By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 14 in. long and 6 in. wide, find the dimensions of the box that will yield the maximum volume. (Round your answers to two decimal places.)

The the side of the cut-out square be x inches

base of box = 14-2x by 6-2x
(clearly x < 3 )

volume = V = x(14-2x)(6-2x)
= 84x - 40x^2 + 4x^3
dV/dx = 84 - 80x + 12x^2
= 0 for a max V
divide by 4
3x^2 - 20x + 21 = 0
x = (20 ± √148)/6
= 5.36 or 1.306 inches, but remember x < 3

so x = 1.31 correct to 2 decimals

check: pick a value slightly greater and slightly less than our answer

let x = 1.3 , V = 1.3(14-2.6)(6-2.6) = 50.388
let x = 1.31 , V = 1.31(14-2.62)(6-2.62) = 50.38836
let x = 1.4 , V = 1.4(14-2.8)(6-2.8) = 50.176

x = 1.31 yields the greatest volume

Well, if we're cutting away identical squares from the corners of the cardboard, we can call the side length of these squares "x".

So, by cutting away x from each corner, the length of the resulting box will be (14 - 2x) inches and the width will be (6 - 2x) inches.

Now, to find the height of the box, we need to fold up the flaps. Each flap will have a width of "x" and a height of "x" since it's a square. So the total height of the box will be x inches.

So, the volume of the box can be calculated as (14 - 2x) * (6 - 2x) * x.

Now, let me think about how to find the maximum volume... hmm... Ah! To find the maximum, we can take the derivative of the volume function, set it equal to zero, and solve for x.

The derivative of the volume function is a bit messy, but I suppose math can be messy sometimes. Solving for x gives you x ≈ 0.82 inches.

So, the dimensions of the box that will yield the maximum volume are approximately:
Length: 12.36 inches
Width: 4.36 inches
Height: 0.82 inches.

That's one box that knows how to maximize its volume!

To find the dimensions of the box that will yield the maximum volume, we need to maximize the volume function.

Let's start by assuming that a square of side length x is cut from each corner.

Step 1: Determine the dimensions of the resulting box.
Since the original cardboard is 14 inches long and 6 inches wide, when we cut x inches from each corner, the length of the box will be reduced by 2x inches and the width will be reduced by 2x inches.

So, the length of the box will be (14 - 2x) inches and the width will be (6 - 2x) inches.

Step 2: Determine the height of the resulting box.
When we fold the flaps, the height of the box will be x inches.

Step 3: Determine the volume of the resulting box.
The volume of a rectangular box is given by V = length * width * height. Substituting the dimensions we found in step 1 and the height from step 2, we get:
V = (14 - 2x) * (6 - 2x) * x

Step 4: Maximize the volume function.
To find the maximum volume, we need to find the value of x that maximizes the volume function V.

To do this, we can use calculus. We will take the derivative of V with respect to x, set it equal to zero, and solve for x. The value of x that satisfies this condition will give us the dimensions that yield the maximum volume.

Differentiating V with respect to x:
dV/dx = (6 - 2x)(14 - 2x) + x(-4)(14 - 2x)

Setting dV/dx equal to zero and solving for x:
0 = (6 - 2x)(14 - 2x) + x(-4)(14 - 2x)

After solving for x, we find the value(s) of x that satisfies this equation.

Step 5: Calculate the dimensions of the box.
Once we have the value of x, we can substitute it back into the expressions we found in step 1 to get the dimensions of the box.

Round the dimensions to two decimal places to get the final answer.

Note: It is recommended to use a calculator or a computer algebra system to solve the equation in step 4, as it can be quite complex.

The demand function for a certain make of replacement cartridges for a water purifier is given by the following equation where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.


-0.01x^2-0.3x+19
Determine the consumers' surplus if the market price is set at $1/cartridge. (Round your answer to two decimal places.)

Find the point on the graph where the tangent line is horizontal.

x/(x^2+25)

I know you have to do the quotient rule to find the derivative of the function, but I do not know what to do after that.