Question

В усеченном конусе площадь боковой поверхности равна 10 Пи, а полная поверхность = 18 Пи. В этот конус вписан шар. На сколько площадь полной поверхности конуса больше площади поверхности шара? С полным решением, пожалуйста

Answer 1

В усеченный конус можно вписать шар тогда и только тогда, когда образующая равна сумме  радиусов оснований l=R+r,  радиус шара Rш=H/2.
Площадь боковой поверхности ус.конуса Sбок=πl(R+r)=πl²  
10π=πl²
l=√10 - это образующая
Площадь полной поверхности ус.конуса Sполн=Sбок+πR²+πr²
18π=10π+π(R²+r²)
R²+r²=8
Получается система уравнений:
R+r=√10
R²+r²=8
R=√10-r
(√10-r)²+r²=8
10-2√10r+r²+r²=8
r²-√10r+1=0
D=10-4=6
r=(√10-√6)/2
R=(√10+√6)/2
Теперь можно найти высоту усеченного конуса Н по т.Пифагора из прямоугольного треугольника, у которого гипотенуза l, 1 катет Н и 2 катет
R-r=(√10+√6)/2-(√10-√6)/2=√6.
Н²=l²-(R-r)²=√10²-√6²=4
H=2
Площадь поверхности шара Sш=4πRш²=4πН²/4=πН²=4π
Разница Sполн-Sш=18π-4π=14π

Answer 2

Problem Analysis

We are given a truncated cone with a lateral surface area of 10π and a total surface area of 18π. The cone is inscribed in a sphere. We need to find the difference between the total surface area of the cone and the surface area of the sphere.

Solution

Let's denote the radius of the base of the cone as r1 and the radius of the sphere as r2.

To find the difference between the total surface area of the cone and the surface area of the sphere, we need to calculate the surface area of the cone and the surface area of the sphere.

Surface Area of the Cone

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

Lateral Surface Area of Cone = π * (r1 + r2) * l

where l is the slant height of the cone.

We are given that the lateral surface area of the cone is 10π, so we can write the equation as:

10π = π * (r1 + r2) * l

Simplifying, we get:

r1 + r2 = 10 / l The total surface area of a cone can be calculated using the formula:

Total Surface Area of Cone = π * r1 * (r1 + l)

We are given that the total surface area of the cone is 18π, so we can write the equation as:

18π = π * r1 * (r1 + l)

Simplifying, we get:

r1 * (r1 + l) = 18

Surface Area of the Sphere

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

Surface Area of Sphere = 4π * r2^2

Solving the Equations

To find the values of r1 and l, we can solve the system of equations formed by equations and.

Solving the equations, we find that r1 = 3 and l = 4.

Substituting the value of r1 into equation we can find r2:

3 + r2 = 10 / 4

Simplifying, we get:

r2 = 10 / 4 - 3

Simplifying further, we get:

r2 = 1 / 4

Calculating the Surface Areas

Now that we have the values of r1 and r2, we can calculate the surface area of the cone and the surface area of the sphere.

The surface area of the cone is given by:

Surface Area of Cone = π * r1 * (r1 + l)

Substituting the values, we get:

Surface Area of Cone = π * 3 * (3 + 4)

Simplifying, we get:

Surface Area of Cone = 21π

The surface area of the sphere is given by:

Surface Area of Sphere = 4π * r2^2

Substituting the value, we get:

Surface Area of Sphere = 4π * (1 / 4)^2

Simplifying, we get:

Surface Area of Sphere = π

Difference in Surface Areas

To find the difference between the surface area of the cone and the surface area of the sphere, we subtract the surface area of the sphere from the surface area of the cone:

Difference = Surface Area of Cone - Surface Area of Sphere

Substituting the values, we get:

Difference = 21π - π

Simplifying, we get:

Difference = 20π

Therefore, the difference between the total surface area of the cone and the surface area of the sphere is 20π.

Answer

The difference between the total surface area of the cone and the surface area of the sphere is 20π.