The length of a rectangle is 1 inch greater than its width. If the dimensions are doubled, its area increases by 36 square inched. Which equation could be used to find its dimensions?
L = W + 1
Since area = L * W
2L(2W) - LW = 36
Substitute W + 1 for L in second equation to find L. Put that value in the first equation to find W. Check by putting both values into the second equation.
Let's assume the width of the rectangle is "x" inches. According to the given information, the length of the rectangle would be "x + 1" inches.
The area of a rectangle can be calculated by multiplying its length and width. So, the area of the rectangle with width "x" and length "x + 1" would be:
A = (x + 1) * x
Now, it is mentioned that when the dimensions are doubled, the area increases by 36 square inches. So, the new area would be:
New area = 2x * 2(x + 1)
To find the equation, we equate the two areas:
(x + 1) * x = 2x * 2(x + 1) - 36
Simplifying the equation further:
x^2 + x = 4x^2 + 8x - 36
Rearranging the terms:
3x^2 + 7x - 36 = 0
Therefore, the equation that could be used to find the dimensions of the rectangle is:
3x^2 + 7x - 36 = 0
To solve this problem, let's break it down step by step.
Step 1: Define the variables:
Let's say that the width of the rectangle is 'w' inches.
According to the problem, the length of the rectangle is 1 inch greater than its width. So, the length would be 'w + 1' inches.
Step 2: Calculate the area of the rectangle:
The area of a rectangle is given by the formula: Area = length × width.
So, the original area of the rectangle is: Area = (w + 1) × w = w^2 + w
Step 3: Double the dimensions and find the new area:
If the dimensions are doubled, the new width would be '2w' inches, and the new length would be '2w + 1' inches.
The new area of the rectangle is: New area = (2w + 1) × (2w) = 4w^2 + 2w
Step 4: Set up the equation:
According to the problem, the new area is 36 square inches more than the original area.
Therefore, we can set up the following equation: New area = original area + 36
(4w^2 + 2w) = (w^2 + w) + 36
Step 5: Simplify the equation:
Simplify the equation by multiplying and combining like terms:
4w^2 + 2w = w^2 + w + 36
4w^2 + 2w = w^2 + w + 36
Step 6: Solve the equation:
To solve this equation, we need to group the variables on one side and the constants on the other side:
4w^2 + 2w - w^2 - w - 36 = 0
Combine like terms:
3w^2 - 2w - 36 = 0
This is a quadratic equation. To solve it, you can either factor it or use the quadratic formula.
The factored form of the equation is:
(3w + 6)(w - 6) = 0
Setting each factor equal to zero, we get:
3w + 6 = 0 or w - 6 = 0
Solving these equations gives us:
w = -2 or w = 6
Since the width cannot be negative, we discard the solution w = -2.
Therefore, the width of the original rectangle is 6 inches.
To find the length, we can substitute the value of the width into the equation for the length:
Length = w + 1 = 6 + 1 = 7 inches
Therefore, the dimensions of the original rectangle are: Width = 6 inches and Length = 7 inches.
The equation that could be used to find its dimensions is: 3w^2 - 2w - 36 = 0.