Calculate 19443

Calculate the height of the cylinder when r = 10 mm and S = 800 mm².
Calculate the radius / r / of the cylinder when the height is 20 mm and S = 1000 mm².

Correct answer:

h₁ =  2.7324 mm
r₂ =  6.0983 mm

Step-by-step explanation:

$$\begin{gathered} r_1=10\text{mm} \\ {S}_1=800{\text{mm}}^2 \\ S=2\pi r(r+h) \\ {h}_1=\frac{S_1}{2\pi \cdot r_1}-r_1=\frac{800}{2\cdot 3.1416\cdot 10}-10=2.7324\text{mm} \end{gathered}$$

$$\begin{gathered} h_2=20\text{mm} \\ {S}_2=1000{\text{mm}}^2 \\ {S}_2=2\pi r_2(r_2+h_2) \\ 1000=2\cdot pi\cdot x(x+20) \\ 1000=2\cdot 3.1415926\cdot x(x+20) \\ -6.2831851999999x^2-125.664x+1000=0 \\ 6.2831851999999x^2+125.664x-1000=0 \\ a=6.283185;b=125.664;c=-1000 \\ D=b^2-4ac=125.664^2-4\cdot 6.283185\cdot (-1000)=40924.107302999 \\ D>0 \\ {x}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-125.66\pm \sqrt{40924.11}}{12.56637} \\ {x}_{1,2}=-10\pm 16.09829 \\ {x}_1=6.098290152 \\ {x}_2=-26.098290152 \\ {r}_2=x_1=6.0983\doteq 6.0983\text{mm} \\ \text{ Verifying Solution: } \\ {S}_3=2\pi \cdot r_2\cdot (r_2+h_2)=2\cdot 3.1416\cdot 6.0983\cdot (6.0983+20)=1000{\text{cm}}^2 \\ {S}_3=S_2 \end{gathered}$$

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