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Answer:
The dimensions of the box are 38.48 x 14.48 x 5.26 in.
Step-by-step explanation:
We will have a piece of cardboard with squares of side x cut from the corners to make a open box.
The volume of the box can be written as:
[tex]V=A\cdot B\cdot x\\\\V=(49-2x)\cdot (25-2x)\cdot x\\\\V=x\cdot (4x^2-2x*(49+25)+49*25)\\\\V=4x^3-148x^2+1225x[/tex]
To calculate the maximum volume, we derive and equal to zero
[tex]dV/dx=12x^2-296x+1225=0[/tex]
We apply the quadratic equation to know the roots of the equation
[tex]x=\frac{-b\pm\sqrt{b^24ac}}{2a}\\\\x=\frac{296\pm\sqrt{296^2-4*12*1225}}{2*12} =\frac{296\pm\sqrt{87616-58800}}{24}= \frac{296\pm\sqrt{28816}}{24}\\\\x=\frac{296\pm169.75}{24}= 12.333\pm7.073\\\\\\x_1=12.333+7.073=19.406\\\\x_2=12.333-7.073=5.26[/tex]
The first solution is physically impossible, as the side that is cut would be bigger than 25.
So the real solution is for x=5.26 in.
Then, the dimensions of the box are:
[tex]A=49-2x=49-2*5.26=49-10.52=38.48\\\\B=25-2x=25-10.52=14.48[/tex]
The dimensions of the box of largest volume you can make with the given cardboard is;
Length = 38.48 in
Width = 14.48 in
Height = 5.26 in
We are given the dimensions of the cardboard as; 25-in. by 49-in.
Let us denote the box as follows;
Length = l
Width = w
Height = h
If we cut congruent squares from the corners, it means that if the length of each square cut is x, then the new dimensions are;
Length = 49 - 2x
Width = 25 - 2x
Height = x
Formula for volume of a box is;
V = Length × width × height
V = (49 - 2x) × (25 - 2x) × (x)
V = 4x³ - 148x² + 1225x
Now, the largest volume will happen at the dimensions when dV/dx = 0.
dV/dx = 12x² - 256x + 1225
At dV/dx = 0;
12x² - 256x + 1225 = 0
Using online quadratic equation solver, we have;
x = 5.26 or 19.406
We will take 5.26 in instead of 19.406 because 19.406 would mean that the length cut out would be longer than 25 in.
Thus, putting x = 5.26 into the dimensions expression gives;
Length = 49 - 2(5.26) = 38.48 in
Width = 25 - 2(5.26) = 14.48 in
Height = 5.26 in
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