Cone - bases

The volume of the cut cone is V = 38000π cm³. The radius of the lower base is 10 cm larger than the radius of the upper base. Determine the radius of the base if height v = 60 cm.

Correct answer:

r₁ =  20 cm
r₂ =  30 cm

Step-by-step explanation:

$$\begin{gathered} V=38000\pi =38000\cdot 3.1416\doteq 119380.5208{\text{cm}}^3 \\ v=60\text{cm} \\ {r}_2=10+r_1 \\ V=\frac{1}{3}\pi \cdot v\cdot (r_1^2+r_1\cdot r_2+r_2^2) \\ V=pi/3\cdot v\cdot (x^2+x\cdot (10+x)+(10+x{)}^2) \\ 119380.52083641=3.1415926/3\cdot 60\cdot (x^2+x\cdot (10+x)+(10+x{)}^2) \\ -188.49555599999x^2-1884.956x+113097.336=0 \\ 188.49555599999x^2+1884.956x-113097.336=0 \\ a=188.495556;b=1884.956;c=-113097.336 \\ D=b^2-4ac=1884.956^2-4\cdot 188.495556\cdot (-113097.336)=88826438.114787 \\ D>0 \\ {x}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-1884.96\pm \sqrt{88826438.11}}{376.991112} \\ {x}_{1,2}=-5\pm 25 \\ {x}_1=20.000000216 \\ {x}_2=-30.000000216 \\ {r}_1=x_1=20=20\text{cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

$$\begin{gathered} r_2=10+r_1=10+20=30=30\text{cm} \\ \text{ Verifying Solution: } \\ {V}_2=\frac{1}{3}\cdot \pi \cdot v\cdot (r_1^2+r_1\cdot r_2+r_2^2)=\frac{1}{3}\cdot 3.1416\cdot 60\cdot (20^2+20\cdot 30+30^2)\doteq 119380.5208{\text{cm}}^3 \end{gathered}$$

Try another example