Answer:
Width of lawn = 35 ft
Dimensions of factory = length: 210 ft, width: 140 ft
Step-by-step explanation:
The total area of the lot can be calculated as:
[tex]A_{lot} = 210 * 280\\A_{lot} = 58800 ft^{2}[/tex]
Since, the area of factory should be equal to area of lawn:
[tex]A_{lot} = A_{factory} + A_{lawn}\\58800 = 2 A_{factory or lawn}\\\\A_{factory or lawn} = \frac{58800}{2}\\A_{factory or lawn} = 29400 ft^{2}[/tex]
Now, let 'x' be the width of lawn, the dimensions of factory can be written as:
[tex](210-2x)\\(280-2x)\\[/tex]
Since, area is equal to length x width:
[tex](210-2x)*(280-2x) = 29400\\Simplifying:\\210*280 - 210*2x - 2x*280 + 4x^{2} = 29400\\58800 - 420x - 560x +4x^{2} = 29400\\4x^{2} - 980x +58800 = 29400\\4x^{2} - 980x + 29400 = 0\\[/tex]
Divide whole equation by 4,
[tex]x^{2} - 245 + 7350 = 0\\[/tex]
Solving above quadratic equation, we get,
[tex]x = 210\\x = 35\\[/tex]
x = 35 seems realistic width of the lawn.
Now, finding the dimension of factory:
[tex](210-2x) = 210 - 2(35) = 140 ft\\(280-2x) = 280 - 2(35) = 210ft[/tex]
We can also reconfirm the area of factory by multiplying the above two lengths:
140 * 210 = 29400 ft
