Question

Помогите решить пожалуйста. Диагональ осевого сечения цилиндра,равная 4√3 см образует с плоскостью основания угол 30 градусов. Найдите боковую поверхность цилиндра.

Answer 1

L лежит против угла в 30 гр= 1/2 гип= 2 корня из 3
R= корень из ( 16*3-4*3)= 6
Бок поверхность =2пиRL=2пи* 2 корня из 3* 6

Answer 2

Calculation of the lateral surface area of a cylinder

To find the lateral surface area of a cylinder, we need to know the height (h) and the radius (r) of the cylinder. However, in this case, we are given the diagonal of the cross-section of the cylinder and the angle it forms with the base.

Let's assume that the diagonal of the cross-section is represented by the letter d and the angle it forms with the base is represented by θ.

From the given information, we can form a right triangle with the diagonal (d) as the hypotenuse, the radius (r) as one of the legs, and the height (h) as the other leg.

Using trigonometry, we can relate the diagonal (d), the radius (r), and the angle (θ) as follows:

sin(θ) = r / d

Since we know the angle (θ) is 30 degrees, we can substitute this value into the equation:

sin(30) = r / d

Simplifying the equation, we have:

1/2 = r / d

From this equation, we can solve for the radius (r) in terms of the diagonal (d):

r = (1/2) * d

Now, we need to find the height (h) of the cylinder. We can use the Pythagorean theorem to relate the radius (r), the height (h), and the diagonal (d) as follows:

d^2 = r^2 + h^2

Substituting the value of the radius (r) we found earlier, we have:

d^2 = ((1/2) * d)^2 + h^2

Simplifying the equation, we have:

d^2 = (1/4) * d^2 + h^2

Multiplying both sides of the equation by 4 to eliminate the fraction, we have:

4 * d^2 = d^2 + 4 * h^2

Simplifying further, we have:

3 * d^2 = 4 * h^2

Dividing both sides of the equation by 4, we have:

(3/4) * d^2 = h^2

Taking the square root of both sides of the equation, we have:

h = √((3/4) * d^2)

Now that we have the radius (r) and the height (h) of the cylinder, we can calculate the lateral surface area (A) using the formula:

A = 2 * π * r * h

Substituting the values we found earlier, we have:

A = 2 * π * ((1/2) * d) * √((3/4) * d^2)

Simplifying the equation, we have:

A = π * d * √(3/4) * d

Finally, we can simplify the equation further:

A = (π * √3 * d^2) / 2

Therefore, the lateral surface area of the cylinder is given by (π * √3 * d^2) / 2.

Please note that this calculation assumes the cylinder is right circular and the diagonal of the cross-section is the hypotenuse of a right triangle formed with the radius and height of the cylinder.

Let me know if you need any further assistance!