Question

Дано: S боковой поверхности= 100 пи Диаметр основания = 16 Найти: H-?

Answer 1

Problem Analysis

We are given the lateral surface area (S) of a cone and the diameter of its base. We need to find the height (H) of the cone.

Solution

To find the height of the cone, we can use the formula for the lateral surface area of a cone and solve for H.

The formula for the lateral surface area (S) of a cone is given by:

S = π * r * l

Where: - S is the lateral surface area of the cone - π is a mathematical constant approximately equal to 3.14159 - r is the radius of the base of the cone - l is the slant height of the cone

The radius (r) of the base of the cone is half the diameter (D) of the base. So, we have:

r = D/2

Substituting this value of r in the formula for the lateral surface area, we get:

S = π * (D/2) * l

Now, we can solve this equation for the slant height (l):

l = (2 * S) / (π * D)

The slant height (l) is the hypotenuse of a right triangle formed by the height (H) and the radius (r) of the cone. Using the Pythagorean theorem, we have:

l^2 = r^2 + H^2

Substituting the value of r from above, we get:

l^2 = (D/2)^2 + H^2

Simplifying this equation, we have:

l^2 = (D^2/4) + H^2

Now, we can substitute the value of l from the previous equation into this equation:

((2 * S) / (π * D))^2 = (D^2/4) + H^2

Simplifying further, we have:

4 * S^2 = π^2 * D^2 + 4 * H^2 * π * D

Finally, we can solve this equation for H:

H = sqrt((4 * S^2 - π^2 * D^2) / (4 * π * D))

Let's substitute the given values and calculate the height (H).

Calculation

Given: - S = 100π - D = 16

Substituting these values into the equation for H, we have:

H = sqrt((4 * (100π)^2 - π^2 * 16^2) / (4 * π * 16))

Simplifying further, we have:

H = sqrt((4 * 100^2 * π^2 - π^2 * 16^2) / (4 * π * 16))

**H = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) / 16) = sqrt((4 * 100^2 - 16^2) /