Axial section of the cone

The axial section of the cone is an isosceles triangle in which the ratio of cone diameter to cone side is 2:3. Calculate its volume if you know its area is 314 cm square.

Correct answer:

V =  366.4297 cm³

Step-by-step explanation:

$$\begin{gathered} D:s=2:3 \\ S=312{\text{cm}}^2 \\ 2r:s=2:3 \\ r:s=1:3 \\ 3r=s \\ S=\pi r(r+s) \\ S=\pi r(r+3r) \\ S=4\pi r^2 \\ r=\sqrt{\frac{S}{4\pi }}=\sqrt{\frac{312}{4\cdot 3.1416}}\doteq 4.9828\text{cm} \\ D=2\cdot r=2\cdot 4.9828\doteq 9.9656\text{cm} \\ s=3\cdot r=3\cdot 4.9828\doteq 14.9484\text{cm} \\ \text{ Verifying Solution: } \\ k=D/s=9.9656/14.9484=\frac{2}{3}\doteq 0.6667 \\ h=\sqrt{s^2-r^2}=\sqrt{14.9484^2-4.9828^2}\doteq 14.0935\text{cm} \\ {S}_1=\pi \cdot r^2=3.1416\cdot 4.9828^2=78{\text{cm}}^2 \\ V=\frac{1}{3}\cdot S_1\cdot h=\frac{1}{3}\cdot 78\cdot 14.0935=366.4297{\text{cm}}^3 \end{gathered}$$

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