Maximum Volume An open box of maximum volume is made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). (a) The table shows the volumes V (in cubic centimeters) of the box for various heights x (in centimeters). Use the table to estimate the maximum volume. Height, x 1 2 3 4 5 6 Volume, V 484 800 972 1024 980 864 (b) Plot the points ( x , V ) from the table in part ( a ) . Does the relation defined by the ordered pairs represent V as a function of x ? (c) Given that V is a function of x , write the function and determine its domain.
Maximum Volume An open box of maximum volume is made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). (a) The table shows the volumes V (in cubic centimeters) of the box for various heights x (in centimeters). Use the table to estimate the maximum volume. Height, x 1 2 3 4 5 6 Volume, V 484 800 972 1024 980 864 (b) Plot the points ( x , V ) from the table in part ( a ) . Does the relation defined by the ordered pairs represent V as a function of x ? (c) Given that V is a function of x , write the function and determine its domain.
Solution Summary: The author explains that a function f is well-defined relation between two variables such that each and every element in set A is mapped with exactly one element.
Maximum Volume An open box of maximum volume is made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure).
(a) The table shows the volumes V (in cubic centimeters) of the box for various heights x (in centimeters). Use the table to estimate the maximum volume.
Height, x
1
2
3
4
5
6
Volume, V
484
800
972
1024
980
864
(b) Plot the points
(
x
,
V
)
from the table in part
(
a
)
.
Does the relation defined by the ordered pairs represent V as a function of
x
?
(c) Given that
V
is a function of
x
, write the function and determine its domain.
4. Use method of separation of variable to solve the following wave equation
მłu
J²u
subject to
u(0,t) =0, for t> 0,
u(л,t) = 0, for t> 0,
=
t> 0,
at²
ax²'
u(x, 0) = 0,
0.01 x,
ut(x, 0) =
Π
0.01 (π-x),
0
Solve the following heat equation by method of separation variables:
ди
=
at
subject to
u(0,t) =0, for
-16024
ძx2 •
t>0, 0 0,
ux (4,t) = 0, for
t> 0,
u(x, 0) =
(x-3,
\-1,
0 < x ≤2
2≤ x ≤ 4.
ex
5.
important aspects.
Graph f(x)=lnx. Be sure to make your graph big enough to easily read (use the space given.) Label all
6
33
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