A factory is to be built on a lot measuring 210 ft by 280 ft. A local building code specifies that a lawn of uniform width and equal in area to the factory must surround the factory.

a) What must the width of the lawn be?
b) And If the dimensions of the factory are A ft by B ft with A<B, then A=? B=?

Are you going to wait the four hours until the quiz expires or do you have a plan B?

I'm trying to solve them but I kept getting them wrong, so my plan B is to get the answer from somewhere else, because I'm struggling whit this quiz.

let the width be x ft

dimensions of the factory are:
210-2x by 280-2x

So what is the area of the factory ?
set that equal to 1/2 the area of the lot and solve that quadratic

To find the width of the lawn surrounding the factory, we need to first calculate the area of the factory. The area of the factory lot can be found by multiplying its length by its width. In this case, the length is 210 ft and the width is 280 ft, so the area of the factory is 210 ft × 280 ft = 58,800 sq ft.

Since the lawn surrounding the factory must have the same area as the factory itself, we can divide the factory's area by 2 to find the area of the lawn. Therefore, the area of the lawn is 58,800 sq ft ÷ 2 = 29,400 sq ft.

To find the width of the lawn, we need to determine the dimensions of the rectangular lawn. Let's assume the width of the lawn is x ft. Since the length and width of the factory will decrease by 2x (x ft on each side) when the lawn is added, the dimensions of the factory will be (210 ft - 2x) by (280 ft - 2x).

Using the formula for the area of a rectangle (length × width), we can set up the equation:

(210 ft - 2x) × (280 ft - 2x) = 29,400 sq ft

Expanding the equation:

(210 ft × 280 ft) - (420 ft × x) - (560 ft × x) + (4x^2) = 29,400 sq ft

58,800 sq ft - 980x - 1120x + 4x^2 = 29,400 sq ft

Rearranging the equation and simplifying:

4x^2 - 2100x + 29,400 = 0

To solve this quadratic equation, we can either use factoring, completing the square, or the quadratic formula. Let's solve it using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 4, b = -2100, and c = 29,400. Substituting these values into the quadratic formula:

x = (-(-2100) ± √((-2100)^2 - 4(4)(29,400))) / (2(4))

Simplifying:

x = (2100 ± √(4,410,000 - 470,400)) / 8

x = (2100 ± √3,939,600) / 8

Now, we can calculate the two possible values for x:

x₁ = (2100 + √3,939,600) / 8

x₂ = (2100 - √3,939,600) / 8

Since the width of the lawn cannot be negative, we discard the negative value (x₂) and only consider the positive value (x₁). Hence, the width of the lawn is:

a) Width of the lawn = x₁ = (2100 + √3,939,600) / 8

For part b, if the dimensions of the factory are A ft by B ft, with A < B, then A = 210 ft - 2x₁ and B = 280 ft - 2x₁.

b) A = 210 ft - 2x₁
B = 280 ft - 2x₁

You can substitute the value of x₁ that we calculated in part a to find the specific values of A and B.