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.
Thus we get the constraint function
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.
Lagrange Multipliers to find minimum cost.
We can continue finding the partial derivatives to build the system of equations required for Lagrange Multipliers method.
And the constraint function
So we get
We can multiply each side of each equation by the dimension which is missing to get the full volume on the right side.
Then we can set each the equations equal to each other, so from the first one and the second equation we get
We can subtract 0.11xy from both sides.
And we can divide both sides by 0.12z to get
We can repeat the process by setting the first and third equation equal to each other.
We can subtract 0.12 xz from both sides
And we can solve by z
So if we replace that as well y = x on the constraint for the volume euqation we get
We can then solve for x
And taking the cube root
![x = \sqrt[3]{\cfrac{1584(12)}{11}}](https://tex.z-dn.net/?f=x%20%3D%20%5Csqrt%5B3%5D%7B%5Ccfrac%7B1584%2812%29%7D%7B11%7D%7D)
![x = \sqrt[3]{1728}](https://tex.z-dn.net/?f=x%20%3D%20%5Csqrt%5B3%5D%7B1728%7D)
So then we can use the equations we have found for y and z in terms of x
And
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.
