A cone is 8 cm high and it's vertical angle is 62 degrees. Find the diameter of it's base
If by vertical angle you mean the angle formed by the point of the cone, then
r/8 = tan31°
d = 2r
Why did the cone go to therapy?
Because it had a "vertex" identity crisis!
Alright, let's get back to your question. To find the diameter of the base, we need to use a little bit of trigonometry. The vertical angle of the cone is 62 degrees, which means we have a right triangle.
The height of the cone is the opposite side of the angle, and the radius of the base (which is half the diameter) is the adjacent side. We can use the trigonometric function tangent to find the radius.
We have:
tan(62 degrees) = height / radius
Plugging in the values we know, we have:
tan(62 degrees) = 8 cm / radius
Now, let's solve for the radius:
radius = 8 cm / tan(62 degrees)
Using some mathematical magic, we get approximately:
radius ≈ 6.785 cm
Since the diameter is twice the radius, we can multiply the radius by 2 to get the diameter:
diameter ≈ 2 * 6.785 cm ≈ 13.57 cm
So, the approximate diameter of the base of the cone is 13.57 cm. Keep in mind that these are rounded values, so they may not be 100% precise. Now, isn't it fascinating how cones can square things away?
To find the diameter of the base of the cone, we can use trigonometry and the given information.
Step 1: Draw a vertical line from the apex of the cone, perpendicular to the base. This line represents the height of the cone.
Step 2: Label the height of the cone as 8 cm.
Step 3: The vertical angle of the cone is 62 degrees. The vertical angle is opposite the height of the cone and forms a right angle with the diameter of the base.
Step 4: Now, consider the right triangle formed by the height (8 cm), the radius (half of the diameter), and the slant height of the cone.
Step 5: In a right triangle, the sine of an angle is equal to the ratio of the opposite side to the hypotenuse. In this case, we need to find the sine of 62 degrees.
sin(62°) = opposite/hypotenuse
Step 6: Rearrange the formula to solve for the hypotenuse:
hypotenuse = opposite / sin(62°)
Step 7: Substitute the given values into the formula:
hypotenuse = 8 cm / sin(62°)
Step 8: Use a calculator to find the value of sin(62°) ≈ 0.8839.
hypotenuse ≈ 8 cm / 0.8839
hypotenuse ≈ 9.05 cm
Step 9: The hypotenuse of the right triangle represents the slant height of the cone, which is also the radius of the base.
Step 10: Since the diameter is equal to twice the radius, we can multiply the radius by 2 to find the diameter of the base:
diameter = 2 * 9.05 cm
diameter ≈ 18.1 cm
Therefore, the diameter of the base of the cone is approximately 18.1 cm.
To find the diameter of the base of the cone, we need to use trigonometric ratios. In this case, we will use the tangent function.
The tangent of an angle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, the side opposite the angle is the radius of the base, and the side adjacent to the angle is the height of the cone.
Let's denote the diameter of the base as d. Since the diameter is twice the radius, we can write the equation as:
d = 2r
Given that the height of the cone is 8 cm, we can use the tangent of the angle to find the radius:
tan(62°) = r / 8
To solve for r, we rearrange the equation:
r = 8 * tan(62°)
Now, we can find the diameter by doubling the radius:
d = 2 * (8 * tan(62°))
Calculating this expression will give you the diameter of the base of the cone.
h = height
r = radius
d = diameter
tan 62° = h / r
Multiply both sides by r
r ∙ tan 62° = h
Divide both sides by tan 62°
r = h / tan 62°
r = 8 / 1.88072646535
r = 4.25369 cm
d = 2 r = 2 ∙ 4.25369 cm = 8.50738 cm
d ≈ 8.5 cm