Question

2)Лист ватмана имеет форму квадрата, от этого листа отрезали прямоугольник со стороной 2см и осталось 120 см². Найдите первоначальные размеры листа ватмана.(выполни рисунок к задаче). 3)Бассейн глубиной 2м, шириной 10м и длиной 20м нужно обложить кафелем, который имеет квадратную форму со стороной 20 см. Сколько потребуется плиток кафеля? Сколько денег необходимо заплатить за кафель, если цена одной плитки 500тенге? 4)Куб с ребром 1м разрезали на кубики с ребром 1 см. Если эти кубики уложить в ряд, ставя их рядом друг с другом. Какова длина этого ряда? 5)Из кубиков сложили прямоугольный параллелепипед. Его длина состоит из 7 кубиков, а ширина из 6 кубиков, а высота из 4 кубиков. Найдите объем прямоугольного параллелепипеда, если поверхность одного кубика равна 54 см².(поверхность кубика это есть сумма площадей всех граней, а грань это квадраты из которых состоит поверхность кубика) 6)Известно, что периметры прямоугольника и квадрата равны. Площадь квадрата равна 81см². Длина прямоугольника в 2 раза больше его ширины. Найти площадь прямоугольника.​

Answer 1

Problem 2: Finding the Initial Dimensions of a Sheet of Paper

To find the initial dimensions of the sheet of paper, we can use the given information that a rectangle with a side length of 2 cm was cut from a square sheet of paper, leaving behind an area of 120 cm².

Let's assume the side length of the square sheet of paper is x cm. After cutting the rectangle, the remaining area is given by:

Remaining Area = Area of Square Sheet - Area of Cut Rectangle

The area of the square sheet is x * x = x² cm², and the area of the cut rectangle is 2 * x = 2x cm².

Therefore, the equation becomes:

x² - 2x = 120

To solve this quadratic equation, we can rearrange it to:

x² - 2x - 120 = 0

Now, we can factorize the equation:

(x - 12)(x + 10) = 0

This gives us two possible solutions: x = 12 and x = -10. Since the side length of a sheet of paper cannot be negative, we discard the negative solution.

Therefore, the initial dimensions of the sheet of paper are 12 cm by 12 cm.

Problem 3: Calculating the Number of Tiles and Cost

To calculate the number of tiles required to cover the pool and the cost of the tiles, we need to find the total area of the pool and then divide it by the area of each tile.

The pool has a depth of 2 m, a width of 10 m, and a length of 20 m. To find the total area, we need to calculate the surface area of the pool, excluding the bottom.

The surface area of the pool is given by:

Surface Area = 2 * (Length * Width + Length * Depth + Width * Depth)

Substituting the given values, we get:

Surface Area = 2 * (20 * 10 + 20 * 2 + 10 * 2) = 2 * (200 + 40 + 20) = 2 * 260 = 520 m²

Now, let's calculate the number of tiles required. Each tile has a side length of 20 cm, which is equal to 0.2 m. The area of each tile is:

Tile Area = Side Length * Side Length = 0.2 * 0.2 = 0.04 m²

To find the number of tiles required, we divide the total area of the pool by the area of each tile:

Number of Tiles = Surface Area / Tile Area = 520 / 0.04 = 13,000 tiles

Therefore, you will need 13,000 tiles to cover the pool.

To calculate the cost of the tiles, we multiply the number of tiles by the price of each tile, which is 500 tenge:

Total Cost = Number of Tiles * Price per Tile = 13,000 * 500 = 6,500,000 tenge

Therefore, you will need to pay 6,500,000 tenge for the tiles.

Problem 4: Finding the Length of the Row of Cubes

To find the length of the row of cubes, we need to determine how many 1 cm cubes can be arranged in a row when the original cube with a side length of 1 m is divided into smaller cubes with a side length of 1 cm.

Since the original cube has a side length of 1 m, it contains 100 cm cubes in each dimension (1 m = 100 cm). Therefore, the length of the row of cubes is 100 cm.

Problem 5: Calculating the Volume of the Rectangular Prism

To calculate the volume of the rectangular prism, we need to multiply its length, width, and height.

The length of the rectangular prism is given as 7 cubes, the width is given as 6 cubes, and the height is given as 4 cubes. Since each cube has a side length of 1 cm, the dimensions of the rectangular prism are:

Length = 7 cm Width = 6 cm Height = 4 cm

Therefore, the volume of the rectangular prism is:

Volume = Length * Width * Height = 7 * 6 * 4 = 168 cm³

Therefore, the volume of the rectangular prism is 168 cm³.

Problem 6: Finding the Area of the Rectangle

To find the area of the rectangle, we need to know the relationship between the perimeter of the rectangle and the side length of the square.

Let's assume the side length of the square is s cm. The perimeter of the square is then 4s cm.

Given that the area of the square is 81 cm², we have:

s² = 81

Taking the square root of both sides, we find:

s = 9

Since the length of the rectangle is twice its width, we have:

Length = 2 * Width

Substituting the value of the width, we get:

Length = 2 * 9 = 18 cm

Therefore, the length of the rectangle is 18 cm.

To find the area of the rectangle, we multiply its length and width:

Area = Length * Width = 18 * 9 = 162 cm²

Therefore, the area of the rectangle is 162 cm².