Math Problem Statement

Знайдіть площу повної поверхні прямокутного паралелепіпеда, якщо його діагональ більша за ребра паралелепіпеда на 10 см, 9 см і 1 см відповідно.

Solution

The problem in the image asks the following:

"Знайдіть площу повної поверхні прямокутного паралелепіпеда, якщо його діагональ більша за ребра паралелепіпеда на 10 см, 9 см і 1 см відповідно."

Translated to English:

"Find the total surface area of a rectangular parallelepiped if its diagonal is larger than the edges of the parallelepiped by 10 cm, 9 cm, and 1 cm, respectively."

Let's solve the problem step by step:

Step 1: Define the dimensions

Let the edges of the rectangular parallelepiped be aa, bb, and cc. From the problem, we know the following relationships between the diagonal and the edges:

d=a+10,d=b+9,d=c+1d = a + 10, \quad d = b + 9, \quad d = c + 1

where dd is the diagonal of the parallelepiped.

Step 2: Diagonal formula

The diagonal of a rectangular parallelepiped is given by the formula: d=a2+b2+c2d = \sqrt{a^2 + b^2 + c^2}

Step 3: Use the relationships between the edges and the diagonal

From the relationships in Step 1, we can express aa, bb, and cc in terms of dd: a=d10,b=d9,c=d1a = d - 10, \quad b = d - 9, \quad c = d - 1

Step 4: Substitute into the diagonal formula

Substitute these expressions into the formula for the diagonal: d=(d10)2+(d9)2+(d1)2d = \sqrt{(d - 10)^2 + (d - 9)^2 + (d - 1)^2} Now, we can solve this equation for dd. Let me calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later. If you want, I can guide you through solving the equation manually or answer any other questions related to the problem.

Here are some questions you might find interesting:

  1. How do we simplify expressions with square roots like a2+b2+c2\sqrt{a^2 + b^2 + c^2}?
  2. What are some general properties of a rectangular parallelepiped?
  3. How do different dimensions affect the surface area of a parallelepiped?
  4. How can the Pythagorean theorem be used in 3D geometry?
  5. What happens when one edge of the parallelepiped becomes very small or large?

Tip: Solving quadratic equations is often useful when dealing with geometric problems involving diagonals!

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
3D Shapes
Rectangular Parallelepiped

Formulas

Diagonal of parallelepiped: d = sqrt(a^2 + b^2 + c^2)
Total surface area of parallelepiped: S = 2(ab + ac + bc)

Theorems

Pythagorean theorem in 3D

Suitable Grade Level

Grades 9-11