Question

Стекляную пластину размером 40см×60см обрезали на одинаковом расстоянии от каждой стороны.При этом ее площадь уменьшилась в 3 раза.Найдите новые размеры пластины

Answer 1

10×15 легко и просто

Answer 2

Problem Analysis

We are given a glass plate with dimensions 40 cm × 60 cm. The plate is cut from each side at an equal distance, resulting in the plate's area being reduced by a factor of 3. We need to find the new dimensions of the plate.

Solution

Let's assume the amount cut from each side is 'x' cm. Since the plate is cut from both sides, the reduction in length will be 2x cm, and the reduction in width will also be 2x cm.

The new length of the plate will be (60 cm - 2x) cm, and the new width will be (40 cm - 2x) cm.

We are given that the area of the new plate is 3 times smaller than the original area. The area of the original plate is given by the product of its length and width, which is 40 cm × 60 cm = 2400 cm².

The area of the new plate is given by the product of its new length and new width, which is (60 cm - 2x) cm × (40 cm - 2x) cm.

According to the problem, the new area is 3 times smaller than the original area, so we can set up the following equation:

(60 cm - 2x) cm × (40 cm - 2x) cm = (2400 cm²) / 3

Now, let's solve this equation to find the value of 'x' and then calculate the new dimensions of the plate.

Calculation

(60 cm - 2x) cm × (40 cm - 2x) cm = (2400 cm²) / 3

Expanding the equation:

(2400 cm² - 120 cm²x - 80 cm²x + 4x² cm²) = (2400 cm²) / 3

Multiplying both sides by 3 to eliminate the fraction:

3(2400 cm² - 120 cm²x - 80 cm²x + 4x² cm²) = 2400 cm²

Simplifying:

7200 cm² - 360 cm²x - 240 cm²x + 12x² cm² = 2400 cm²

Rearranging the equation:

12x² cm² - 600 cm²x + 4800 cm² = 0

Dividing the entire equation by 12 cm² to simplify:

x² - 50 cm²x + 400 cm² = 0

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

Using the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

where a = 1, b = -50 cm², and c = 400 cm².

Calculating the discriminant:

D = b² - 4ac = (-50 cm²)² - 4(1)(400 cm²) = 2500 cm⁴ - 1600 cm⁴ = 900 cm⁴

Since the discriminant is positive, we have two real solutions for 'x'.

Calculating the solutions:

x₁ = (-(-50 cm²) + √(900 cm⁴)) / (2(1)) = (50 cm² + 30 cm²) / 2 = 40 cm² x₂ = (-(-50 cm²) - √(900 cm⁴)) / (2(1)) = (50 cm² - 30 cm²) / 2 = 10 cm²

Now we can calculate the new dimensions of the plate:

New length = 60 cm - 2x = 60 cm - 2(10 cm) = 60 cm - 20 cm = 40 cm New width = 40 cm - 2x = 40 cm - 2(10 cm) = 40 cm - 20 cm = 20 cm

Answer

The new dimensions of the glass plate after cutting it on each side at an equal distance are 40 cm × 20 cm.