Question 366108: John is making juice. For this particular juice, he's filling a bottle in the shape of a pentagonal pyramid: the base is a regular pentagon, 15cm to a side, and the height of the "pyramid" is 32cm. Assume these measurements are for the interior of the bottle.
Assume the flow of juice into the bottle starts at zero and increases at a rate of 1 millilitre per second per second. How many seconds will it take to completely fill the bottle? Please round to the nearest whole number.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Let  represent the rate of change of the fill rate; in this case  . Let  be the instantaneous fill rate, and let  be the instantaneous volume filled. And finally, let  represent the elapsed time in seconds since I started to fill the bottle.
But the constant of integration in this case is the initial flow rate, so:
Then the instantaneous volume is:
 .
Since both the initial volume,  , and the initial fill rate,  , are both zero, if  is the volume of the pyramid bottle, then we only need solve:
for , namely:
since we can assume for the precision required for this problem that cubic centimeters are equivalent to milliliters.
The volume of any pyramid is given by  where  is the height, and  is the area of the base.
To find the area of a regular polygon knowing the measure of a side, use:
)
Which, for your regular pentagon reduces to:
)
You should be able to handle it from here.
John
My calculator said it, I believe it, that settles it
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