1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Gnoma [55]
3 years ago
15

A rancher plans to make four identical and adjacent rectangular pens against a​ barn, each with an area of 400 msquared. What ar

e the dimensions of each pen that minimize the amount of fence that must be​ used?

Mathematics
1 answer:
Ilia_Sergeevich [38]3 years ago
6 0

Answer:

base = 10\sqrt{5}m=22.36m and height = 8\sqrt{5}m = 17.89m

Step-by-step explanation:

In order to solve this problem, we must first understand how the pens are going to be placed. Refer to the attached diagram for an image of this.

From this diagram, we can start building the equations we are going to use to solve this. Since the problem doesn't specify the length of the barn, we will have no restriction on this. Taking this into account, we can start by building an equation for the area of each pen.

The area of a rectangle is given by the equation:

A=bh

where b is the base of the rectangle and h is the height of the rectangle.

with the given information we can now build our first equation:

400=bh

Since we have an equation with two variables, or two unknowns, we must have a second equation we can use to find the b and h values we need. In order to get this second equation, we must take into account that we need to minimize the amount of fencing to be used, so we can build an equation that will represent this fencing. When looking at the diagram, we can see that the fencing will be used to cover 4 bases and 5 heights, so our fencing equation should look like this:

F=4b+5h

Where F represents the amount of fencing to be used.

So now we can solve the first equation to substitute it into the second. Let's solve for the base b.

bh=400\\\\b=\frac{400}{h}

Which can now be substituted into the second equation so we get:

F=4(\frac{400}{h})+5h

which simplifies to

F=\frac{1600}{h} +5h

Now, this is the equation we need to minimize. In order to do so, we need to take the derivative of that and set it equal to zero, so we get:

F=\frac{1600}{h}+5h\\ \\F=1600h^{-1}+5h\\ \\F'=-1600h^{-2}+5\\ \\F'=-\frac{1600}{h^{2}}+5

And now we can set all this equal to zero, so we get:

-\frac{1600}{h^{2}}+5=0

Which can now be solved for h, so we get:

-\frac{1600}{h^{2}}+5=0\\ \\-\frac{1600}{h^{2} }=-5\\ \\1600=5h^{2}\\ \\h^{2}=\frac{1600}{5}\\  \\h^{2}=320\\ \\h=\sqrt{320}=8\sqrt{5}m=17.89m

That's where the first answer came from. We can now use this value to find the second answer together with the first equation:

b=\frac{400}{h}\\ \\b=\frac{400}{8\sqrt{5}}\\ \\b=\frac{50}{\sqrt{5}}

which simplifies to:

h=\frac{50}{\sqrt{5}}*\frac{\sqrt{5} }{\sqrt{5} }\\  \\h=\frac{50\sqrt{5} }{5}\\ \\h=10\sqrt{5}m = 22.36m

Aproximately and that's where te second answer came from.

So each  pen must measure:

22.36 m by 17.89 m

When multiplied you'll see that each pen will have an area of 400m^{2}

You might be interested in
Solve the inequality <br> -7&lt;3x+5 less than or equal to(no sign for that)7
olganol [36]

Answer:

x=-1

Step-by-step explanation:

-7 < 3(x) + 5 ≤  7

-7 < 3(-1) + 5 ≤  7

-7 < -3 + 5 ≤  7

-7 < 2 ≤  7

6 0
3 years ago
Rewrite the following the form log (c) log (20) - log (5)
Dmitrij [34]

Answer:log (20/5)

Step-by-step explanation:

When subtracting logs you divide the numbers after the log

4 0
3 years ago
Identify the angle or side that is common to triangle NPM and triangle OPM.
jonny [76]
MP is the best choice as both triangles have MP as the same vector

7 0
3 years ago
The sum of three consecutive integers is 33. Find the numbers.
Rufina [12.5K]

Answer:

A

Step-by-step explanation:

x+x+1+x+2=33

3x+3=33

3x=30

x=10

so x+1 =11

x+2 =12

5 0
3 years ago
To divide 1.89 ÷ 0.9, we need to change it. Choose the correct problem below to show what the “new” look would be.
Arlecino [84]

Answer:

D. 18.9 ÷ 9

Step-by-step explanation:

we need to divide

1.89 ÷ 0.9 but write it in different form

so ,we need to eliminate decimal from 0.9

as 0.9*10 = 9

thus,we multiply both  1.89 and  0.9 with 10, then we will have

(1.89*10) ÷ (0.9*10)

=> 18.9 ÷ 9

Thus, based on above calculation new look would be D. 18.9 ÷ 9

8 0
3 years ago
Other questions:
  • Hello! I graduated in 2008 but forgot how to solve multiplication. It's been so long. However I want to be refreshed on solving
    8·1 answer
  • A section of rope is 1 inch short of being 1 yard long. The rope is cut into tenths. How long is each section?
    13·1 answer
  • A pail holds 2 3/4 gallons of water. How much is this in cups?
    15·1 answer
  • The average expenditure on Valentine's Day was expected to be $100.89 (USA Today, February 13, 2006). Do male and female consume
    9·1 answer
  • Help me and the 1st 2 answer will get brainliest and i need asap im in the test
    9·2 answers
  • Pls help with this please
    13·2 answers
  • Line AB contains points A (-2, 3) and B (4, 5). Line AB has a slope that is
    10·2 answers
  • I NEED ANSWERS AND HOW DO I HSOW MY WORK FOR THIS HELP
    5·1 answer
  • HELPPPPP PLESSEEE THIS IS FOR MY EXAMMM
    10·1 answer
  • Which value is the solution of
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!