Question

Образующая конуса наклонена к плоскости основания под углом 45*. Найти площадь поверхности и объем конуса, если образующая конуса дм.

Answer 1

Всё подробно описала в решении.

Answer 2

Finding the Surface Area and Volume of a Cone

To find the surface area and volume of a cone, we need to know the slant height (or generatrix) and the angle between the generatrix and the base plane. In this case, the generatrix is inclined to the base plane at an angle of 45 degrees.

Let's denote the slant height of the cone as h and the radius of the base as r. The height of the cone can be found using the Pythagorean theorem, which states that the square of the height is equal to the square of the slant height minus the square of the radius:

h^2 = r^2 + r^2 = 2r^2

From this equation, we can solve for h:

h = sqrt(2r^2)

Now, let's calculate the surface area and volume of the cone.

Surface Area of the Cone

The surface area of a cone can be calculated using the formula:

Surface Area = πr(r + h)

Substituting the value of h from above, we get:

Surface Area = πr(r + sqrt(2r^2))

Simplifying further, we have:

Surface Area = πr^2(1 + sqrt(2))

Volume of the Cone

The volume of a cone can be calculated using the formula:

Volume = (1/3)πr^2h

Substituting the value of h from above, we get:

Volume = (1/3)πr^2(sqrt(2r^2))

Simplifying further, we have:

Volume = (1/3)πr^3sqrt(2)

Now, let's calculate the surface area and volume of the cone using the given slant height.

Calculation

Given that the slant height of the cone is d dm, we can calculate the radius r using the formula:

r = d / sqrt(2)

Substituting this value into the formulas for surface area and volume, we get:

Surface Area = π(d^2/2)(1 + sqrt(2))

Volume = (1/3)π(d^3/2)sqrt(2)

Now, you can substitute the value of d into these formulas to find the surface area and volume of the cone.

Please note that the units of measurement used in the calculations should be consistent. If the slant height is given in decimeters (dm), the resulting surface area and volume will be in square decimeters (dm^2) and cubic decimeters (dm^3), respectively.

I hope this helps! Let me know if you have any further questions.