Question

Составте систему уравнений по условию задачи прямоугольный газон обнесен изгородью длина которой 30 м площадь газона 56 м в квадрате найдите длины сторон газона

Answer 1

Длина газона - х м. ширина газона - у м
2(х + у) = 30
ху = 56
Вот такая система уравнений, т.к. периметр прямоугольника Р = 2(а + b), а площадь S = ab.

Answer 2

Problem Analysis

We are given a rectangular lawn that is surrounded by a fence. The length of the fence is 30 meters, and the area of the lawn is 56 square meters. We need to find the lengths of the sides of the lawn.

Solution

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

From the given information, we can form the following equations:

1. The perimeter of the fence is equal to the length of the fence: - 2L + 2W = 30 meters

2. The area of the lawn is equal to the product of its length and width: - L * W = 56 square meters

Now, we have a system of two equations with two variables. We can solve this system to find the lengths of the sides of the lawn.

Solving the System of Equations

We can solve the system of equations using various methods, such as substitution or elimination. Let's use the substitution method to solve this system.

From equation 1, we can express L in terms of W: - 2L = 30 - 2W - L = (30 - 2W) / 2

Substituting this value of L into equation 2, we get: - (30 - 2W) / 2 * W = 56

Simplifying the equation: - (30 - 2W) * W = 112

Expanding and rearranging the equation: - 30W - 2W^2 = 112

Rearranging the equation in standard form: - 2W^2 - 30W + 112 = 0

Now, we have a quadratic equation in terms of W. We can solve this equation to find the possible values of W. Once we have the value of W, we can substitute it back into equation 1 to find the corresponding value of L.

Solving the Quadratic Equation

We can solve the quadratic equation using various methods, such as factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to find the values of W.

The quadratic formula is given by: - W = (-b ± √(b^2 - 4ac)) / (2a)

For our equation 2W^2 - 30W + 112 = 0, the coefficients are: - a = 2 - b = -30 - c = 112

Substituting these values into the quadratic formula, we get: - W = (-(-30) ± √((-30)^2 - 4 * 2 * 112)) / (2 * 2) - W = (30 ± √(900 - 896)) / 4 - W = (30 ± √4) / 4 - W = (30 ± 2) / 4

Simplifying the equation, we get two possible values for W: - W = (30 + 2) / 4 = 8 - W = (30 - 2) / 4 = 7

Finding the Length of the Lawn

Now that we have the values of W, we can substitute them back into equation 1 to find the corresponding values of L.

For W = 8: - 2L + 2(8) = 30 - 2L + 16 = 30 - 2L = 30 - 16 - 2L = 14 - L = 14 / 2 - L = 7

For W = 7: - 2L + 2(7) = 30 - 2L + 14 = 30 - 2L = 30 - 14 - 2L = 16 - L = 16 / 2 - L = 8

Therefore, the lengths of the sides of the lawn are: - L = 7 meters and W = 8 meters - L = 8 meters and W = 7 meters

Conclusion

The lengths of the sides of the rectangular lawn can be either 7 meters by 8 meters or 8 meters by 7 meters, depending on the orientation of the lawn.