Cylinder diameter height

A cylinder has the same diameter as its height. Calculate these data if the surface is 200 cm square. Report the results to the nearest millimeter.

Final Answer:

d =  6.5 cm
h =  6.5 cm

Step-by-step explanation:

$$\begin{gathered} S=200{\text{cm}}^2 \\ d=h \\ d=2r \\ r=d/2=h/2 \\ S=2\pi r^2+2\pi rh \\ S=2\pi (h/2{)}^2+2\pi (h/2)h \\ S=\pi h^2/2+\pi h^2 \\ h=\sqrt{\frac{S}{\pi \cdot (1+1/2)}}=\sqrt{\frac{200}{3.1416\cdot (1+1/2)}}\doteq 6.5147\text{cm} \\ d=h=6.5147=6.5\text{cm} \end{gathered}$$

$$\begin{gathered} \text{ Verifying Solution: } \\ r=h/2=6.5147/2\doteq 3.2574\text{cm} \\ {S}_2=2\pi \cdot r^2+2\pi \cdot r\cdot h=2\cdot 3.1416\cdot 3.2574^2+2\cdot 3.1416\cdot 3.2574\cdot 6.5147=200{\text{cm}}^2 \\ h=6.5147=6.5\text{cm} \end{gathered}$$

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