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 30 inches long and 14 inches wide find the dimensions of the box that will yield the maximum volume.
To find the dimensions of the box that will yield the maximum volume, we can set up an equation for the volume of the box.
Let's assume that the length of the squares cut from each corner is "x" inches.
The dimensions of the resulting box can be calculated as follows:
Length = (30 - 2x) inches
Width = (14 - 2x) inches
Height = x inches
The volume of the box can be calculated by multiplying these dimensions:
Volume = Length * Width * Height
Now, our goal is to find the value of "x" that maximizes the volume.
To do that, we can take the derivative of the volume function with respect to "x", set it equal to zero, and solve for "x". The value of "x" that we find will correspond to the critical point where the volume is maximized.
Let's find the derivative of the volume function:
dV/dx = (d/dx)(Length * Width * Height)
Using the product rule of differentiation, this becomes:
dV/dx = (d/dx)(30 - 2x) * (14 - 2x) * x
+ (30 - 2x) * (d/dx)(14 - 2x) * x
+ (30 - 2x) * (14 - 2x) * (d/dx)(x)
Simplifying this expression:
dV/dx = -4(7 - x)(15 - 2x) + x(14 - 2x)(-2)
Setting the derivative equal to zero:
-4(7 - x)(15 - 2x) + x(14 - 2x)(-2) = 0
Solving this equation will give us the value of "x" that maximizes the volume.
After finding the value of "x", we can substitute it back into the expressions for Length, Width, and Height to get the dimensions of the box that yields the maximum volume.