Answer and step-by-step explanation:
For any rectangle, the one with the largest area will be the one whose dimensions are as close to a square as possible.
However, the dividers change the process to find this maximum somewhat.
Letting x represent two sides of the rectangle and the 3 parallel dividers, we have 2x+3x = 5x.
Letting y represent the other two sides of the rectangle, we have 2y.
We know that 2y + 5x = 750.
Solving for y, we first subtract 5x from each side:
2y + 5x - 5x = 750 - 5x
2y = -5x + 750
Next we divide both sides by 2:
2y/2 = -5x/2 + 750/2
y = -2.5x + 375
We know that the area of a rectangle is given by
A = lw, where l is the length and w is the width. In this rectangle, one dimension is x and the other is y, making the area
A = xy
Substituting the expression for y we just found above, we have
A = x(-2.5x+375)
A = -2.5x² + 375x
This is a quadratic equation, with values a = -2.5, b = 375 and c = 0.
To find the maximum, we will find the vertex. First we find the axis of symmetry, using the equation
x = -b/2a
x = -375/2(-2.5) = -375/-5 = 75
Substituting this back in place of every x in our area equation, we have
A = -2.5x² + 375x
A = -2.5(75)² + 375(75) = -2.5(5625) + 28125 = -14062.5 + 28125 = 14062.5
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