a cone has a circular base, a perpendicular height of 42cm and asemivertical angle of of 30. Calculate the slant height of the cone
Draw a diagram. The slant height is
s = 42/cos30°
To calculate the slant height of a cone, we can use the formula:
Slant height (l) = √(r^2 + h^2)
where:
- r is the radius of the circular base of the cone,
- h is the height of the cone.
In this specific case, we are given the height of the cone (h = 42 cm) and the semi-vertical angle, which is half of the angle formed by the height and the slant height. Since the semi-vertical angle is 30 degrees, the full angle would be 60 degrees.
To find the radius (r) of the base of the cone, we can use trigonometry as follows:
sin(θ) = opposite/hypotenuse
In this case, θ is the angle between the radius and the slant height, opposite is the radius (r), and hypotenuse is the slant height (l).
sin(60) = r / l
Rearranging the equation:
r = l * sin(60)
Now we can substitute the expression for r into the formula for the slant height:
l = √((l * sin(60))^2 + 42^2)
Simplifying the equation and isolating l:
l^2 = (l * sin(60))^2 + 42^2
l^2 = l^2 * sin^2(60) + 42^2
l^2 = l^2 * (3/4) + 42^2
l^2 - (3/4) * l^2 = 42^2
l^2 * (1 - 3/4) = 1764
l^2 * (1/4) = 1764
l^2 = 1764 * (4/1)
l^2 = 7056
l = √7056
l ≈ 84 cm
Therefore, the slant height of the cone is approximately 84 cm.