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3 years ago

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A farmer will build a rectangular pen for his sheep. A wall will form one side of the pen. The farmer has 28 m of fencing to for

m the other three sides. The farmer plans to build the pen so that it has its maximum possible area. What will be the dimensions of the farmer’s sheep pen? Enter your answers in the boxes.

2 answers:

umka2103 [35]3 years ago

8 0

its <span> 7m x 14m </span> hope this helps you

Aleks04 [339]3 years ago

8 0

Answer with explanation:

→The farmer want to build a rectangular pen for his sheep.

→A wall will form one side of the pen.

→The farmer has 28 m of fencing to form the other three sides.

Let length of rectangle = x m

and, Breadth of rectangle = y m

Perimeter of rectangle = ( 2 x + 2 y) meter

Let the side through which wall has been built is along breadth.then equation can be written as

→ 2 x+ 2 y= 28+y

2 x+y=28

y=28-2 x--------(1)

Area of rectangle = A=Length ×Breadth

A= x y

A= x×(28-2 x)

A= 28 x -2 x²

Differentiating with respect to x, we get

A'=28 -4 x

For Maximal or Minimal value,

A'=0

→28 - 4 x=0

4 x= 28

→→Dividing both side by ,4, we get

x=7

putting the value of x , in equation 1

→y=2 8-2×7

→y=28-14

→y=14

So, length= 7 m

Breadth = 1 4 m

if wall has been side opposite to length, then Dimension of rectangle will be,then equation can be written as

2 x + 2 y= 28+x

x+2 y=28

→x=28-2 y

Applying same method done above,we will get

Length= 14 m

Breadth = 7 m

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A norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular. Find the dimensions of a norman

Yanka [14]

Answer:

$W\approx 8.72$ and $L\approx 15.57$ .

Step-by-step explanation:

Please find the attachment.

We have been given that a norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular. The total perimeter is 38 feet.

The perimeter of the window will be equal to three sides of rectangle plus half the perimeter of circle. We can represent our given information in an equation as:

$2L+W+\frac{1}{2}(2\pi r)=38$

We can see that diameter of semicircle is W. We know that diameter is twice the radius, so we will get:

$2L+W+\frac{1}{2}(2r\pi)=38$

$2L+W+\frac{\pi}{2}W=38$

Let us find area of window equation as:

$\text{Area}=W\cdot L+\frac{1}{2}(\pi r^2)$

$\text{Area}=W\cdot L+\frac{1}{2}(\pi (\frac{W}{2})^2)$

$\text{Area}=W\cdot L+\frac{\pi}{2}(\frac{W}{2})^2)$

$\text{Area}=W\cdot L+\frac{\pi}{2}(\frac{W^2}{4})$

$\text{Area}=W\cdot L+\frac{\pi}{8}W^2$

Now, we will solve for L is terms W from perimeter equation as:

$L=38-(W+\frac{\pi }{2}W)$

Substitute this value in area equation:

$A=W\cdot (38-W-\frac{\pi }{2}W)+\frac{\pi}{8}W^2$

Since we need the area of window to maximize, so we need to optimize area equation.

$A=W\cdot (38-W-\frac{\pi }{2}W)+\frac{\pi}{8}W^2$

$A=38W-W^2-\frac{\pi }{2}W^2+\frac{\pi}{8}W^2$

Let us find derivative of area equation as:

$A'=38-2W-\frac{2\pi }{2}W+\frac{2\pi}{8}W$

$A'=38-2W-\pi W+\frac{\pi}{4}W$

$A'=38-2W-\frac{4\pi W}{4}+\frac{\pi}{4}W$

$A'=38-2W-\frac{3\pi W}{4}$

To find maxima, we will equate first derivative equal to 0 as:

$38-2W-\frac{3\pi W}{4}=0$

$-2W-\frac{3\pi W}{4}=-38$

$\frac{-8W-3\pi W}{4}=-38$

$\frac{-8W-3\pi W}{4}*4=-38*4$

$-8W-3\pi W=-152$

$8W+3\pi W=152$

$W(8+3\pi)=152$

$W=\frac{152}{8+3\pi}$

$W=8.723210$

$W\approx 8.72$

Upon substituting $W=8.723210$ in equation $L=38-(W+\frac{\pi }{2}W)$ , we will get:

$L=38-(8.723210+\frac{\pi }{2}8.723210)$

$L=38-(8.723210+\frac{8.723210\pi }{2})$

$L=38-(8.723210+\frac{27.40477245}{2})$

$L=38-(8.723210+13.70238622)$

$L=38-(22.42559622)$

$L=15.57440378$

$L\approx 15.57$

Therefore, the dimensions of the window that will maximize the area would be $W\approx 8.72$ and $L\approx 15.57$ .

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3 years ago

What is the reason for each step in the solution of the inequality?

Citrus2011 [14]

step 1: Given

step 2: Distributive Property

step 3: Combine Like terms

step 4: Subtraction Property of Order

step 5: Addition property of order

step 6: Division property of order

Step-by-step explanation:

We are given the inequality: $-38-3(x+5)>6x-17$

We need to match each step with the reasons.

The given steps are:

1.

−38−3(x+5)>6x−17

So, Reason for step 1: Given

2.

−38−3x−15>6x−17

So, Reason for step 2: Distributive Property

3.

−3x−53>6x−17

So, Reason for step 3: Combine Like terms

4.

−9x−53>−17

So, Reason for step 4: Subtraction Property of Order

5.

−9x>36

So, Reason for step 5: Addition property of order

6.

x<−4

So, Reason for step 6: Division property of order

Keywords: Solving inequalities

Learn more about Solving inequalities at:

#learnwithBrainly

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3 years ago

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Irina-Kira [14]

Answer:

C

Step-by-step explanation:

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3 years ago

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What is the first step to Solving Word Problems?

asambeis [7]

The first step in solving any word problem is to read the problem carefully.

Steps would be as follows:
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3- Make a plan
4- Solve the problem
5- Check that your answer makes sense

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3 years ago

Jim bought a fishing rod for $42. This price was 70% of the original price. What was the original price?

Dmitry [639]

Answer:

54.6

Step-by-step explanation:

42 = 70%

100% - 70% = .30%

42 x 1.3 = 54.6

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