If a ball is thrown directly upward with a velocity of 14 m/s, its height (in meters) after t seconds is given by y = 14t − 4.9t2. What is the maximum height attained by the ball? (Round to two decimal places.)
______ m
After how many seconds is that height attained? (Round to three decimal places.)
______ s
Well, well, well, let's solve this math problem with a touch of humor, shall we?
To find the maximum height attained by the ball, we need to determine the vertex of this fancy quadratic equation. Imagine the ball going up and then coming back down like a yo-yo; we're interested in that peak point.
So, pay attention math enthusiasts! The equation that describes the height of this poor little ball is y = 14t - 4.9t^2. To find the maximum height, we can use the formula for the x-coordinate of the vertex: t = -b/2a.
Here's the fun part: a = -4.9 (the coefficient of t^2), and b = 14 (the coefficient of t). Plug those values in, and what do you get? A freshly baked result!
t = -b/2a = -14 / (2 * -4.9) = 14 / 9.8
Now, let's not leave our answer hanging without a proper unit, shall we? So, the time it takes the ball to reach its maximum height is approximately 1.43 seconds (rounded to three decimal places).
Hold your applause, folks! We're not done yet. Now that we know the time, we can find the height by plugging that cheeky t value back into the equation:
y = 14t - 4.9t^2 = 14 * 1.43 - 4.9 * (1.43)^2
Some quick calculations later, and we have our final answer – drumroll, please – the maximum height attained by the ball is approximately 8.18 meters (rounded to two decimal places).
Huzzah! The ball reached for the stars and reached about 8.18 meters. Just don't ask me if it managed to touch the moon.
To find the maximum height attained by the ball, we need to find the vertex of the quadratic equation y = 14t - 4.9t^2.
The vertex of a quadratic equation in the form y = ax^2 + bx + c, where a, b, and c are constants, can be found using the formula t = -b / (2a).
In this case, a = -4.9, b = 14.
t = -b / (2a)
t = -14 / (2 * -4.9)
t = -14 / -9.8
t = 1.43 seconds (rounded to two decimal places)
So, the maximum height attained by the ball is at 1.43 seconds.
Now, let's substitute this value of t back into the equation to find the height:
y = 14t - 4.9t^2
y = 14 * 1.43 - 4.9 * 1.43^2
y = 20.02 - 9.89
y = 10.13 meters (rounded to two decimal places)
Therefore, the maximum height attained by the ball is 10.13 meters, and it is attained after 1.43 seconds.
To find the maximum height attained by the ball, we need to determine the vertex of the parabolic equation y = 14t − 4.9t^2.
The vertex of a parabola is given by the formula (h, k), where h represents the x-coordinate of the vertex and k represents the y-coordinate of the vertex. In this case, h represents the time after which the maximum height is attained, and k represents the maximum height.
The formula for the x-coordinate of the vertex is given by the equation t = -b/2a, where a, b, and c are coefficients from the general form of a quadratic equation (y = ax^2 + bx + c).
In the given equation y = 14t − 4.9t^2, the coefficient of t^2 is -4.9 and the coefficient of t is 14. Plugging these values into the formula, we get:
t = -14 / (2 * -4.9)
t = 1.42857
Rounding to three decimal places, the ball reaches its maximum height after approximately 1.429 seconds.
To find the maximum height, plug this value of t back into the equation y = 14t − 4.9t^2:
y = 14 * 1.429 − 4.9 * (1.429)^2
y = 20.005 - 9.953
y ≈ 10.05
Therefore, the maximum height attained by the ball is approximately 10.05 meters.
y = 14 t - 4.9 t^2
if you do not know physics or calculus you must complete the square to find the vertex of the parabola.
4.9 t^2 - 14 t = -y
t^2 - 2.857 t = -(1/4.9) y
t^2 - 2.857 t + 1.429^2 = -(1/4.9)y+2.04
(t-1.429)^2 = -(1/4.9)[ y - 10 ]
vertex at height 10 meters and time 1.43 seconds