A rectangle's width is 6 inches less than its length. If it's area is 216 square inches find the dimensions of the rectangle. Write an equation and solve
the length is 6 more than the width, so
w(w+6) = 216
Sure, let's solve it in a humorous way!
To find the dimensions of the rectangle, let's call the length "L" (I know, it sounds like "El", and no, it's not a superhero), and the width "W" (which doesn't stand for "Weirdness" or "Wackiness", unfortunately).
Now, according to the problem, the width is 6 inches less than the length. So if you give me a length of "L", I can calculate the width as "L - 6". Sound good? Great!
Since we know the area of the rectangle is 216 square inches, we can set up an equation:
L * (L - 6) = 216
Now, let's solve it in our own quirky way! Let's distribute the "L" to get:
L^2 - 6L = 216
Now, let's bring all the fun parts to one side of the equation:
L^2 - 6L - 216 = 0
Whaaat?! It's quadratic time! Let's plug it into our quadratic formula helmet and get the answer:
L = (-(-6) ± √((-6)^2 - 4(1)(-216))) / (2(1))
And after performing some hilarious calculations, we find that L ≈ 18.5843 or L ≈ -11.5843.
We know that L represents the length, so we'll ignore the negative value (let's say it escaped from the clown car).
Now, we can find the width by substituting L into the equation "W = L - 6". So, W ≈ 18.5843 - 6.
Therefore, the approximate dimensions of the rectangle are L ≈ 18.5843 inches and W ≈ 12.5843 inches.
Remember, cracks and laughter always make math more enjoyable!
Let's denote the length of the rectangle as "L" and the width as "W". We are given that the width is 6 inches less than the length, so we can write the equation:
W = L - 6
The area of a rectangle is given by the formula A = L * W, and we are given that the area is 216 square inches. Substituting the values we have:
216 = L * (L - 6)
Now, let's solve for L. Expand the equation:
216 = L^2 - 6L
Move all terms to one side of the equation:
L^2 - 6L - 216 = 0
Next, let's factor this quadratic equation. We can look for two numbers whose product is -216 and whose sum is -6. After some trial and error, we find that -18 and 12 satisfy these conditions:
(L - 18)(L + 12) = 0
Setting each factor equal to zero, we have two possibilities:
L - 18 = 0 or L + 12 = 0
Solving for L in each case:
L = 18 or L = -12
Since length cannot be negative, we discard the solution L = -12.
Therefore, the length of the rectangle is L = 18.
We can substitute this value into the equation W = L - 6 to find the width:
W = 18 - 6 = 12
So, the dimensions of the rectangle are:
Length = 18 inches
Width = 12 inches
To find the dimensions of the rectangle, we can set up an equation based on the given information.
Let's assume that the length of the rectangle is "L" inches.
According to the problem, the width of the rectangle is 6 inches less than its length. Therefore, the width can be represented as "L - 6" inches.
The formula for the area of a rectangle is given by:
Area = Length × Width
So, we can write the equation for the given problem as:
216 = L × (L - 6)
To solve this equation, we can follow these steps:
Step 1: Expand the equation:
216 = L² - 6L
Step 2: Move all the terms to one side of the equation to form a quadratic equation:
L² - 6L - 216 = 0
Step 3: Solve the quadratic equation. We can either factorize the quadratic equation or use the quadratic formula. In this case, factoring may not be straightforward, so we will use the quadratic formula:
The quadratic formula is given by:
L = (-b ± √(b² - 4ac)) / 2a
In our equation, the values are:
a = 1
b = -6
c = -216
Plugging in the values into the quadratic formula, we get:
L = (-(-6) ± √((-6)² - 4 * 1 * (-216))) / (2 * 1)
Simplifying further:
L = (6 ± √(36 + 864)) / 2
L = (6 ± √900) / 2
L = (6 ± 30) / 2
Now, we have two possible values for L:
L₁ = (6 + 30) / 2 = 36 / 2 = 18
L₂ = (6 - 30) / 2 = -24 / 2 = -12
Since the dimensions of the rectangle cannot be negative, we can disregard the negative value.
Therefore, the length of the rectangle is 18 inches.
Now, we can find the width by substituting the length into the expression "L - 6":
Width = L - 6 = 18 - 6 = 12 inches
So, the dimensions of the rectangle are:
Length = 18 inches
Width = 12 inches