SOLUTION: Alec has written an award winning short story. His mother wants to frame it with a uniform border. She wants the finished product to have an area of 315 in^2. The writing portion o
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Question 1149545: Alec has written an award winning short story. His mother wants to frame it with a uniform border. She wants the finished product to have an area of 315 in^2. The writing portion occupies an area that is 11 inches wide and 17 inches long. How wide is the border? Found 2 solutions by greenestamps, ikleyn:Answer by greenestamps(13200) (Show Source):
You should know how to set up a problem like this for solving using formal algebra, so let's do that.
Let the width of the border by x. Then the dimensions of the finished product are 11+2x and 17+2x. So we need to find the value of x for which (11+2x) times (17+2x) is equal to 315.
To solve from here, we need to factor that quadratic, by finding two numbers whose product is 32 and whose difference is 14.
or
Clearly the negative answer makes no sense in the problem, so we choose the positive solution.
ANSWER: The width of the border is x=2 inches.
So with the algebraic solution, we end up having to find two numbers with a product of 32 and a difference of 14.
But the original problem asks us to find two numbers with a product of 315 and a difference of 17-11 = 6.
So the formal algebraic solution doesn't simplify the task of finding the answer to the problem; it leads us to the same kind of task as the original problem.
So if a formal algebraic solution is not required, we might as well solve the original problem directly.
The prime factorization of 315 is 3*3*5*7. We need to combine those factors into two numbers with a difference of 6. A little trial and error leads us to 15*21; since the original dimensions were 11 and 17, twice the width of the border is 4 inches, so the width of the border is 2 inches.
The dimensions of the writing portion are 11 inches by 17 inches.
If the width of the uniform border is x inches, then the outer dimensions of the finished product are (11+2x) by (17+2x) inches.
The equation for the area of the finished product is
(11+2x)*(17+2x) = 315 square inches.
Simplify and solve for x
11*17 + 56x + 4x^2 = 315
4x^2 + 56x - 128 = 0
x^2 + 14x - 32 = 0
(x+16)*(x-2) = 0
Of the two roots, x= -16 and x= 2, only positive root x= 2 makes sense.
ANSWER. The uniform width of the border is 2 inches.
