SOLUTION: The length of a rectangle is 3 inches longer than it is wide. If the area is 70 square inches, what are the dimensions of the rectangle?

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Question 1060021: The length of a rectangle is 3 inches longer than it is wide. If the area is 70 square inches, what are the dimensions of the rectangle?
Answer by acw1213(28) About Me  (Show Source):
You can put this solution on YOUR website!
We need to write a system of equations here.

One equation will include the formula for the area/
The formula for the area of a rectangle is
Lw = a
where
L = length
W = width
a = area
Plug in the area where it is necessary.
Lw = 70
One equation is finished!
The next equation will be the situation described in your response.
The length is 3 inches added to the width.
In an equation, it is
3 + w = L
Now use your systems,
3 + w = L
Lw = 70
to solve for the length and width.



I'm going to solve using substitution.
Substitute 3 + w in for "L" in the second equation.

(3 + w)(w)=70
Distribute.
3w+%2B+w%5E2+=+70
Now we have a quadratic equation on our hands. Let's get this into standard form.

Subtract 70 from both sides.

3w+%2B+w%5E2+-+70+=+0
We cannot factor this, so solve using the quadratic formula.
w+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
a = 3
b = 1
c = -70

Plug in and solve for w.

w+=+%28-1+%2B-+sqrt%28+1%5E2-4%2A3%2A-70+%29%29%2F%282%2A3%29+
Simplify. Please view my steps.

w+=+%28-1+%2B-+sqrt%28+1%2B840+%29%29%2F%286%29+
I squared 1 and multiplied -4 by 3, and then by -70.
I multiplied 2 times 3.
w+=+%28-1+%2B-+sqrt%28841%29%29%2F%286%29+
I added one to 840.
w+=+%28-1+%2B-+29%29%29%2F%286%29+
I found the square root of 841, which is 29.

Now, we need to solve for "w". -1 can be ADDED or SUBTRACTED from 29, so we need to solve for both

Addition:
w+=+%28-1+%2B+29%29%29%2F%286%29+
-1 + 29 is 28.
28 / 6 can be simplified to 14/3.
w = 14/3


Subtraction:
w+=+%28-1+-+29%29%29%2F%286%29+
-1 - 29 is -30.
-30 / 6 is -5.
w = -5

Now we have a problem. w can equal -5 or 14/3.
-5 is the more logical answer since 14/3 will turn out to be a repeating decimal.
Our width is -5.

Don't worry, our system is not old news yet!
Let's return to it.
Lw = 70
w + 3 = L
Let's plug in -5 for "w" in the second equation and solve for "L".
-5 + 3 = L
-5 + 3 is -2.
-2 = L
YAY! We are finished!
Our width is -5 and our length is -2. These are our DIMENSIONS.