Question

Периметр прямоугольника равен 36 см, а площадь 80 см^найти стороны прямоугольника.

Answer 1

Problem Analysis

We are given the perimeter and area of a rectangle and we need to find the sides of the rectangle.

Solution

Let's assume the length of the rectangle is L and the width is W.

The perimeter of a rectangle is given by the formula: P = 2(L + W).

Given that the perimeter is 36 cm, we can write the equation as: 36 = 2(L + W).

The area of a rectangle is given by the formula: A = L * W.

Given that the area is 80 cm^2, we can write the equation as: 80 = L * W.

We have two equations with two unknowns (L and W). We can solve these equations simultaneously to find the values of L and W.

Solving the Equations

Let's solve the equations using substitution.

From the first equation, we can express L in terms of W: L = (36 - 2W) / 2.

Substituting this value of L in the second equation, we get: 80 = [(36 - 2W) / 2] * W.

Simplifying the equation, we have: 160 = 36W - 2W^2.

Rearranging the equation, we get: 2W^2 - 36W + 160 = 0.

We can solve this quadratic equation to find the values of W.

Using the quadratic formula, we have: W = (-b ± √(b^2 - 4ac)) / (2a).

Substituting the values of a, b, and c, we get: W = (-(-36) ± √((-36)^2 - 4 * 2 * 160)) / (2 * 2).

Simplifying the equation, we have: W = (36 ± √(1296 - 1280)) / 4.

Simplifying further, we have: W = (36 ± √16) / 4.

Taking the positive value, we have: W = (36 + 4) / 4 = 10.

Substituting this value of W in the first equation, we get: 36 = 2(L + 10).

Simplifying the equation, we have: L + 10 = 18.

Subtracting 10 from both sides, we get: L = 8.

Therefore, the sides of the rectangle are L = 8 cm and W = 10 cm.

Answer

The sides of the rectangle are 8 cm and 10 cm.

Verification

Let's verify our answer by calculating the perimeter and area using the given sides.

The perimeter of the rectangle is: P = 2(L + W) = 2(8 + 10) = 2(18) = 36 cm.

The area of the rectangle is: A = L * W = 8 * 10 = 80 cm^2.

Therefore, our answer is correct.

Conclusion

The sides of the rectangle are 8 cm and 10 cm.