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
igor_vitrenko [27]
3 years ago
8

A home improvement contractor is painting the walls and ceiling of a rectangular room. The volume of the room is 1584 cubic feet

. The cost of wall paint is $0.06 per square foot and the cost of ceiling paint is $0.11 per square foot. Find the room dimensions that will minimize the cost of the paint. (Enter your answers as a comma-separated list.) Incorrect: Your answer is incorrect. ft
Mathematics
1 answer:
liq [111]3 years ago
5 0

Answer:

The room dimensions that will minimize the cost of the paint are 12 ft x 12 ft x 11 ft.

Step-by-step explanation:

We can find first the volume equation using the formula of the volume of a box.

V= xyz

Thus we get the constraint function

1584 = xyz

Then since we are asked to minimize the cost, we can write the cost function which is the area of each one of the walls and ceiling multiplied by the painting cost.

C=0.11 xy+ 2(0.06)xz+2(0.06yz \\ C =0.11 xy+0.12xz+0.12yz

Lagrange Multipliers to find minimum cost.

We can continue finding the partial derivatives to build the system of equations required for Lagrange Multipliers method.

C_x=\lambda V_x \\ C_y = \lambda V_y \\ C_z = \lambda V_z

And the constraint function

xyz=1584

So we get

0.11y+0.12z=\lambda yz \\ 0.11x+0.12z=\lambda xz \\ 0.12x+0.12y=\lambda xy\\ xyz=1584

We can multiply each side of each equation by the dimension which is missing to get the full volume on the right side.

0.11xy+0.12xz=\lambda xyz \\ 0.11xy+0.12yz=\lambda xyz \\ 0.12xz+0.12yz=\lambda xyz

Then we can set each the equations equal to each other, so from the first one and the second equation we get

0.11xy+0.12xz= 0.11xy+0.12yz

We can subtract 0.11xy from both sides.

0.12xz=0.12yz

And we can divide both sides by 0.12z to get

x=y

We can repeat the process by setting the first and third equation equal to each other.

0.11xy+0.12xz= 0.12xz+0.12yz

We can subtract 0.12 xz from both sides

0.11xy=0.12yz

And we can solve by z

z= \cfrac{0.11x}{0.12}\\ z = \cfrac{11x}{12}

So if we replace that as well y = x on the constraint for the volume euqation we get

1584=x(x)\left(\cfrac{11}{12}x\right) \\ 1584=\cfrac{11}{12}x^3

We can then solve for x

x^3 = \cfrac{1584(12)}{11}

And taking the cube root

x = \sqrt[3]{\cfrac{1584(12)}{11}}

x = \sqrt[3]{1728}

x=12 ft

So then we can use the equations we have found for y and z in terms of x

y = x \\ y = 12 ft

And

z= \cfrac{11x}{12}\\z= \cfrac{11(12)}{12} \\ z=11ft

Then the dimensions of the room that will minimize the cost are 12ft x 12 ft x 11 ft. Since you have to enter using commas you can write 12, 12, 11, please check as well if you have to insert the units that are feet for each.

You might be interested in
Please Solve this Lines and angles question
Kay [80]

Answer:

133°

Step-by-step explanation:

As vertically opposite angles are equal,

< B E D = < F E G

Therefore,

< B E D = 47°

We know that co - interior angles which occur between 2 parallel lines, are added up to 180°.

Given that, < A B E = x

Therefore,

< B E D + < A B E = 180 °

47 + x = 180

x = 180 - 47

x = 133°

Let me know if you have any other questions. :)

8 0
2 years ago
CAN SOMEONE HELP WITH THIS ASAP!!!
jek_recluse [69]

Answer: x=8

Step-by-step explanation:

Since we are given that ∠ABC and ∠DBC are complementary angles, we know that adding them together should give us 90°. We can use this to solve for x.

6x+13+4x-3=90         [combine like terms]

10x+10=90                 [subtract both sides by 10]

10x=80                       [divide both sides by 10]

x=8

Now, we know that x=8.

6 0
3 years ago
Which equation can be used to find the surface area of the cylinder?
Elan Coil [88]

Answer:

D

Step-by-step explanation:

correct

5 0
2 years ago
Read 2 more answers
Is there anyone that can help me with this problem?
ipn [44]

Answer:

A

Step-by-step explanation:

mnp=wxy

8 0
2 years ago
Read 2 more answers
Freeport High School has 240 students and 12 teachers. Write down the ratio of the number teachers to the number of students.​
mash [69]

Answer:

1 : 20

Step-by-step explanation:

ratio of teachers : students = 12 : 240 ( divide both parts by 12 )

= 1 : 20

3 0
2 years ago
Other questions:
  • What is the product of the polynomials below?<br> (7x2+2x+4)(2x+5)
    6·1 answer
  • The total cost of n shirts is $15. The shirts are priced at a constant rate of $3 each.
    6·2 answers
  • If f(x) = x^3, which of the following describes the graph of f(x − 3)?
    14·1 answer
  • Height of Dutch Men. Males in the Netherlands are the tallest, on average, in the world with an average height of 183 centimeter
    12·1 answer
  • India or hot and Kelly visit the same number of museums ha visits to museum doing each of the her 3 days there if Kelly visits 3
    5·1 answer
  • (06.02 mc) the equation of line cd is y=3x-3. Write an equation of a line perpendicular to line cd in slope intercept form that
    10·2 answers
  • What property must you do to eliminate the parentheses in an equation.
    12·2 answers
  • A line passes through the points (2,-4) and (6,10). What is the equation of the line?
    5·1 answer
  • What is 95/18 as a decimal?
    14·2 answers
  • Find k. Write your answer in simplest radical form.
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!