if a cone is 7.5 cm high and has a vertical angle of 70 degree calculate the diameter of its base.

draw a diagram of a vertical cross-section of the cone, and mark off the altitude. You have a right triangle.

The radius of the base is found from

r/7.5 = cot(70)

The answer

To calculate the diameter of the base of a cone based on its height and vertical angle, you can use trigonometry. The vertical angle of a cone is the angle between the height and the slant height.

First, let's identify the relevant sides of the cone. The height of the cone is the distance from the tip of the cone to the base along its central axis. The slant height is the distance from the tip of the cone to any point on the base along the surface. The radius is the distance from the center of the base to any point on the base.

We can use the tangent function, which relates the opposite side (height) to the adjacent side (radius). In this case, the opposite side is the height, and the adjacent side is half the diameter (radius).

Step 1: Find the length of the slant height (hypotenuse) using the height and the vertical angle.
Given the height (opposite side) = 7.5 cm and the vertical angle = 70 degrees, we can use the tangent function:
tan(70°) = height / slant height
slant height = height / tan(70°)
slant height = 7.5 cm / tan(70°)

Step 2: Find the radius using the slant height and vertical angle.
To find the radius, we need to find the opposite side of the right triangle formed between the vertical height, slant height, and radius.
Using the sine function:
sin(70°) = radius / slant height
radius = slant height * sin(70°)

Step 3: Calculate the diameter of the base.
The diameter is twice the radius.
diameter = 2 * radius

By plugging in the given values and performing the calculations, you can find the diameter of the base of the cone.