Let w=width
L= length
3w+1=l
Perimeter -2(l+w)
2(W+3w+1)=42
Divide both sides by 2
W+3w+1=21
Subtract 1
W+3w=20
Add like terms
4w=20
Divide both sides by 4
W=5
The width is 5 inches
3w+1=l
Input the width into the equation
3(5)+1=l
15+1=l
16=l
The length of the rectangle is 16 inches
The rectangle is 5 x 16 inches
<u>Given</u>:
Given that the graph OACE.
The coordinates of the vertices OACE are O(0,0), A(2m, 2n), C(2p, 2r) and E(2t, 0)
We need to determine the midpoint of EC.
<u>Midpoint of EC:</u>
The midpoint of EC can be determined using the formula,

Substituting the coordinates E(2t,0) and C(2p, 2r), we get;

Simplifying, we get;

Dividing, we get;

Thus, the midpoint of EC is (t + p, r)
Hence, Option A is the correct answer.
i. The Lagrangian is

with critical points whenever



- If
, then
. - If
, then
. - Either value of
found above requires that either
or
, so we get the same critical points as in the previous two cases.
We have
,
,
, and
, so
has a minimum value of 9 and a maximum value of 182.25.
ii. The Lagrangian is

with critical points whenever
(because we assume
)



- If
, then
. - If
, then
, and with
we have
.
We have
,
,
, and
. So
has a maximum value of 61 and a minimum value of -60.
9/9 = 1 and 8/9 is only 1/9 less than 9/9 which is 1.
Answer: B) 1 8/9 is closer to 1