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Start by recognising that this is a separable differential equation; it can be written with all of the x's on one side of the equals sign, and all of the y's on the other. The first step is to rearrange s...

The function of x has two separate x terms multiplied by each other. Therefore, the appropriate way to integrate this function is by parts. The equation for integration by parts is [integral (u.dv/dx)] = ...

We can turn this into a constrained optimization problem. Let's denote x the side of the parking lot that is perpendicular to the highway, and y the side that is parallel to the highway. We therefore need...

Equation A: 2x + 5y = 8

Equation B: 4x + y = 7

2A: 4x + 10y = 16

2A - B: 9y = 9, Therefore y = 1

Sub y back into equation A to find x:  2x + 5 = 8,  2x ...

times both sides by both denominators

= (2x+3)(2x+1) - (2x-8)(x-4) = (2x+1)(x-4)

times out brackets

= (4x² + 8x +3) - (2x² -16x +32) = 2x² -7x -4