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Question

The length of a rectagular fenced enclosure is 12 feet more than the width. If Farmer Dan has 100 feet of encing, write an inequality to find the dimensions of the rectangle with the largest priementer that can be created using 100 feet or fencing. Part 2: What are the dimensions of the rectagular with the largest priementer? What is the area enclosed bu this recatgle

Answer 1

(For writing purposes I am going to call the Length of the fence L and the Width of the fence W)

 

First, Lets break down what this word problem gives us.

 

We know that the fence is a rectangle

and the Length of the fence is 12 more than the Width

 

when you see the phrase "more than" you should think addition. We can set this up as  L = 12 + W

 

Next, we see that Farmer Dan has a total of 100 feet of fencing to work with

 

What is the equation we should use to find the Perimeter (P) of a rectangle?  

P = L + L + W + W -->  P  = 2L +2W  

 

and we know that the Length is 12 more than the Width (L = 12 + W) so let's substitute this into our Perimeter equation 

 

 

P = 2 ( 12 + W ) + 2W 

 

 

now we have to set this up in an inequality to get the largest perimeter

 

 

2(12+W) + 2W ≤ 100

 

( Note: we use the "less than or equal to" inequality because we know that we cannot have more than 100 feet of fencing because that is all Farmer Dan has on him so we can either try to use all of the fencing material he has or less than what he has. )

 

Simplifying this inequality we get

 

24 + 2W + 2W  ≤ 100

 

24 + 4W  ≤ 100

 

4W  ≤ 100 - 24

 

4W ≤ 76

 

W ≤ (76/4)

 

W ≤ 19

 

if we say the Width, W = 19  

 

What should our Length be?  L = 12 + W

 

L = 12 + 19

L = 31

 

 

 

What is the perimeter?  

 

P = 2L + 2W

P = 2 (31) + 2 (19)  

 

P = 62 + 38

P = 100 feet

 

 

the largest fence farmer Dan can build with the rule given ( length of a rectangular fenced enclosure is 12 feet more than the width) is 100 feet perimeter 

 

the dimensions of the fence would be the Length = 31 feet and the Width = 19 feet

 

the Area (A) of this fence is found by multiplying the Length and the Width together

 

A = L*W

 

A = 31 * 19 = 589 square-feet 

 

 

 

 

Answer 2

let's setup equation for the length and width

 

let say

 

width = x

 

length = x+12

 

since the fence is a rectangular perimeter

 

the formula is  2(length + width)

 

2(x+ 12 + x) = 2(2x + 12) = 4x + 24

 

there is 100 ft of fence to use, so the inequality will be

 

a.) 4x + 24 ≤ 100 

 

this is true, because it doesn't need all 100 feet of fence to make the perimeter and cannot use over 100 feet

 

b.) for biggest perimeter then the

 

4x + 24 = 100 

 

4x = 76

 

x = 19 feet which is the width

 

length is 19+12= 31 feet

 

c.) area is 19*31 = 589 square feet