A rectangle was 25 cm longer than it was wide. A new rectangle was formed by decreasing the length by 6 cm and decreasing the width by 5 cm. The area of the new rectangle was 585 cm squared. Find the dimensions of the original rectangle.
original:
width --- x
length -- x+25
new one:
width --- x-5
length --- x+25-6 = x+19
(x-5)(x+19) = 585
x^2 + 14x - 95 = 585
x^2 + 14x - 680 = 0
(x-20)(x+34) = 0
x = 20 or x is a negative, which would be bogus
original
width is 20 and length is 45
To solve this problem, we can set up equations based on the given information.
Let's denote the width of the original rectangle as "w" cm. Since the length is 25 cm longer than the width, the length can be represented as "w + 25" cm.
The area of a rectangle is calculated by multiplying its length by its width. So, the area of the original rectangle is:
Area = length * width
Area = (w + 25) * w
Area = w^2 + 25w
Now, let's consider the new rectangle. The length of the new rectangle is obtained by decreasing the length of the original rectangle by 6 cm, which gives us "w + 25 - 6 = w + 19" cm. Similarly, the width of the new rectangle is "w - 5" cm.
We are given that the area of the new rectangle is 585 cm^2. Therefore, we have the equation:
Area = length * width
585 = (w + 19) * (w - 5)
585 = w^2 + 19w - 5w - 95
585 = w^2 + 14w - 95
Rearranging the equation gives us:
w^2 + 14w - 95 - 585 = 0
w^2 + 14w - 680 = 0
To solve this quadratic equation, we can either factorize it or use the quadratic formula.
Since the coefficients of w^2, w, and the constant term are not easily factorable, let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 14, and c = -680. Substituting these values into the formula, we get:
w = (-14 ± √(14^2 - 4 * 1 * -680)) / (2 * 1)
w = (-14 ± √(196 + 2720)) / 2
w = (-14 ± √2916) / 2
w = (-14 ± 54) / 2
Now, we have two possible solutions:
1. w = (-14 + 54) / 2 = 40 / 2 = 20
2. w = (-14 - 54) / 2 = -68 / 2 = -34
Since the width of a rectangle cannot be negative, the second solution is not valid.
Therefore, the width of the original rectangle is 20 cm.
We can now find the length of the original rectangle by adding 25 cm to the width:
Length = width + 25 = 20 + 25 = 45 cm
Thus, the dimensions of the original rectangle are 20 cm (width) and 45 cm (length).