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A rectangular box is to be made from a sheet of 24 inch length and 9 inch width cutting out identical squares of side length x from the four corners and turning up the sides
What is the maximum volume of the box?
- a)200 cubic inch
- b)400 cubic inch
- c)100 cubic inch
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
A rectangular box is to be made from a sheet of 24 inch length and 9 i...
length o f the box = 24 - 2x
width of the box = 9 - 2x
Volume of box = (24 - 4) (9 - 4),2
= 20 x 5 x 2 = 200 cu inch
= 20 x 5 x 2 = 200 cu inch
Most Upvoted Answer
A rectangular box is to be made from a sheet of 24 inch length and 9 i...
To find the maximum volume of the box, we need to determine the dimensions of the box that will maximize the volume.
Let's assume that each side length of the square cut out from the corners is x.
We know that the length of the rectangular sheet is 24 inches and the width is 9 inches.
So, the length of the base of the box after the squares are cut out and the sides are turned up will be (24 - 2x) inches and the width will be (9 - 2x) inches.
The height of the box will be x inches.
Therefore, the volume V of the box can be calculated as follows:
V = (24 - 2x)(9 - 2x)x
To find the maximum volume, we need to find the value of x that maximizes V.
To do this, we can take the derivative of V with respect to x and set it equal to zero.
dV/dx = 2(24 - 2x)(9 - 2x) + (24 - 2x)(-4x) + (9 - 2x)(-4x)
= 2(24 - 2x)(9 - 2x) - 8x(24 - 2x) - 8x(9 - 2x)
= 2(24 - 2x)(9 - 2x) - 8x(24 - 2x - 9 + 2x)
= 2(24 - 2x)(9 - 2x) - 8x(15)
= 2(24 - 2x)(9 - 2x) - 120x
Setting this expression equal to zero and solving for x will give us the value of x that maximizes V.
2(24 - 2x)(9 - 2x) - 120x = 0
Expanding and simplifying this equation, we get:
4x^2 - 66x + 216 = 0
Solving this quadratic equation using the quadratic formula, we find that x = 6 or x = 9/2.
Since we are cutting out squares from the corners, the side length x cannot be greater than half of the width or length of the sheet. Therefore, x = 6 is the valid solution.
Substituting this value of x into the volume equation, we get:
V = (24 - 2(6))(9 - 2(6))(6)
= (24 - 12)(9 - 12)(6)
= (12)(-3)(6)
= -216
Since volume cannot be negative, we discard this solution.
Therefore, the maximum volume of the box is obtained when x = 6.
Substituting this value into the volume equation, we get:
V = (24 - 2(6))(9 - 2(6))(6)
= (24 - 12)(9 - 12)(6)
= (12)(-3)(6)
= 216
Therefore, the maximum volume of the box is 216 cubic inches, which corresponds to option A.
Let's assume that each side length of the square cut out from the corners is x.
We know that the length of the rectangular sheet is 24 inches and the width is 9 inches.
So, the length of the base of the box after the squares are cut out and the sides are turned up will be (24 - 2x) inches and the width will be (9 - 2x) inches.
The height of the box will be x inches.
Therefore, the volume V of the box can be calculated as follows:
V = (24 - 2x)(9 - 2x)x
To find the maximum volume, we need to find the value of x that maximizes V.
To do this, we can take the derivative of V with respect to x and set it equal to zero.
dV/dx = 2(24 - 2x)(9 - 2x) + (24 - 2x)(-4x) + (9 - 2x)(-4x)
= 2(24 - 2x)(9 - 2x) - 8x(24 - 2x) - 8x(9 - 2x)
= 2(24 - 2x)(9 - 2x) - 8x(24 - 2x - 9 + 2x)
= 2(24 - 2x)(9 - 2x) - 8x(15)
= 2(24 - 2x)(9 - 2x) - 120x
Setting this expression equal to zero and solving for x will give us the value of x that maximizes V.
2(24 - 2x)(9 - 2x) - 120x = 0
Expanding and simplifying this equation, we get:
4x^2 - 66x + 216 = 0
Solving this quadratic equation using the quadratic formula, we find that x = 6 or x = 9/2.
Since we are cutting out squares from the corners, the side length x cannot be greater than half of the width or length of the sheet. Therefore, x = 6 is the valid solution.
Substituting this value of x into the volume equation, we get:
V = (24 - 2(6))(9 - 2(6))(6)
= (24 - 12)(9 - 12)(6)
= (12)(-3)(6)
= -216
Since volume cannot be negative, we discard this solution.
Therefore, the maximum volume of the box is obtained when x = 6.
Substituting this value into the volume equation, we get:
V = (24 - 2(6))(9 - 2(6))(6)
= (24 - 12)(9 - 12)(6)
= (12)(-3)(6)
= 216
Therefore, the maximum volume of the box is 216 cubic inches, which corresponds to option A.
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A rectangular box is to be made from a sheet of 24 inch length and 9 inch width cutting out identical squares of side length x from the four corners and turning up the sidesWhat is the maximum volume of the box?a)200 cubic inchb)400 cubic inchc)100 cubic inchd)None of theseCorrect answer is option 'A'. Can you explain this answer?
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A rectangular box is to be made from a sheet of 24 inch length and 9 inch width cutting out identical squares of side length x from the four corners and turning up the sidesWhat is the maximum volume of the box?a)200 cubic inchb)400 cubic inchc)100 cubic inchd)None of theseCorrect answer is option 'A'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about A rectangular box is to be made from a sheet of 24 inch length and 9 inch width cutting out identical squares of side length x from the four corners and turning up the sidesWhat is the maximum volume of the box?a)200 cubic inchb)400 cubic inchc)100 cubic inchd)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A rectangular box is to be made from a sheet of 24 inch length and 9 inch width cutting out identical squares of side length x from the four corners and turning up the sidesWhat is the maximum volume of the box?a)200 cubic inchb)400 cubic inchc)100 cubic inchd)None of theseCorrect answer is option 'A'. Can you explain this answer?.
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