SOLUTION: Solve using the 5 step problem-solving process. You are pulling a stone border of uniform width around a rectangular garden that measures 6 yards by 15 yards. You only have enough

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Question 250919: Solve using the 5 step problem-solving process. You are pulling a stone border of uniform width around a rectangular garden that measures 6 yards by 15 yards. You only have enough stone to cover 100 square yards. How wide should the border be?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the measurement of your yard is:

length 15 yards.
width = 6 years.

the area of your yard is 15 * 6 = 90 square yards.

you want to put a stone border around this yard.

you have 100 square yards of stone available to you.

once you ring your yard with the stone border, the total area will be:

90 square yards + 100 square yards = 190 square yards.

the border will be the same width all around, so we let the width of the border be equal to x.

this means that the length of the garden will be increased by 2*x, and the width of the garden will be increased by 2*x (x added to each end equals a total of 2*x).

our equation for the area of the garden plus the border becomes:

(2x + 6) * (2x + 15) = 190

we multiply these factors together to get:

4x^2 + 30x + 12x + 90 = 190

we subtract 190 from both sides of this equation and we simplify by combining like terms to get:

4x^2 + 42x - 100 = 0

we use the quadratic formula to solve this equation to get:

x = 2 or x = -12.5

x has to be positive, so we take x = 2 and substitute in the original equation to confirm that the equation is true meaning that the value of x is good.

original equation is:

(2x + 6) * (2x + 15) = 190

replace x with 2 to get:

(2*2 + 6) * (2*2 + 15) = 190

simplify to get:

10 * 19 = 190

since this equation is true, the value of x = 2 is good.

the answer to the problem is:

the width of our border will be 2 yards.

USE OF QUADRATIC FORMULA

the quadratic formula is:

x+=+%28%28-b%29+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282%2Aa%29

the standard form of a quadratic equation is:

ax^2 + bx + c = 0

your formula was:

4x^2 + 42x - 100 = 0

this meant that:

a = 4
b = 42
c = -100

using these values, the quadratic formula became:

x+=+%28%28-42%29+%2B-+sqrt%2842%5E2-%284%2A4%2A%28-100%29%29%29%29%2F%282%2A4%29

simplifying made the equation look like:

x+=+%28%28-42%29+%2B-+sqrt%281764+%2B+1600%29%29%2F%282%2A4%29

simplifying further made the equation look like:

x+=+%28%28-42%29+%2B-+sqrt%283364%29%29%2F%288%29

simplifying further made the equation look like:

x+=+%28%28-42%29+%2B-+58%29%2F%288%29

the rest was just finishing up work by performing the indicated operations.

FIVE STEP PROCESS

We followed the 5 step process even though you may not have realized it.

The 5 step process is as follows:

1. REad the problem
2. Choose a variable to represent one quantity. Use that variable to represent every quantity in the problem.
3. Write an equation relating the quantities in the problem.
4. Solve your equation and answer the question asked.
5. Check your answer.

We read the problem. (step 1)
We chose the variable x to represent the width of the border (step 2)
We wrote an equation relating the quantities in the problem. That was the equation of (2x+6) * (2x+15) = 190 (step 3)
WE solved the equation and answered the question asked (step 4).
We checked our answer to confirm that it was good (step 5).