Question

Помогите / Длина прямоугольно параллелепипеда в 2 раза больше его ширины и на 2 см больше высоты .Если каждое ребро параллелепипеда увеличить на 1 см , то его объем увеличится на 68 см^3. Найдите длину ,ширину и высоту параллелепипеда

Answer 1

Пусть a - длина, b - ширина, c - высота.

a₁ = 2b₁ ⇒ b₁ = a₁ /2

a₁ = c₁ + 2 ⇒ c₁ = a₁ - 2

V₁ = a₁b₁c₁ = a₁*a₁ /2*(a₁ - 2) = a₁²(a₁ - 2)/2

a₂ = a₁ + 1

b₂ = a₁ /2 + 1

c₂ = a₁ - 1

V₂ = a₂b₂c₂ = (a₁ + 1)(a₁ - 1)(a₁ /2 + 1)

V₂ = V₁ + 68

(a₁ + 1)(a₁ - 1)(a₁ /2 + 1) = a₁²(a₁ - 2)/2 + 68

(a₁² - 1)(a₁ + 2)/2 = a₁²(a₁ - 2)/2 + 136/2

a₁³ - a₁ + 2a₂ - 2 = a₁³ - 2a₁² + 136

4a₁² - a₁ - 138 = 0

D = 1 - 4*4*(-138) = 2209 = 47²

a = (1 + 47)/2*4 = 48/8 = 6 (см)

a = (1 - 47)/2*4 < 0 -- не удовл.

b = 6/2 = 3 (см)

c = 6 - 2 = 4 (см)

Ответ: 6 см, 3 см, 4 см.

Answer 2

Problem Analysis

We are given the following information about a rectangular parallelepiped: - The length is twice the width. - The length is 2 cm more than the height. - If each edge of the parallelepiped is increased by 1 cm, the volume increases by 68 cm^3.

We need to find the length, width, and height of the parallelepiped.

Solution

Let's assume: - Width = x cm - Length = 2x cm - Height = 2x - 2 cm

To find the volume of the parallelepiped, we use the formula: Volume = Length * Width * Height

Given that increasing each edge by 1 cm increases the volume by 68 cm^3, we can write the equation: (Length + 1) * (Width + 1) * (Height + 1) - Length * Width * Height = 68

Now, let's substitute the values of length, width, and height in terms of x into the equation and solve for x.

Calculation

Substituting the values into the equation: (2x + 1) * (x + 1) * (2x - 1) - 2x * x * (2x - 2) = 68

Simplifying the equation: (2x + 1) * (x + 1) * (2x - 1) - 4x^2 * (2x - 2) = 68

Expanding and simplifying further: 4x^3 - 4x^2 - 2x + 1 - 8x^3 + 16x^2 - 16x = 68

Combining like terms: -4x^3 + 12x^2 - 18x - 67 = 0

To solve this cubic equation, we can use numerical methods or a graphing calculator.