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
To determine the angle between the vertex of the cone and the bottom of the can, we need to first calculate the slant height of the cone.
The slant height of a cone can be calculated using the Pythagorean theorem:
Slant height (l) = √(r^2 + h^2)
where r is the radius of the cone and h is the height of the cone.
Given that the diameter of the can is 7.6 cm, the radius of the cone (r) is half of this value, so r = 3.8 cm. The height of the cone (h) is not given, but it is the same as the height of the can, which is 10.4 cm.
Substitute these values into the formula:
l = √(3.8^2 + 10.4^2)
l = √(14.44 + 108.16)
l = √122.6
l ≈ 11.07 cm
Now, we can determine the angle between the vertex of the cone and the bottom of the can using trigonometry. The tangent of this angle is equal to the opposite side (radius) divided by the adjacent side (slant height).
tan(θ) = r / l
tan(θ) = 3.8 / 11.07
tan(θ) ≈ 0.3432
Now we need to find the angle whose tangent is approximately 0.3432. This can be found using the arctan function on a calculator:
θ ≈ arctan(0.3432)
θ ≈ 19.7 degrees
Therefore, the angle between the vertex of the cone and the bottom of the can is approximately 19.7 degrees.