4.6 ± sqrt(1.56)] / (2*4.9)
t ≈ [-4.6 ± 1.25] / 9.8
t ≈ (-4.6 + 1.25) / 9.8 or t ≈ (-4.6 - 1.25) / 9.8
t ≈ -3.35 / 9.8 or t ≈ -5.85 / 9.8
t ≈ -0.34 or t ≈ -0.60
Since time cannot be negative in this context, we disregard the negative solution.
Therefore, both balls are at the same height at approximately t = 0.6 seconds.
To find when both balls are at the same height, we need to set their height equations equal to each other and solve for t.
For the baseball, the height equation is h = -4.9t² + 9.5t + 1.
For the golf ball, the height equation is h = -4.9t² + 12t.
Setting the two equations equal to each other, we get:
-4.9t² + 9.5t + 1 = -4.9t² + 12t.
Let's simplify this equation:
-4.9t² + 9.5t + 1 - (-4.9t² + 12t) = 0,
-4.9t² + 9.5t + 1 + 4.9t² - 12t = 0,
-2.5t - 1 = 0.
Solving for t:
-2.5t = 1,
t = 1 / (-2.5),
t ≈ -0.4.
Since time cannot be negative in this context, we discard the negative solution.
Therefore, both balls are at the same height after approximately 0.4 seconds.