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An open box is to be made out of square cardboard of 18 cm by cutting offs equal squares from the corners and turning up the sides. What is the maximum volume of the box in cm3?
- a)423
- b)432
- c)412
- d)424
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
An open box is to be made out of square cardboard of 18 cm by cutting ...
Let x be the corner side which is cut off and v be volume.
v = l × b × h
v = l × b × h
v = (18 − 2x) × (18 − 2x) × x
v = 4x3 − 72x2 + 324x
v′ = 12x2 − 144x + 324v′
v′′ = 24x − 144
v′ = 12x2 − 144x + 324 = 0
x2 − 12x + 27=0
(x−3) (x−9) = 0
(x=3) or (x = 9)
atx = 3
v′= 24x −144 = 24(3) −144 = −72<0
at x = 3 we have volume maximum
v = 4(3)3 − 72(2)2 + 324(2)
∴ v = 432
Most Upvoted Answer
An open box is to be made out of square cardboard of 18 cm by cutting ...
Problem Overview
An open box is created by cutting equal squares from the corners of an 18 cm square cardboard and folding up the sides. To find the maximum volume of this box, we will determine the optimal size of the squares to cut out.
Volume Formula
The volume V of the box can be expressed as:
- V = length × width × height
- Here, height is the side of the square cut out, denoted as x.
After cutting squares of side x from each corner, the dimensions of the box become:
- Length = 18 - 2x
- Width = 18 - 2x
- Height = x
Thus, the volume can be rewritten as:
- V(x) = (18 - 2x)(18 - 2x)x
Maximizing Volume
To find the maximum volume, we need to:
- Expand the volume formula:
- V(x) = x(18 - 2x)(18 - 2x)
- V(x) = x(324 - 72x + 4x^2)
- Differentiate V(x) and set the derivative to zero to find critical points.
Finding Critical Points
- The first derivative dV/dx = 324 - 144x + 12x^2
- Setting this to zero will give us the critical points.
After solving, we find that the maximum volume occurs at x = 3 cm.
Calculating Maximum Volume
Now substituting x = 3 into the volume formula:
- V(3) = (18 - 2(3))(18 - 2(3))(3)
- V(3) = (12)(12)(3) = 432 cm³
Conclusion
Thus, the maximum volume of the open box is 432 cm³, corresponding to option B.
An open box is created by cutting equal squares from the corners of an 18 cm square cardboard and folding up the sides. To find the maximum volume of this box, we will determine the optimal size of the squares to cut out.
Volume Formula
The volume V of the box can be expressed as:
- V = length × width × height
- Here, height is the side of the square cut out, denoted as x.
After cutting squares of side x from each corner, the dimensions of the box become:
- Length = 18 - 2x
- Width = 18 - 2x
- Height = x
Thus, the volume can be rewritten as:
- V(x) = (18 - 2x)(18 - 2x)x
Maximizing Volume
To find the maximum volume, we need to:
- Expand the volume formula:
- V(x) = x(18 - 2x)(18 - 2x)
- V(x) = x(324 - 72x + 4x^2)
- Differentiate V(x) and set the derivative to zero to find critical points.
Finding Critical Points
- The first derivative dV/dx = 324 - 144x + 12x^2
- Setting this to zero will give us the critical points.
After solving, we find that the maximum volume occurs at x = 3 cm.
Calculating Maximum Volume
Now substituting x = 3 into the volume formula:
- V(3) = (18 - 2(3))(18 - 2(3))(3)
- V(3) = (12)(12)(3) = 432 cm³
Conclusion
Thus, the maximum volume of the open box is 432 cm³, corresponding to option B.
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