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
How do I do this <br> X+y=-4<br> 3x-6y=15
avanturin [10]

Answer:

x=-1, y=-3. (-1, -3).

Step-by-step explanation:

x+y=-4

3x-6y=15

---------------

x=-4-y

3(-4-y)-6y=15

-12-3y-6y=15

-12-9y=15

9y=-12-15

9y=-27

y=-27/9

y=-3

x+(-3)=-4

x-3=-4

x=-4+3

x=-1

8 0
3 years ago
PLZZ HELLO ILL GIVE BRAINLIST
sp2606 [1]

Answer:

X= 1

Step-by-step explanation:

2, + the 2 her brother gives her is too high, its 4.

which means we go lower.

0 can't be right, because 0 + 2 = 2, not 3.

lets try 1.

1 + 2 = 3.

So that means X = 3.

3 0
3 years ago
Which ratio is equal to 27 : 81?
son4ous [18]
3:9 and if you reduce it again, 1:3
4 0
2 years ago
Read 2 more answers
Fill in the table using this function rule.​
Elodia [21]

Answer:

1) 16
2) 12
4) 4
5) 0

Step-by-step explanation:

Plug in the value for x in the equation above and solve for y for all 5.

3 0
2 years ago
What is the perimeter of the actual garden?
Lilit [14]

Answer: 576 inches or 48 feet

Step-by-step explanation:

set up a proportion where \frac{drawing AB}{actual AB} =\frac{drawing perimeter}{actual perimeter}

so you get drawingAB = 24 in

actualAB = 18 feet = 18 * 12 in = 216 in

drawing perimeter = 64 in

so plug these in to get\frac{24}{216} =\frac{64}{actual perimeter}

rearrange to get actual perimeter = \frac{216 * 64}{24}

and solve to get 576 inches or 576/12 = 48 feet

4 0
3 years ago
Other questions:
  • Q is directly proportional to r.<br> Q is 76 when r is 20.<br> Work out q when r is 45
    8·1 answer
  • 128.09 to the nearest tenth
    11·1 answer
  • What is the diameter of a 30 inch circle?
    12·1 answer
  • The population of a town increases 14% in 2 years. if the population was 6000 originally, what is the population after the incre
    7·1 answer
  • Solutions to y=2(5)-x and x+y=2
    13·1 answer
  • 1 5/8 write as a decimal
    15·2 answers
  • Which of the following are exterior angles? Check 2 that apply.
    15·1 answer
  • Need help finding the area
    8·2 answers
  • Consider functions f and g<br><br> vvvv
    9·1 answer
  • Peter was thinking of a number. peter adds 6 to it, then doubles it and gets an answer of 48.5. what was the original numbe
    14·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!