Algebra question! ( Please show steps thank you!)

Question

A small rectangular vineyard has dimensions of 100 feet by 200 feet. Surrounding the vineyard there will be a grassy border of uniform width x.

a) Write an equation for total area A (of both the vineyard and grassy area) in terms of the uniform width (x) of the grassy area:

A(x)=

b) If the total area is 23421 square feet, determine the width of the grassy area. The width of the grassy area is
feet.

Answer 1

The area of the vineyard itself is 100ft * 200ft

a) With the border, each dimension is greater by 2x, since there is an added length of x on all sides. So one dimension is 100+2x and the other is 200+2x.
(Drawing a picture will help you visualize what is happening here)
The total area is:
A(x) = (2x+100)(2x+200)
If you multiply out the righthand side - you will end up with a quadratic equation:
A(x) = 4x^2 + 400x +200x +20000
A(x) = 4x^2 + 600x + 20000

b) Substitute 23421 for A:
23421 = 4x^2 + 600x + 20000

subtract 23421 from both sides
4x^2 + 600x -3421 = 0

solve for x:
This doesn't look like it will factor nicely..so use the quadratic equation to find the roots
Because this is a quadratic you will end up with two answers for x, but you'll find that only one makes sense (one root will be negative which can be rejected since you can't have a negative length.

Hope that helps!