If the length of one side of a square is increased by 4 cm and the length of an adjacent side is decreased by 8 cm, the area of the figure is decreased by 20%. Find the length of each side of the square to the nearest tenth.
(s + 4) (s - 8) = (1 - .20) s^2
s^2 - 4 s - 32 = .8 s^2 ... .2 s^2 - 4 s - 32 = 0 ... s^2 - 20 s - 160 = 0
use the quadratic formula to find s
Thanks!
To solve this problem, let's start by assigning variables to the length of one side of the square. Let's call this variable "x" cm.
According to the problem, if we increase the length of one side by 4 cm, it becomes (x + 4) cm. Similarly, if we decrease the length of an adjacent side by 8 cm, it becomes (x - 8) cm.
The original area of the square can be calculated by squaring the length of one side, so the original area is x^2 square cm.
According to the problem, the area of the figure after these changes is decreased by 20%. To calculate the new area, we multiply the original area by 0.8 (1 - 0.2) since a decrease of 20% is equivalent to 80% of the original area. Therefore, the new area is (0.8 * x^2) square cm.
Now, we can set up an equation:
(0.8 * x^2) = (x + 4)(x - 8)
Let's solve this equation:
0.8x^2 = x^2 - 4x - 8x + 32
0.8x^2 = x^2 - 12x + 32
0.8x^2 - x^2 + 12x - 32 = 0
Simplifying:
0.2x^2 + 12x - 32 = 0
To solve this quadratic equation, we can either factor it, complete the square, or use the quadratic formula. In this case, it's easier to use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation, a = 0.2, b = 12, and c = -32.
Substituting these values into the quadratic formula:
x = (-12 ± sqrt(12^2 - 4 * 0.2 * -32)) / (2 * 0.2)
x = (-12 ± sqrt(144 + 25.6)) / 0.4
x = (-12 ± sqrt(169.6)) / 0.4
x = (-12 ± 13) / 0.4
Now, we have two possible solutions:
1. x = (-12 + 13) / 0.4 = 1 / 0.4 = 2.5
2. x = (-12 - 13) / 0.4 = -25 / 0.4 = -62.5
Lengths cannot be negative, so the length of each side of the square is 2.5 cm (to the nearest tenth). Hence, the square's sides are approximately 2.5 cm each.