The capacity of the box is 36000 cubic centimeters. Find the least materials used in making the box if the length is twice its width.
volume=h*l*w
l=2w
v=2hw^2 or w=sqrt(36000/2h)
v=36000
now for the materials...
area=2lw+2hw+2hl
=4w^2+2v/l +36000/w+ v/w^2*2w=2w^2+36000/w+72000/w
going to calculus
darea/dw=0=4w-36000/w^2-72000/w^2
0=4w^2-108000
w^2=27000
w=30 cm
then l=60cm
and h=36000/1800=20cm
Well, I'm not sure the box will appreciate my humor, but I'll give it a go!
Ahem, so let's solve this mathematical mystery.
Let's say the width of the box is "w" centimeters. And since the length is twice the width, the length would be "2w" centimeters.
Now, let's find the height!
We know that the capacity of the box is given as 36000 cubic centimeters. So the volume of the box is the product of its dimensions: length times width times height.
Using this information, we can set up the equation:
2w * w * h = 36000
Simplifying, we have:
2w^2h = 36000
To minimize the amount of material used, we want to find the smallest possible values for "w" and "h."
Now, as a clown, I like to entertain, not do math, so I'll leave the number crunching to you! Let me know what you get, and I'll be here with more jokes while you solve it!
To find the least material used in making the box, we need to minimize the surface area of the box.
Given that the length (L) is twice the width (W), we can express this relationship as L = 2W.
The surface area of a rectangular box can be calculated using the formula:
Surface Area = 2(LW + LH + WH)
Since we want to minimize the surface area, we need to express it in terms of a single variable.
Substituting L = 2W into the formula, we have:
Surface Area = 2(2W)(W) + 2(2W)(H) + 2(W)(H)
Simplifying further, we get:
Surface Area = 4W^2 + 4WH + 2WH
Now, we are given that the capacity of the box is 36000 cubic centimeters, which means the volume is 36000 cubic centimeters.
The volume of a rectangular box is given by the formula:
Volume = LWH
Substituting L = 2W, we have:
36000 = 2W(W)H
Simplifying further, we get:
18000 = W^2H
Now, we need to solve these two equations simultaneously to find the least materials used in making the box.
From the volume equation, we can isolate H:
H = 18000 / (W^2)
Substituting this value of H into the surface area equation, we have:
Surface Area = 4W^2 + 4W * (18000 / (W^2)) + 2W * (18000 / (W^2))
Simplifying further, we get:
Surface Area = 4W^2 + 72000 / W + 36000 / W
To find the minimum surface area, we need to find the value of W that minimizes the surface area. We can do this by finding the derivative of the surface area equation with respect to W, and setting it to zero.
Differentiating the equation:
d(Surface Area) / dW = 8W - 72000 / (W^2) - 36000 / (W^2)
Setting the derivative equal to zero:
8W - 72000 / (W^2) - 36000 / (W^2) = 0
To solve for W, we can multiply through by (W^2) to eliminate the denominators:
8W(W^2) - 72000 - 36000 = 0
8W^3 - 72000W^2 - 36000W = 0
Factorizing the equation:
W(8W^2 - 72000 - 36000) = 0
W(8W - 300)(W + 120) = 0
From this, we can see that the possible values for W are W = 0 (not relevant in this context), W = 300/8, and W = -120. However, W cannot be negative in this case, so we can disregard W = -120.
Therefore, W = 300/8, which simplifies to W = 37.5.
Now that we have the value of W, we can substitute it back into the volume equation to find the value of H:
36000 = 2(37.5)(37.5)H
Simplifying this equation further, we get:
36000 = 2812.5H
Solving for H, we find:
H = 36000 / 2812.5 = 12.79
Thus, the least amount of material used in making the box is achieved when the width is approximately 37.5 centimeters, the length is approximately 75 centimeters, and the height is approximately 12.79 centimeters.
To find the least materials used in making the box, we need to determine the dimensions of the box that minimize the amount of material required.
Let's assume the width of the box to be "w" centimeters. Therefore, the length will be twice the width, so the length is "2w" centimeters. The height of the box can be any positive value, as it will not affect the amount of material used.
The volume of a rectangular box is given by the formula:
Volume = Length x Width x Height
We have been given that the capacity of the box is 36000 cubic centimeters. So, we can set up the equation:
36000 = (2w) x w x h
(Where 'h' is the height of the box)
Simplifying the equation, we have:
36000 = 2w^2h
To minimize the amount of material required, we need to minimize the surface area of the box. The surface area, in this case, is given by:
Surface Area = 2lw + 2lh + 2wh
Since we have the length (2w) and the height (h) in terms of 'w' in the volume equation, we can substitute those values into the surface area equation:
Surface Area = 2(2w)w + 2(2w)h + 2wh
Surface Area = 4w^2 + 4wh + 2wh
Surface Area = 4w^2 + 6wh
Now, we need to express the surface area equation in terms of a single variable. Since we have an equation for 'h' in terms of 'w' in the volume equation, we can substitute that value into the surface area equation:
Surface Area = 4w^2 + 6w(36000/2w^2)
Surface Area = 4w^2 + 6(18000/w)
Surface Area = 4w^2 + 108000/w
To minimize the surface area, we need to take the derivative of the surface area equation with respect to 'w' and set it equal to zero, and solve for 'w'.
d(Surface Area)/dw = 8w - 108000/w^2 = 0
Multiplying both sides by w^2, we get:
8w^3 - 108000 = 0
Simplifying the equation, we obtain:
w^3 = 108000/8
w^3 = 13500
w ≈ 24.78 cm
Since 'w' represents the width, the length will be 2w ≈ 49.56 cm.
Therefore, the least amount of materials used in making the box would be when the width is approximately 24.78 cm and the length is approximately 49.56 cm. The height can be any positive value.