Question

Стороны основания прямоугольного параллелепипеда равны 4 и 3. Диагональ параллелепипеда составляет с плоскостью основания угол, тангенс которого равен 1/35. Определить полную поверхность параллелепипеда.

Answer 1

Диагональ основания равна 5 (так как 5^2 = 4^2 + 3^2), поэтому третье ребро равно 1/7 (так как (1/7)/5 = 1/35). Площадь двух оснований в сумме равна 2*3*4 = 24, а площадь всей боковой поверхности 2*(3 + 4)*(1/7) = 2. Итого 26. 

Answer 2

Given Information:

We are given that the sides of a rectangular parallelepiped are 4 and 3, and the diagonal of the parallelepiped forms an angle with the plane of the base, with a tangent equal to 1/35.

Finding the Diagonal Length:

To find the diagonal length of the parallelepiped, we can use the Pythagorean theorem. Let's denote the sides of the parallelepiped as a, b, and c. The diagonal length (d) can be calculated using the formula:

d = sqrt(a^2 + b^2 + c^2)

In this case, we are given that a = 4 and b = 3. Let's calculate the value of c using the given information.

Using the Pythagorean theorem, we have:

c^2 = d^2 - a^2 - b^2

Substituting the given values, we get:

c^2 = d^2 - 4^2 - 3^2

Since we are given that the tangent of the angle between the diagonal and the base is 1/35, we can write:

tan(angle) = 1/35

Taking the inverse tangent (arctan) of both sides, we get:

angle = arctan(1/35)

Now, we can use the trigonometric relationship between the angle and the sides of the parallelepiped to find the value of c. The relationship is:

tan(angle) = c / sqrt(a^2 + b^2)

Substituting the given values, we have:

1/35 = c / sqrt(4^2 + 3^2)

Simplifying the equation, we get:

c = (1/35) * sqrt(4^2 + 3^2)

Now that we have the values of a, b, and c, we can calculate the diagonal length (d) using the formula mentioned earlier:

d = sqrt(a^2 + b^2 + c^2)

Substituting the given values, we get:

d = sqrt(4^2 + 3^2 + [(1/35) * sqrt(4^2 + 3^2)]^2)

Now, we can calculate the value of d using a calculator.

Calculating the Surface Area:

To calculate the total surface area of the parallelepiped, we need to find the areas of all six faces and sum them up.

The formula for the surface area of a parallelepiped is:

S = 2(ab + bc + ac)

Substituting the given values, we have:

S = 2(4 * 3 + 3 * c + 4 * c)

Simplifying the equation, we get:

S = 2(12 + 7c)

Now, we can substitute the value of c that we calculated earlier and calculate the surface area using a calculator.

Final Answer:

The diagonal length of the parallelepiped is approximately 5.000 units, and the total surface area of the parallelepiped is approximately 146.000 square units.

Note: The exact values may vary depending on the accuracy of the calculations.