Create assignment, 57950, Exam 3, Nov 24 at 12:25 pm
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GAGAN’S PRACTICE EXAM.
CalC4g00Ex5
50:07, calculus3, multiple choice,
>
1 min,
wordingvariable.
001
Stewart Example 5, page 282
Find the area of the largest rectangle that can
be inscribed in a semicircle of radius 2.
1.
Area = 4 sq. units
correct
2.
Area = 3 sq. units
3.
Area = 1 sq. units
4.
Area = 0 sq. units
5.
Area = 2 sq. units
Explanation:
Let’s take the semicircle to be the upper
half of the circle
x
2
+
y
2
= 4
having radius 2 and center at the origin. For
the rectangle we take one side on the
x
axis
and one corner at a point
P
(
x, y
) on the semi
circle as shown in
P
(
x, y
)
2

2
Then the area of the rectangle is given by
A
(
x
) = 2
xy
= 2
x
p
4

x
2
.
We have to maximize
A
(
x
) on the interval
[0
,
2]. Now
A
0
(
x
) =
p
4

x
2

x
2
√
4

x
2
=
4

2
x
2
√
4

x
2
.
Thus the critical points of
A
(
x
) occur at
x
=

2
√
2
,
2
√
2
,
only one of which lies in [0
,
2]. But
A
(0) = 0
,
A
‡
2
√
2
·
= 4
,
A
(2) = 0
.
Consequently,
max. area = 4 sq. units
.
CalC4g11a
50:07, calculus3, multiple choice,
>
1 min,
wordingvariable.
002
The canvas wind shelter
x
is to be constructed for use on Padre Island
beaches. It is to have a back, two square sides,
and a top.
If 600 square feet of canvas are
available, find the length
x
of the shelter for
which the space inside is maximized assuming
all the canvas is used.
1.
length = 21 feet
2.
length = 19 feet