Question

Осевое сечение конуса – прямоугольный равнобедренный треугольник, площадь которого  64 см2. Найдите площадь полной поверхности конуса. ​подробно распишите

Answer 1

Ответ:  ≈154,5 π см²

Подразумевается, что конус прямой, – для другого в условии недостаточно данных.  

    Проекция вершины прямого конуса на плоскость совпадает с центром основания. Осью конуса называется прямая, соединяющая вершину и центр основания.

     Пусть АВ - диаметр конуса, СО его высота.

Осевое сечение - ∆ АСВ, где угол С=90°, АС=ВС.

    СО - высота и медиана равнобедренного прямоугольного ∆ АВС, h=СО=АО=ОВ; h=R.

AB=2R  

S(ABC)=h•2R/2=R²

R²=64 => R=√64=8

Острые углы ∆ АВС равны 45°.

           AC=BC=AO:sin45°=8√2

S(осн)=πR²=π•8²= 64 см²

       S(бок)=πRL=π•8•8√2=64√2•π

S(полн)=S(осн)+ S(бок)

Ѕ(полн)=64π+64√2•π=64π•(1+√2) = ≈154,5 π см²

Answer 2

Calculating the Total Surface Area of a Cone

To find the total surface area of a cone, we can use the formula:

Total Surface Area = Base Area + Lateral Area

Where: - Base Area = πr^2 - Lateral Area = πrl

Here, r is the radius of the base of the cone, l is the slant height, and π is a mathematical constant approximately equal to 3.14159.

Given Information

The given information is that the cross-section of the cone is a right-angled isosceles triangle with an area of 64 cm^2.

Finding the Radius and Slant Height

To find the radius and slant height, we can use the properties of a right-angled isosceles triangle. In such a triangle, the sides are in the ratio 1:1:√2.

Let's assume the base and height of the triangle are both equal to x. Then, the area of the triangle can be expressed as:

Area = (1/2) * base * height

Given that the area is 64 cm^2, we have:

64 = (1/2) * x * x 64 = (1/2) * x^2 x^2 = 64 * 2 x^2 = 128 x = √128 x = 8√2

So, the base and height of the triangle are both 8√2 cm.

The radius of the cone is half the base of the triangle, so the radius (r) is 8√2 / 2 = 4√2 cm.

The slant height (l) of the cone can be found using the Pythagorean theorem, where l is the hypotenuse and r is one of the other sides:

l^2 = r^2 + h^2 l^2 = (4√2)^2 + (8√2)^2 l^2 = 32 + 128 l^2 = 160 l = √160 l = 4√10 cm

Calculating the Total Surface Area

Now that we have the radius and slant height, we can calculate the total surface area of the cone using the formula mentioned earlier:

Total Surface Area = Base Area + Lateral Area Total Surface Area = πr^2 + πrl

Substituting the values: Total Surface Area = π(4√2)^2 + π(4√2)(4√10) Total Surface Area = 32π + 16√10π

So, the total surface area of the cone is 32π + 16√10π square centimeters.

This calculation is based on the given information and the standard formulas for the total surface area of a cone.