Flight of a Ball If a ball is thrown upward at 96 feet per second from the top of a building that is 100 feet high, the height of the ball can be modeled by S(t)=100 + 96t -16t^2 feet, where t is the number of seconds after the ball is thrown. How long after the ball is thrown is the height 228 feet?
228=100+96t-16t^2
You're almost there. Tidy up the equation to get:
16t^2 - 96t + 128 = 0 which simplifies to
t^2 - 6t + 8 = 0
Now you have a simple quadratic. You'll get two answers because there are two times when it reaches 228ft.
S=120+96t-16t^2
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Well, this is a classic case of a ball trying to reach new heights (quite literally). Let's solve this equation together and find out when the ball hits the height of 228 feet.
We have the equation:
228 = 100 + 96t - 16t^2
To make things easier, let's rearrange the equation in standard form:
16t^2 - 96t + (100 - 228) = 0
Simplifying further:
16t^2 - 96t - 128 = 0
Now, it's time to solve for t. Let me grab my mathematical circus gear and work some magic!
Using the remarkable quadratic formula, t = (-b ± √(b^2 - 4ac)) / (2a), we can calculate the values of t.
Plugging in our values:
t = (-(-96) ± √((-96)^2 - 4(16)(-128))) / (2(16))
Calculating...
t = (96 ± √(9216 + 8192)) / 32
t = (96 ± √17408) / 32
Oh, it seems we have a square root situation here. The square root of 17408 is approximately 131.96.
So, t = (96 ± 131.96) / 32
Now, let's explore both possibilities:
t1 = (96 + 131.96) / 32
t1 ≈ 7.31 seconds
t2 = (96 - 131.96) / 32
t2 ≈ -1.62 seconds
Now, we have our two solutions. If a negative time is not part of your ball-throwing dreams, we can safely discard t2.
Hence, the time it takes for the ball to reach a height of 228 feet is approximately 7.31 seconds.
Remember, I always aim to deliver laughs along with solutions. I hope I added some humor to the height calculation!
To find out how long after the ball is thrown the height is 228 feet, we need to solve the equation:
228=100+96t-16t^2
To do this, we first move all terms to one side of the equation to have a quadratic equation in standard form:
16t^2 - 96t + 100 - 228 = 0
Now, simplify the equation:
16t^2 - 96t - 128 = 0
Divide each term by 16 to make the equation easier to work with:
t^2 - 6t - 8 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's factor the equation:
(t - 8)(t + 1) = 0
Setting each factor to zero:
t - 8 = 0 or t + 1 = 0
Solving for t:
t = 8 or t = -1
Since time cannot be negative, we discard the negative solution and conclude that the ball reaches a height of 228 feet 8 seconds after it is thrown.