a rectangle is 4 times as long as it is wide. a second rectangle is 5 centimeters longer and 2 centimeters wider than the first. the area of the second rectangle is 530 square centimeters greater than the first. what are the dimensions of the original rectangle?

L = 4 B

L2=L + 5
B2=B + 2

(L+5)(B+2) = LB +530
L B + 5B + 2 L + 10 = L B + 530

5 B + 2 L = 520
but L = 4 B
so
5 B + 8 B = 520
13 B = 530
B = 40.8
L = 163

LB = 6648
(L+5)(B+2) = 7174
difference = 525.6 pretty close

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To solve this problem, let's break it down into steps:

Step 1: Define the variables
Let's assume the width of the original rectangle is "w" centimeters.
Since the length is 4 times the width, the length of the original rectangle is "4w" centimeters.

Step 2: Calculate the area of the original rectangle
The area of a rectangle is given by the formula: Area = Length × Width.
So, the area of the original rectangle is: Original_area = (4w) × w = 4w^2 square centimeters.

Step 3: Calculate the dimensions of the second rectangle
According to the problem, the second rectangle is 5 centimeters longer and 2 centimeters wider than the first.
So, the width of the second rectangle is "w + 2" centimeters, and the length is "4w + 5" centimeters.

Step 4: Calculate the area of the second rectangle
Using the same formula as in Step 2, the area of the second rectangle is: Second_area = (4w + 5) × (w + 2) = 4w^2 + 13w + 10 square centimeters.

Step 5: Set up and solve the equation
According to the problem, the area of the second rectangle is 530 square centimeters greater than the area of the first rectangle.
So, we can set up the equation: Second_area = Original_area + 530
Substituting the area formulas from Steps 2 and 4 into the equation:
4w^2 + 13w + 10 = 4w^2 + 530.

Step 6: Simplify and solve the equation
By canceling out the common terms on both sides of the equation, we get:
13w + 10 = 530

Next, subtracting 10 from both sides, we have:
13w = 520

Finally, dividing both sides by 13 gives us:
w = 40

Step 7: Find the dimensions of the original rectangle
Since the width of the original rectangle is "w = 40" centimeters, the length is "4w = 160" centimeters.

Therefore, the original rectangle has dimensions of 40 centimeters by 160 centimeters.