Question

Детская площадка имеет форму прямоугольника, площадь которого равна 210 м2. Одна его сторона на 1 метр(-ов, -а) больше, чем другая. Детской площадке необходимо построить бордюр. В магазине продаётся материал для бордюра в упаковках. В одной упаковке имеется 5 метров(-а) материала.

Answer 1

Problem Analysis

We are given a rectangular children's playground with an area of 210 m². One side of the playground is 1 meter longer than the other side. We need to calculate the dimensions of the playground and determine how many packages of border material, where each package contains 5 meters, are needed.

Solution

Let's assume the shorter side of the rectangular playground is x meters. Then the longer side will be x + 1 meters.

The area of a rectangle is given by the formula: Area = length × width.

In this case, the area of the playground is 210 m², so we can set up the following equation:

x × (x + 1) = 210

To solve this equation, we can expand it and rearrange it into a quadratic equation:

x² + x - 210 = 0

Now we can solve this quadratic equation to find the value of x.

Solving the Quadratic Equation

We can solve the quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula, which states that for an equation of the form ax² + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b² - 4ac)) / (2a)

In our case, a = 1, b = 1, and c = -210. Substituting these values into the quadratic formula, we get:

x = (-1 ± √(1² - 4(1)(-210))) / (2(1))

Simplifying further:

x = (-1 ± √(1 + 840)) / 2

x = (-1 ± √841) / 2

x = (-1 ± 29) / 2

So we have two possible solutions for x: x = (-1 + 29) / 2 = 14 and x = (-1 - 29) / 2 = -15.

Since the length cannot be negative, we discard the solution x = -15.

Therefore, the shorter side of the rectangular playground is 14 meters, and the longer side is 14 + 1 = 15 meters.

Calculating the Number of Border Material Packages

Now that we know the dimensions of the playground, we can calculate the perimeter of the playground, which is the sum of all four sides of the rectangle.

The perimeter of a rectangle is given by the formula: Perimeter = 2 × (length + width).

In this case, the perimeter of the playground is:

Perimeter = 2 × (14 + 15) = 2 × 29 = 58 meters

Since each package of border material contains 5 meters, we divide the perimeter by 5 to determine the number of packages needed:

Number of packages = Perimeter / 5 = 58 / 5 = 11.6

Since we cannot have a fraction of a package, we round up to the nearest whole number.

Therefore, we need 12 packages of border material to build the border for the children's playground.

Answer

The shorter side of the rectangular children's playground is 14 meters and the longer side is 15 meters. To build the border for the playground, 12 packages of border material are needed.

Verification

Let's verify the solution by calculating the area of the rectangular playground and the perimeter.

The area of the playground is 14 × 15 = 210 m², which matches the given area.

The perimeter of the playground is 2 × (14 + 15) = 58 meters, which matches the calculated perimeter.

Therefore, the solution is correct.