a rectangle has a width of 4 cm less than it's length. if a new rectangle is formed by increasing the width 5 cm and decreasing the length 3 cm, the area of this resulting rectangle is 177 cm squared. what are the dimensions of the original rectangle?
let x = length of the original rectangle
let x-4 = width of the original rectangle
using the conditions given by the 2nd statement,
new width: (x-4) + 5
new length: x - 3
using the next condition given in the problem, and recalling that the area of a rectangle is just length*width:
(x-3)[(x-4) + 5] = 177
(x-3)(x+1) - 177 = 0
expanding this using FOIL:
x^2 - 2x - 3 - 177 = 0
x^2 - 2x - 180 = 0
we see that this one's not factorable,, so you use quadratic formula~
hope this helps :)
Why did the rectangle go on a diet? Because it wanted to decrease its width! Now let's solve this puzzling problem together.
Let's assume the length of the original rectangle is "L" cm. According to the problem, the width is 4 cm less than the length, so the width would be (L - 4) cm.
Now, if we create a new rectangle by adding 5 cm to the width and subtracting 3 cm from the length, the width of the new rectangle would be (L - 4 + 5) cm, and the length would be (L - 3) cm.
The area of a rectangle is calculated by multiplying the length by the width, so we have:
(L - 4)(L - 3) = 177
Expanding this equation:
L^2 - 7L + 12 = 177
Bringing all terms to one side:
L^2 - 7L - 165 = 0
Now, let's solve this quadratic equation using factoring or the quadratic formula. However, I must warn you, math problems involving clowns tend to be a bit tricky!
Let's solve this problem step by step:
Step 1: Assign variables
Let's assign variables to the dimensions of the original rectangle. Let's call the length "L" and the width "W".
Step 2: Express the given information in terms of variables
According to the problem, the width of the original rectangle is 4 cm less than its length. This can be expressed as: W = L - 4.
Step 3: Express the area of the original rectangle in terms of variables
The formula for the area of a rectangle is A = L * W. Substituting W = L - 4, we can express the area as: A = L * (L - 4).
Step 4: Set up the equation based on the given area
According to the problem, the resulting rectangle has an area of 177 cm^2. Therefore, we can set up the equation: 177 = L * (L - 4).
Step 5: Solve the equation
To solve the equation, we can simplify it and bring it to the standard quadratic form. Let's multiply out the brackets: 177 = L^2 - 4L.
Step 6: Rearrange the equation
To solve a quadratic equation, we need it to be in the form of "ax^2 + bx + c = 0". Let's rearrange the equation: L^2 - 4L - 177 = 0.
Step 7: Factor or use the quadratic formula to solve the equation
To solve the quadratic equation, we can factor it or use the quadratic formula. In this case, factoring is not apparent, so let's use the quadratic formula: L = (-b ± √(b^2 - 4ac)) / 2a.
Plugging in the values, we get: L = (-(-4) ± √((-4)^2 - 4(1)(-177))) / 2(1).
Simplifying further, we have: L = (4 ± √(16 + 708)) / 2.
Simplifying the square root, we get: L = (4 ± √724) / 2.
Step 8: Solve for L
To find the possible values of L, we can evaluate the expression with both the positive and negative square roots.
L1 = (4 + √724) / 2 ≈ 17.54 cm.
L2 = (4 - √724) / 2 ≈ -13.54 cm.
Since a length cannot be negative, we disregard L2.
Step 9: Solve for W
Now that we have the value for L, we can substitute it back into the expression W = L - 4.
Substituting L1 into the equation, we get: W = 17.54 - 4 = 13.54 cm.
So, the dimensions of the original rectangle are approximately:
Length = 17.54 cm
Width = 13.54 cm
To solve this problem, we can set up two equations based on the given information.
Let's assume the length of the original rectangle is "x" cm. According to the problem, the width of the rectangle is 4 cm less than its length, meaning the width is (x - 4) cm.
Now, let's form the new rectangle by increasing the width by 5 cm and decreasing the length by 3 cm. The width of the new rectangle would be (x - 4 + 5) = (x + 1) cm, and the length would be (x - 3) cm.
Using these dimensions, we can find the area of the new rectangle, which is given as 177 cm squared:
Area of the new rectangle = Length × Width
177 = (x - 3) × (x + 1)
Expanding the equation:
177 = x^2 - 2x - 3
Rearranging the terms:
x^2 - 2x - 3 - 177 = 0
Simplifying:
x^2 - 2x - 180 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula.
By factoring, we can rewrite the equation as:
(x - 12)(x + 15) = 0
Setting each factor equal to zero:
x - 12 = 0 or x + 15 = 0
Solving for x, we get two possible solutions:
x = 12 or x = -15
Since the length cannot be negative, we discard the solution x = -15.
Therefore, the original length is x = 12 cm.
Using this value, we can find the width of the original rectangle:
Width = x - 4
Width = 12 - 4
Width = 8 cm
So, the dimensions of the original rectangle are length = 12 cm and width = 8 cm.