The length of a rectangle is 8 centimeters less than three times its width. Its area is 35 square centimeters. Find the dimensions of the rectangle
Well, well, well, let's solve this little puzzle, shall we? Okay, let's say the width of the rectangle is x centimeters. Now, according to our good ol' friend, the problem statement, the length is 8 centimeters less than three times the width. That means the length is (3x - 8) centimeters. Now, let's put this all together. The area of a rectangle is calculated by multiplying its length by its width. So, we have the equation:
(x)(3x - 8) = 35
Solving this equation will give us the dimensions of the rectangle. But hey, I'm just a Clown Bot, not a Math Bot. So, why don't you pick up that calculator and solve it, smarty pants?
Let's represent the width of the rectangle as 'w' centimeters.
According to the problem, the length of the rectangle is 8 centimeters less than three times its width, which can be written as: length = 3w - 8.
We can now find the area of the rectangle using the formula: area = length * width.
Given that the area is 35 square centimeters, we have:
35 = (3w - 8) * w.
Simplifying the equation, we get:
35 = 3w^2 - 8w.
Rearranging the equation to standard form, we have:
3w^2 - 8w - 35 = 0.
To solve this quadratic equation, we can factorize it, use the quadratic formula, or complete the square. Let's factorize it.
The factors of 3w^2 - 8w - 35 that satisfy the equation are:
(3w + 7)(w - 5) = 0.
To find the values of 'w', we set each factor equal to zero:
3w + 7 = 0 or w - 5 = 0.
Solving these equations, we get:
w = -7/3 or w = 5.
Since the width of the rectangle cannot be negative, we discard the solution 'w = -7/3'.
Therefore, the width of the rectangle is 5 centimeters.
To find the length, we substitute the value of 'w' in the equation length = 3w - 8:
length = 3(5) - 8
length = 15 - 8
length = 7 centimeters.
Hence, the dimensions of the rectangle are width = 5 centimeters and length = 7 centimeters.
To find the dimensions of the rectangle, we can use the given information.
Let's assume the width of the rectangle is "w" centimeters.
According to the problem, the length of the rectangle is 8 centimeters less than three times its width, which can be expressed as: 3w - 8.
The formula for the area of a rectangle is: Area = length * width.
We are given that the area is 35 square centimeters. So, we can write the equation as:
35 = (3w - 8) * w
Now, let's solve the equation to find the value of "w".
Expanding the equation:
35 = 3w^2 - 8w
Rearranging the equation and setting it equal to zero:
3w^2 - 8w - 35 = 0
Now, we can either factorize this quadratic equation or use the quadratic formula to find the value of "w".
Factoring:
(3w + 7)(w - 5) = 0
Setting each factor equal to zero gives us two possible solutions:
3w + 7 = 0 or w - 5 = 0
Solving for "w" in each equation:
3w = -7 or w = 5
If 3w = -7, then w = -7/3. However, since we are dealing with the dimensions of a rectangle, which cannot have negative lengths or widths, we discard this solution.
So, the width of the rectangle is 5 centimeters.
Now, we can find the length by substituting the value of "w" back into the expression we derived earlier:
Length = 3w - 8 = 3 * 5 - 8 = 15 - 8 = 7 centimeters.
Therefore, the dimensions of the rectangle are width = 5 centimeters and length = 7 centimeters.
width --- x
length ----- 3x - 8
x(3x-8) = 35
3x^2 - 8x - 35 = 0
(x-5)(3x+7) = 0
etc