Team Homework 7 explanations
All work and diagrams shown on attached sheet
Ch 8 review 60
A line a(x^2) is rotated about the y axis to make a bowl.
The bowl
a)
The bowl is filled with water to height H. Find the volume of the water.
To find
the volume of the water, we used horizontal slices.
The volume of water can be
sliced into circles with thickness delta h and area pi(r^2).
a.
Since the radius changes varying on the height of the bowl, we needed to
find a general formula for radius in terms of height.
In this case, radius is
also equal to the X coordinate on the graph.
We were given the formula y
= ax^2, so we simply had to solve that for x to get the radius.
b.
X=sqrt(y/a), so now we know radius in terms of height (y)
c.
Using this, we got the area of a circle with thickness delta h.
Area = pi
(r^2) = pi (y/a) delta y.
Now we just integrated the area from 0 to a height
h, adding up all the slices to get the total volume.
b) Area of water surface at height h.
We had already found the formula for area in
terms of height in the last part of the problem, so for this step we just replaced y
with the height of (h) given to us to find the area of the top slice.
c)
Differential equation for height as a function of time.
a.
The rate of evaporation for the bowl is equal to a constant times the area
that evaporates in a given unit of time.
In this case, we know the area and
are told that the constant for rate of evaporation is “k.”
Therefore, by
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 Winter '07
 Irena
 Algebra, Geometric Series, Expression, Qn, Closedform expression

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