Section
2.3
Modeling
with
First
Order:
Problem
2
(1
point)
Water
leaks
from
a
vertical
cylindrical
tank
through
a
small
hole
in
its
base
at
a
rate
proportional
to
the
square
root
of
the
volume
of
water
remaining.
The
tank
initially
contains
175
liters
and
16
liters
leak
out
during
the
first
day.
days
B.
How
much
water
will
remain
in
the
tank
after
2
days?
volume
=
-
i
A.
When
will
the
tank
be
half
empty?
¢
Solution:
SOLUTION
Let
V(¢)
be
the
volume
of
water
in
the
tank
at
time
¢,
then
This
is
a
separable
equation which
has
the
solution
kt
Vi
=+
@2
Since
V(0)
=
175
this
gives
175
=
C2
so
kt
2
V()
=
(7
+
4/175)°.
However,
V(1)
=
159,
and
so
k
2
159
=
(E
+v/175)%,
so
that
k
=
2(v/I59
—
+/T75)
=
—1.2384.
Therefore,
V(t)
=
(—0.6192t
+
/T75)%.
The
tank
will
be
half-empty
when
V()
=
87.5,
so
we
solve
87.5
=
(—0.6192¢
+
/T75)*
to
obtain
t
=
6.257
days.
The
tank
will
be
half
empty
in
6.257
days.
The
volume
after
2
days
is
¥(2)
which
is
approximately
143.769
liters.
