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A rancher has 1,800 linear feet of fencing and wants to enclose a rectangular field and then divide it into two equal pastures with one internal fence parallel to one of the rectangular sides. what is
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Aug 18, 2019
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A rancher has 1,800 linear feet of fencing and wants to enclose a rectangular field and then divide it into two equal pastures with one internal fence parallel to one of the rectangular sides. what is the maximum area of each pasture? 5400
Mathematics
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RoshanKumar Mutha
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RoshanKumar Mutha
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Answer: 67.500 ft²
Step-by-step explanation:
1) Name the
dimensions
using
variables
:
y: length of the rectangular field
x: widht of the rectangular field
2)
Model
the amount of fence used by the two equal pastures:
two sides and one internal fence: 2x + x = 3x
two lengths: 2y
⇒ 3x + 2y = 1800 ← linear feet of fence
y = 1800 / 2 - 3x/2 ← solving for y
y = 900 - 3x/2
3)
Area
of each pasture
A = x(y/2) ← half ot xy
A = x (900 - 3x/2) ← replacing y with 900 - 3x/2
A = 900x - 3x² / 2 ← using distributive property
4)
Maximum area ⇒ A' = 0
A' = 900 - 3x ← derivative of the polynomial 900x - 3x² / 2
900 - 3x = 0
⇒ 3x = 900
⇒ x = 900/3
⇒ x = 300
4)
Determine y
y = 900 - 3x/ 2 = 900 - 3(300)/2 = 900 - 450 = 450
5)
Area of each pasture
A = xy/2 = 300 × 450 /2 = 67500 ← final answer
Isaac Abraham
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Aug 24, 2019
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Isaac Abraham
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