Question

Стороны основания прямоугольного параллелепипеда равны 3 см и 4 см. найти его объем, если параллелепипед прямоугольный и его диагональ наклонена к плоскости основания 30°

Answer 1

Объяснение:

V=a*b*c

a=4

b=3

<A=30

tgA=c/d1

c=d1×tg30=d1×1/корень3=

=(d1×корень3) /3

c=(d1×корень3) /3

d1=корень(а^2+b^2)=

=корень(4^2+3^2)=5 см

с=(5×корень3)/3=5/3×корень3

V=4×3×5/3×корень3=20корень3

Ответ : V= 20корень3

Answer 2

Calculation of the Volume of a Rectangular Parallelepiped

To find the volume of a rectangular parallelepiped, we need to know the lengths of its sides. In this case, the sides of the parallelepiped are given as 3 cm and 4 cm.

The formula for calculating the volume of a rectangular parallelepiped is:

Volume (V) = Length (L) × Width (W) × Height (H)

In this case, the length and width of the parallelepiped are 3 cm and 4 cm, respectively. We need to find the height of the parallelepiped.

Finding the Height of the Parallelepiped

To find the height of the parallelepiped, we are given that the diagonal is inclined at an angle of 30° to the plane of the base

Using trigonometry, we can determine the height of the parallelepiped. Let's denote the height as H.

We know that the diagonal is the hypotenuse of a right triangle formed by the height, length, and width of the parallelepiped. The angle between the diagonal and the length is 30°.

Using the trigonometric relationship:

cos(30°) = adjacent / hypotenuse

We can rearrange the equation to solve for the adjacent side, which is the height of the parallelepiped:

adjacent = cos(30°) × hypotenuse

In this case, the hypotenuse is the diagonal of the parallelepiped, which we need to calculate.

Calculating the Diagonal of the Parallelepiped

To calculate the diagonal of the parallelepiped, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, the other two sides are the length and width of the parallelepiped, which are 3 cm and 4 cm, respectively.

Using the Pythagorean theorem:

diagonal^2 = length^2 + width^2

We can rearrange the equation to solve for the diagonal:

diagonal = sqrt(length^2 + width^2)

Now that we have the diagonal, we can substitute it into the equation for the adjacent side to find the height of the parallelepiped.

Calculation Steps

1. Calculate the diagonal of the parallelepiped using the formula: diagonal = sqrt(length^2 + width^2). 2. Substitute the diagonal into the equation: adjacent = cos(30°) × diagonal. 3. Calculate the height of the parallelepiped using the formula: height = adjacent.

Let's perform the calculations:

1. Calculate the diagonal: - diagonal = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 cm.

2. Calculate the adjacent side: - adjacent = cos(30°) × 5 cm = 0.866 × 5 cm = 4.33 cm.

3. Calculate the height of the parallelepiped: - height = adjacent = 4.33 cm.

Answer

Therefore, the height of the rectangular parallelepiped is 4.33 cm. Now, we can calculate the volume of the parallelepiped using the formula:

Volume (V) = Length (L) × Width (W) × Height (H)

Substituting the given values:

Volume = 3 cm × 4 cm × 4.33 cm = 51.96 cm^3

Hence, the volume of the rectangular parallelepiped is 51.96 cm^3.