Answer:
m∠KSM = 59.55°
Step-by-step explanation:
The formula of the lateral area of a cone is LA = [tex]\frac{1}{2}[/tex] Cl, where c is the circumference of the base and l is the slant height
The formula of the circumference of a circle is C = 2πr
From the figure
SO is the height of the cone
KM is the diameter of the base
SO⊥ KM
SK and SM are slant heights
We can use the trigonometry ratio in triangle SOK to find angle KSO, then multiply its measure by 2 to find the measure of ∠KSM, because m∠KSO is equal to m∠MSO
∵ LA = [tex]\frac{1}{2}[/tex] Cl
∵ C = 2πr
∴ LA = [tex]\frac{1}{2}[/tex] (2πr) l
∴ LA = πrl
∵ LA = 156
- l is the side SK
∵ SK = 10 units
∴ l = 10
- Substitute the values of LA and l in the formula above
∵ 156 = 10πr
- Divide both sides by 10π
∴ [tex]\frac{156}{10\pi }[/tex] = r
∴ r = [tex]\frac{78}{5\pi }[/tex]
∵ sin(∠KSO) = [tex]\frac{OK}{SK}[/tex]
∵ OK is the radius of the base
∴ sin(∠KSO) = [tex]\frac{\frac{78}{5\pi }}{10}[/tex]
∴ sin(∠KSO) = [tex]\frac{39}{25\pi }[/tex]
- Use [tex]sin^{-1}[/tex] to find m∠KSO
∴ ∠KSO = [tex]sin^{-1}\frac{39}{25\pi }[/tex]
∴ ∠KSO = 29.77°
∵ m∠KSO = m∠MSO
∴ m∠KSM = 2 m∠KSO
∴ m∠KSM = 2(29.77°)
∴ m∠KSM = 59.55°
