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The dimensions of the box are 10 ft and 5 ft
The maximum volume is 500 ft³
Step-by-step explanation:
A rectangle box with
- A square base and no top
- It needs to be made using 300 ft² of material
- It has greatest volume
Surface area of a box without top (SA) = perimeter of base × height + area of the base
Volume of a box (V) = base area × height
Assume that the length of the side of the square base is x and the height of the box is y
∵ It needs to be made using 300 ft² of material
∴ The surface area of the box is 300 ft²
∵ Its base is a square of side length x ft
∴ Perimeter of the base = 4 × x = 4 x
∴ Area of the base = x²
∵ The height of the box = y ft
∵ SA = perimeter of base × height + area of the base
∵ SA = (4x)(y) + x²
∴ SA = 4xy + x²
∵ SA of the box = 300 ft²
- Equate the two expressions of SA
∴ 4xy + x² = 300
Now let us find y in terms of x
- Subtract x² from both sides
∴ 4xy = 300 - x²
- Divide each term by 4x to find y
∴
∵ V = area of the base × height
∴ V = x² × y = x²y
- Substitute y by the equation of it above
∴
∴
∵ The volume of the box is greatest
- That means differentiate V and equate it by 0
∵
∵ ⇒ greatest volume
∴
- Subtract 75 from both sides
∴
- Divide both sides by
∴ x² = 100
- Take √ for both sides
∴ x = 10
Substitute the value of x in the equation of y
∵
∴ y = 5
The dimensions of the box are 10 ft and 5 ft
∵
∵ x = 10
∴
∴
∴ V = 750 - 250
∴ V = 500 ft³
The maximum volume is 500 ft³
Learn more:
You can learn more about the volume in brainly.com/question/6443737
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