Please help me solve this and show me how you got the answer? Thank you.

Question

Judy has a rectangular garden 9 by 22 feet. She wants to put a grass border around the garden with a uniform width on all sides. if she has enough grass seed to cover 312 square feet, how wide can the grass border be?

Answer 1

I call this a doughnut hole problem. First find the size of the doughnut - the garden with the border - and then subtract the hole - the garden.

 

b = width of the border

The dimensions of the garden with the border is 9 + 2b by 22 + 2b  (2b because the border is on both sides)

 

(9 + 2b)(22 + 2b) 

198 + 44b + 18b + 4b²

198 + 62b + 4b²

 

Now subtract the garden that is 9(22) or 198.

 

62b + 4b² = 312   The area of the border must equal the amount of seed

 

4b² + 62b - 312 = 0

(b - 4 )(4b + 78) = 312

 

b ∈{4, -78}

Length cannot be negative

 

The width of the border is 4 feet.