A cone just fits inside a can. The diameter of the can is 7.6 cm and the height is 10.4 cm. Determine the angle between the vertex of the cone and the bottom of the can, to the nearest tenth of a degree.

Answer 1

To determine the angle between the vertex of the cone and the bottom of the can, we first need to calculate the slant height of the cone.

The slant height of the cone can be found using the Pythagorean theorem, where the slant height (l), radius (r) and height (h) of the cone are related by:

l^2 = r^2 + h^2

Given that the radius of the cone is half of the diameter of the can, the radius (r) of the cone is 7.6/2 = 3.8 cm.

Substitute the values into the formula:

l^2 = 3.8^2 + 10.4^2
l^2 = 14.44 + 108.16
l^2 = 122.6
l = √122.6
l ≈ 11.1 cm

Now that we have the slant height of the cone, we can determine the angle (θ) between the vertex of the cone and the bottom of the can using trigonometry.

sin(θ) = opposite/hypotenuse
sin(θ) = 10.4/11.1
θ = sin^(-1)(10.4/11.1)
θ ≈ 67.5 degrees

Therefore, the angle between the vertex of the cone and the bottom of the can is approximately 67.5 degrees.