Find the most economical proportions for a covered box of fixed volume whose base is a rectangle with one side three times the other.
Given Volume:
$V = x \, (3x) \, y
V = 3x^2 \, y$
0=3x2y′+6xy
0=3x2y′+6xy
y′=−2y/x
y′=−2y/x
Total Area:
AT=2(3x2)+2(3xy)+2(xy)
AT=2(3x2)+2(3xy)+2(xy)
AT=6x2+8xy
AT=6x2+8xy
dAT/dx=12x+8(xy′+y)=0
dAT/dx=12x+8(xy′+y)=0
12x+8[x(−2y/x)+y]=0
12x+8[x(−2y/x)+y]=0
12x+8[−2y+y]=0
12x+8[−2y+y]=0
12x=8y
12x=8y
y=32x
y=32x
Altitude = 3/2 × shorter side of base. answer
To find the most economical proportions for a covered box of fixed volume with a rectangular base, we need to determine the dimensions of the rectangle that minimize the surface area.
Let's assume the dimensions of the rectangle are "x" and "3x" (where x is the smaller side). The height of the box will be denoted as "h".
The volume of the box can be calculated as the product of the dimensions: V = x * 3x * h = 3x^2 * h.
The surface area of the box consists of the area of the rectangular base, the area of the top cover, and the four side walls.
The area of the base is given by: A_base = x * 3x = 3x^2.
The area of the top cover is the same as the base: A_top = 3x^2.
The area of each of the four side walls is given by: A_walls = 2h * x + 2h * 3x = 8hx.
The total surface area is the sum of these areas: A_total = A_base + A_top + A_walls = 3x^2 + 3x^2 + 8hx.
Since we want to minimize the surface area while keeping the volume constant, we can express the surface area as a function of a single variable, x, and then differentiate it with respect to x to find its critical points.
A_total = 6x^2 + 8hx.
To eliminate the height (h), we can use the equation for volume V = 3x^2 * h and solve for h: h = V / (3x^2).
Substituting h in terms of x into the equation for surface area: A_total = 6x^2 + 8(V / (3x^2)) * x.
Simplifying this equation: A_total = 6x^2 + 8V / (3x).
To minimize the surface area, we can differentiate A_total with respect to x and set it equal to zero:
dA_total/dx = 12x - 8V / (3x^2) = 0.
Multiplying through by x^2 gives: 12x^3 - 8V = 0.
Now, solving for x^3: x^3 = (8V) / 12 = 2V / 3.
Finally, taking the cube root of both sides: x = (2V / 3)^(1/3).
Therefore, the most economical proportions for the covered box occur when one side of the rectangular base is (2V / 3)^(1/3) and the other side is 3 times larger, or 3 * (2V / 3)^(1/3).
Note that the height of the box in terms of these dimensions can be found using the volume equation: h = V / (3x^2).
To find the most economical proportions for a covered box of fixed volume, we need to consider the relationship between the dimensions and volume of the box.
Let's assume the dimensions of the rectangle base are length (L) and width (W). According to the given information, we have the following relationship:
L = 3W (one side is three times the other)
The volume of the box is given as a fixed value, let's call it V. The formula for the volume of a rectangular box is:
V = L * W * H
Where H represents the height of the box. Since we are looking for the most economical proportions, we want to minimize the surface area of the box. The surface area is calculated as follows:
Surface Area = 2LW + 2LH + 2WH
Now, we need to express the surface area in terms of a single variable to minimize it. Let's rewrite the equation for the surface area using the relationship between L and W:
Surface Area = 2(3W)(W) + 2(3W)(H) + 2(W)(H)
Simplifying this equation:
Surface Area = 6W^2 + 6WH + 2WH
Next, we substitute the value of H in terms of V. From the volume formula, we have:
V = LWH
L = 3W
V = 3W^2 * H
H = V / (3W^2)
Now we can substitute this into the surface area equation:
Surface Area = 6W^2 + 6W(V / 3W^2) + 2(V / 3W^2) * W
Simplifying the expression:
Surface Area = 6W^2 + 2V / W + 2V / W
Surface Area = 6W^2 + 4V / W
The task now is to minimize the surface area. To do that, we can take the derivative of the surface area equation with respect to W, set it equal to zero, and solve for W. However, since this is a lengthy process better suited for mathematical software, let's consult a calculator or graphing tool to find the minimum.
By plugging in various values for W and observing the corresponding surface areas, we can determine the value of W that gives the minimum surface area. Keep in mind that W should be a positive value, and it can be any real number greater than zero.
Once we find the value of W that minimizes the surface area, we can substitute it back into the equation L = 3W to find the corresponding value of L. The height (H) can then be calculated using the volume formula.
By following this process, you will be able to find the most economical proportions for the covered box of fixed volume with a rectangular base.