The perimeter of the rectangle is 180 m. If its length is increased by 20 m and its width reduced by 10 m

The perimeter of the rectangle is 180 m. If its length is increased by 20 m and its width reduced by 10 m, then the area will increase by 100 m2. Find the length and width of the given rectangle.

Let’s write the expression for the perimeter of our original rectangle, if a is the length, b is the width:

P = 2 * (a + b) = 180.

Let’s express “a” through “b”:

a + b = 180/2;

a + b = 90;

a = 90 – b.

Let us express the area of ​​the original rectangle in terms of “b”:

S = a * b = (90 – b) * b.

Let’s write the expressions for the length, width and area of ​​the rectangle after the change:

(a + 20) – length;

(b – 10) – width;

(S + 100) – area.

Let’s compose an expression:

(a + 20) * (b – 10) = S + 100.

Substitute “a” and “S” expressed through “b”. Let’s compose and solve the equation:

((90 – b) + 20) * (b – 10) = (90 – b) * b + 100;

(110 – b) * (b – 10) = 90b – b2 + 100;

110b – b2 – 1100 + 10b = 90b – b2 + 100;

110b – b2 + 10b – 90b + b2 = 100 + 1100;

30b = 1200;

b = 40 (m).

a = 90 – 40 = 50 (m).

Answer: the length of the rectangle is 50 m, the width is 40 m.