SOLUTION: Solve using derivatives - A dart is shot straight up from 1.5 metres above ground level. The distance (d), in metres, the dart is above the ground at time (t), in seconds is:

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Question 861872: Solve using derivatives - A dart is shot straight up from 1.5 metres above ground level. The distance (d), in metres, the dart is above the ground at time (t), in seconds is:
d%28t%29=+-4.9t%5E2+%2B+20t+%2B+1.5

1) What is the maximum height the dart reaches? I think this is 21.92 m
2) for how many seconds is the dart in the air?
Found 3 solutions by ewatrrr, cigany29, Alan3354:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi

 d%28t%29=+-4.9t%5E2+%2B+20t+%2B+1.5

took the derivative as directed and set to zero to find x-value at that point

 -9.8t + 20 = 0,   t = -20/9.8 = 2.0408

substituted that x-value found into d(t) to find the height

 d%282.0408+%29=+21.9082m

Answer by cigany29(23) About Me  (Show Source):
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Solve using derivatives - A dart is shot straight up from 1.5 metres above ground level. The distance (d), in metres, the dart is above the ground at time (t), in seconds is:
d%28t%29=+-4.9t%5E2+%2B+20t+%2B+1.5

1) What is the maximum height the dart reaches? I think this is 21.92 m
Already answered.
-------------
2) for how many seconds is the dart in the air?
From t = 0 until d(t) = 0 at impact.
d%28t%29=+-4.9t%5E2+%2B+20t+%2B+1.5+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case -4.9x%5E2%2B20x%2B1.5+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2820%29%5E2-4%2A-4.9%2A1.5=429.4.

Discriminant d=429.4 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-20%2B-sqrt%28+429.4+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2820%29%2Bsqrt%28+429.4+%29%29%2F2%5C-4.9+=+-0.0736703080011585
x%5B2%5D+=+%28-%2820%29-sqrt%28+429.4+%29%29%2F2%5C-4.9+=+4.15530296106238

Quadratic expression -4.9x%5E2%2B20x%2B1.5 can be factored:
-4.9x%5E2%2B20x%2B1.5+=+%28x--0.0736703080011585%29%2A%28x-4.15530296106238%29
Again, the answer is: -0.0736703080011585, 4.15530296106238. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-4.9%2Ax%5E2%2B20%2Ax%2B1.5+%29

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Ignore the negative solution.
t =~ 4.155 seconds