Vanilla Box Company is going to make open-topped boxes out of 17 × 13-inch rectangles of cardboard by cutting squares out of the corners and folding up the sides. What is the largest volume box it can make this way? (Round your answer to one decimal place.)
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To find the largest volume box that can be made, we need to determine the dimensions of the open-topped box. Let's start by analyzing the problem step by step:
1. Start with the original rectangle: Dimension = 17 × 13 inches.
2. To create an open-topped box, squares will be cut out of each corner. Let's assume the side length of each square is x inches.
3. After cutting the squares, the dimensions of the resulting box will be (17-2x) × (13-2x) inches, considering the reduction of x inches on both sides.
4. The height of the box will be x inches, since the squares cut from the corners will be folded up to form the sides.
5. The volume of the box is calculated by multiplying the length, width, and height: V = (17-2x) × (13-2x) × x.
6. To find the largest volume, we differentiate the volume equation with respect to x and set it equal to zero:
dV/dx = 0
Now, let's calculate the largest volume step by step:
1. Start with the volume equation: V = (17-2x) × (13-2x) × x.
2. Expand and simplify the equation: V = (221x - 56x^2 + 4x^3).
3. Differentiate the equation with respect to x: dV/dx = 221 - 112x + 12x^2.
4. Set the derivative equal to zero and solve for x:
221 - 112x + 12x^2 = 0.
5. This is a quadratic equation, so let's solve it using the quadratic formula: x = (-(-112) ± √((-112)^2 - 4(12)(221)))/(2(12)).
6. Simplify the equation: x = (112 ± √(12544 - 10608))/(24).
7. Continue simplifying: x = (112 ± √1944)/24.
8. Further simplify: x = (112 ± 44√3)/24.
9. Since we're cutting squares out of the corners, the side length cannot be negative or greater than half of the original side length. Therefore, we only consider positive values and restrict the range: 0 < x < 6.5.
Now, we have the possible values for x. Plug them back into the volume equation to find the largest volume:
1. Substitute the value of x into the volume equation to find V:
- For x = 0, V = (17-2(0)) × (13-2(0)) × 0 = 0.
- For x = (112 + 44√3)/24, V = [(17-2((112 + 44√3)/24))] × [(13-2((112 + 44√3)/24))] × ((112 + 44√3)/24).
- For x = (112 - 44√3)/24, V = [(17-2((112 - 44√3)/24))] × [(13-2((112 - 44√3)/24))] × ((112 - 44√3)/24).
2. Calculate the values of V using a calculator or algebraic software like Python.
3. Compare the values of V to find the largest volume.
4. Round the answer to one decimal place, which gives the largest volume box.
Following these steps, you can find the largest volume box that can be made from the given dimensions.