Question

Ящик с крышкой имеет форму куба. Василий Петрович решила открыть декоративной пленкой все грани ящика ,кроме нижней. Если бы каждое ребро ящика было на 1 дм больше,то Василию Петровичу пришлось бы израсходовать 405 дм 2 плёнки.Найдите площадь крышки ящика

Answer 1

У куба 6 граней. В условии дана площадь всех граней куба кроме нижней, т.е. 5-ти граней.

Пусть а - ребро данного куба. Но по условию каждое ребро ящика увеличено на 1 дм.
Получаем формулу:

S = 5·(a+1)²
5*(a+1)² = 405
5(a²+2a+1)=405
a²+2a+1=405:5
а²+2а+1=81
a²+2a-80=0 - решаем квадратное уравнение 

D = b² - 4ac = 2² - 4·1·(-80) = 4 + 320 = 324

√D = √324 = 18

a₁=(-2-18)/2=(-2-18)/2= -10 - отрицательное знач. не удовлетв. условию

a₂=(-2+18)/2=(-2+18)/2=8  
а=8 дм - ребро куба

Sкрышки = a²
S = 8*8 = 64 дм²

Ответ: 64 дм²

Answer 2

Problem Analysis

We are given a box with a lid in the shape of a cube. Vasily Petrovich wants to cover all the faces of the box, except the bottom, with decorative film. If each side of the box is increased by 1 dm, Vasily Petrovich would need 405 dm² of film. We need to find the area of the lid of the box.

Solution

Let's assume the side length of the original cube is x dm. The area of each face of the cube is x² dm². Since there are 6 faces, the total area of the cube is 6x² dm².

If each side of the cube is increased by 1 dm, the new side length would be x + 1 dm. The area of each face of the new cube is (x + 1)² dm². Again, since there are 6 faces, the total area of the new cube is 6(x + 1)² dm².

According to the given information, the difference in film required is 405 dm². This means the area of the new cube minus the area of the original cube is equal to 405 dm²:

6(x + 1)² - 6x² = 405

Simplifying the equation:

6x² + 12x + 6 - 6x² = 405

12x + 6 = 405

12x = 399

x = 33.25

Now that we have the value of x, we can calculate the area of the lid of the box. The lid is a square with side length x dm, so the area of the lid is x² dm².

Substituting the value of x:

Area of the lid = (33.25)² = 1105.06 dm²

Therefore, the area of the lid of the box is approximately 1105.06 dm².

Answer

The area of the lid of the box is approximately 1105.06 dm².