3 0 3600 16 225 1200 16 16 3 0 the rectangle has a

3
= 0
? =
3600
16
= 225
𝐴
′′
= (1200 − 16?)
′
= −
16
3
< 0
The rectangle has a maximum area when x =225
Find x = 225
? = 600 − (
4
3
) 225 = 300
b. . New Horizons Travel advertises a package plan for a Florida vacation. The fare for
the flight is $400/person plus $8/person for each unsold seat on the plane. The
plane holds 120 passengers and the flight will be cancelled if there are fewer than
50 passengers. What number of passengers will maximize revenue?
? = ?????? ?? ?????? ?????
∴ The dimensions of each rectangle has 200 meters and 225
meters
560 − 16? = 0
? = 35

𝑅
′′
= (560 − 16?)
′
= −16 < 0
The revenue will be maximum area when x =35
Number of passengers = 120 -35 = 85
c. An open-topped storage box with a square base has a capacity of 5 m3 . The material
for the sides cost $1.60/m2 , while the material for the bottom costs $2.00/ m2 .
Find the dimensions that will minimize the cost of making the box.
∴ The maximum revenue will be when the flight has 85
passengers.
Unit # 4 – Extensions

𝐶
′
= 4? −
32
?
2
C’= 0
4? −
32
?
2
= 0
4? =
32
?
2
?
3
= 8, ? = 2
𝐶
′′
= (4? −
32
? 2
) = 4 +
64
? 3
At x=2, ?
′′
= 4 +
64
8 3
= 12 > 0
Cost is maximum when x=2
At x =2 , ? =
5 5
4
= 1,25
Unit # 4 – Extensions
d. The perimeter of an isosceles triangle is 38 m. Find the length of the sides of the
triangle of maximum area.

Unit # 4 – Extensions
We know that the idosceles triangle is 38 meters
? + ? + ? = 38
2? + ? = 38
? = 38 − 2?
From a triangle ABD, ?
2
= ℎ
2
+ (
?
2
)
2
ℎ
2
= ?
2
− (19 − ?)
2
ℎ
2
= 38? − 361
ℎ = √38? − 361
𝐴??? 𝐴 =
1
2
ℎ?
𝐴??? 𝐴 = (
1
2
) (38 − 2?)√38? − 361
𝐴??? 𝐴 = (19 − ?)√38? − 361
𝐴
′
= (( 19 − ?)√38? − 361 )′
𝐴
′
= ( 19 − ?)
1
2√38? − 361
(38? − 361)
′
+ √38? − 361 (19? − ?)′
𝐴
′
=
38(19 − ?)
2√38? − 361
− √38? − 361
𝐴
′
=
316 − 19? − 38? + 361
√38? − 361
𝐴
′
=
722 − 57?
√38? − 361
Critical Numbers when A ’ is zero
722 − 57?
√38? − 361
= 0

Unit # 4 – Extensions
? =
722
57
=
38
3
38? − 361 = 0
? =
361
38
=
19
2
𝐴
′′
= (
722 − 57?
√38? − 361
)′
‘
𝐴
′′
=
√38? − 361
(722 − 57?)
′
− (722 − 57?)(√38? − 361 )′
(√38? − 361 )
2
𝐴
′′
=
√38? − 361
(−57) − (722 − 57?)(
38
√38? − 361
)
38? − 361
𝐴
′′
=
20577 − 27436
(38? − 361)
3
2
𝐴
′′
=
−6859
(38? − 361)
3
2
When ? =
38
3
, 𝐴
′′
=
−6859
(38?−361)
3
2
< 0
The triangle has a maximum area when ? =
38
3
When ? =
19
2
, 𝐴
′′
=
−6859
(38(
19
2
)−361)
3
2
> 0
The triangle has a minimum area when ? =
19
2

Unit # 4 – Extensions
At ? =
19
2
, ? = 38 − 2 (
38
2
) =
38
3
∴ The length of the side of the triangle are
38
3
? ,
38
3
? ,
38
3
?

Unit # 4 – Extensions
e. An apple orchard now has 80 trees per hectare and the average yield is 400 apples
per tree. For each additional tree planted per hectare the average yield per tree is
reduced by approximately four apples. How many trees per hectare will give the
largest crop of apples?