Question

Картинка имеет форму пря­мо­уголь­ни­ка со сто­ро­на­ми 11 см и 33 см. Её на­кле­и­ли на белую бу­ма­гу так, что во­круг кар­тин­ки по­лу­чи­лась белая окан­тов­ка оди­на­ко­вой ширины. Площадь, ко­то­рую за­ни­ма­ет кар­тин­ка с окантовкой, равна 779 см2. Ка­ко­ва ши­ри­на окантовки? Ответ дайте в сантиметрах.

Answer 1

Пусть  см — ши­ри­на окантовки. Пло­щадь пря­мо­уголь­ни­ка равна про­из­ве­де­нию сторон., по­лу­ча­ем уравнение:

 



 

Корень −26 не под­хо­дит по усло­вию задачи, следовательно, ши­ри­на окан­тов­ки равна 4 см.

 

Answer 2

Problem Analysis

We are given a rectangular picture with dimensions 11 cm and 33 cm. The picture is pasted on white paper in such a way that there is a white border of equal width around the picture. The total area occupied by the picture and the border is 779 cm². We need to find the width of the border in centimeters.

Solution

Let's assume the width of the border is x cm.

The length of the picture with the border is (11 + 2x) cm, and the width is (33 + 2x) cm.

The total area occupied by the picture with the border is the product of its length and width:

(11 + 2x) cm * (33 + 2x) cm = 779 cm²

Simplifying the equation:

(11 + 2x)(33 + 2x) = 779

Expanding the equation:

363 + 22x + 66x + 4x² = 779

Rearranging the equation:

4x² + 88x + 363 - 779 = 0

4x² + 88x - 416 = 0

Dividing the equation by 4 to simplify:

x² + 22x - 104 = 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 = 22, and c = -104.

Calculating the discriminant:

D = b² - 4ac = 22² - 4(1)(-104) = 484 + 416 = 900

Since the discriminant is positive, there are two real solutions for x.

Using the quadratic formula:

x = (-22 ± √900) / 2

x = (-22 ± 30) / 2

x₁ = (-22 + 30) / 2 = 8 / 2 = 4

x₂ = (-22 - 30) / 2 = -52 / 2 = -26

Since the width of the border cannot be negative, we discard the negative solution.

Therefore, the width of the border is 4 cm.

Answer

The width of the border is 4 centimeters.