The length of a rectangle is 11 cm more than its width. The area of the rectangle is 2040 cm^2.
Find the dimensions of the rectangle.
width --- x
lengyth -- x+11
x(x+11) = 2040
x^2 + 11x - 2040 = 0
(x-4)(x+51) = 0
x = 40 or x = - 51 , negative width makes no sense,
so x = 40
the rectangle is 40 cm by 51 cm
Step 1: Let's assume the width of the rectangle as "x" cm.
Step 2: According to the given information, the length of the rectangle is 11 cm more than its width. So, the length can be expressed as "x + 11" cm.
Step 3: The formula to calculate the area of a rectangle is:
Area = Length * Width
Step 4: Substituting the respective values in the formula, we get:
2040 = (x + 11) * x
Step 5: Expand the equation:
2040 = x^2 + 11x
Step 6: Rearrange the equation into a quadratic form:
x^2 + 11x - 2040 = 0
Step 7: Now we can solve this quadratic equation for x. We can either factorize it or use the quadratic formula.
Step 8: Factoring this equation, we find:
(x + 51)(x - 40) = 0
Step 9: Setting each factor to zero and solving for x:
x + 51 = 0 OR x - 40 = 0
Step 10: Solving for x in each equation, we get:
x = -51 OR x = 40
Step 11: Since we cannot have a negative length or width, we discard the value x = -51.
Step 12: Therefore, the width of the rectangle is x = 40 cm.
Step 13: Using this width, we can find the length of the rectangle:
Length = width + 11 = 40 + 11 = 51 cm
Step 14: So, the dimensions of the rectangle are:
Width = 40 cm, Length = 51 cm
To find the dimensions of the rectangle, let's first set up a system of equations using the given information.
Let's assume the width of the rectangle is "w" cm.
According to the problem, the length of the rectangle is 11 cm more than its width, so the length can be represented as "w + 11" cm.
The area of a rectangle is given by the formula: Area = Length * Width.
We are given that the area is 2040 cm^2, so we can set up our equation as:
w * (w + 11) = 2040
Now, we can solve this quadratic equation to find the value of "w".
w^2 + 11w - 2040 = 0
To solve this equation, we can either factorize it or use the quadratic formula. Let's use the quadratic formula:
The quadratic formula states that for an equation of the form "ax^2 + bx + c = 0", the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = 11, and c = -2040. Substituting these values into the quadratic formula:
w = (-11 ± √(11^2 - 4 * 1 * -2040)) / (2 * 1)
Now, let's solve for "w":
w = (-11 ± √(121 + 8160)) / 2
w = (-11 ± √8281) / 2
Since we are looking for the dimensions of the rectangle, we can discard the negative value. Taking the positive value:
w = (-11 + √8281) / 2
w = (-11 + 91) / 2
w = 80 / 2
w = 40
So, the width of the rectangle is 40 cm.
To find the length, we can substitute this value back into the expression "w + 11":
Length = 40 + 11
Length = 51
So, the dimensions of the rectangle are: width = 40 cm and length = 51 cm.