four pens of equal size should be formed from 220 feet of fencing. the length of the total rectangular area formed should be triple the width. Find the length and width of each individual pen.
Answers
Step-by-step explanation:
Step 1: Set up a picture/diagram to help answer the question and write out the needed equations.
Diagram:
enter image source here
The total length will be
x
and the height will be
y
.
Needed Equations:
Perimeter of this diagram
=
500
=
2
x
+
5
y
Total Area
=
A
=
x
y
Step 2: Solve for
y
using the equation
500
=
2
x
+
5
y
5
y
=
500
−
2
x
y
=
100
−
2
5
x
Step 3: Substitute the equation for
y
into the function for area.
A
=
x
(
100
−
2
5
x
)
A
=
−
2
5
x
2
+
100
x
Step 4: Find the derivative of the equation for area.
A
'
=
−
4
5
x
+
100
Step 5: Use the derivative equation in order to find the critical point(s) that maximize the area.
Critical points are when
A
'
=
0
and when
A
'
does not exist. It is also good to check the endpoints of an equation in order to check for a maximum or minimum.
Since
A
'
always exists, only find where
A
'
=
0
(there will be no endpoints to check since this is a pen).
0
=
−
4
5
x
+
100
−
100
=
−
4
5
x
100
=
4
5
x
x
=
125
A
'
is positive when
x
<
125
and
A
'
is negative when
x
>
125
, therefore meaning that x=125 is a maximum. Since this value is a maximum, the area is maximized when the total length is 125ft.
Step 6: Find the height (
y
) when
x
=
125
.
500
=
2
(
125
)
+
5
y
5
y
=
250
y
=
50
The dimensions that will maximize the area the total area of the pen will be
125
f
t
.
b
y
50
f
t
.
Step 7: Find the total area of the pen.
A
=
x
y
A
=
(
125
)
(
50
)
A
=
6
,
250
f
t
.
2