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julianahe

  • 2 years ago

A cylinder has a surface area of 402 cm^2. The height is three times greater than the radius. What is the height of the cylinder?

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  1. julianahe
    • 2 years ago
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    plz helppp!! i

  2. klimenkov
    • 2 years ago
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    I will help you if you'll write a formula for the surface area for a cylinder.

  3. julianahe
    • 2 years ago
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    SA= 2πr^2 + 2πrh

  4. zepdrix
    • 2 years ago
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    |dw:1359238717297:dw|They told us that \(h=3r\), which we can rewrite as \(\dfrac{1}{3}h=r\). From here, we can substitute this \(h\) value in place of all the \(r\)'s in the formula. \[\large A=2\pi r^2+2\pi r h \qquad \rightarrow \qquad A=2\pi\left(\dfrac{1}{3}h\right)^2+2\pi\left(\dfrac{1}{3}h\right)h\]

  5. zepdrix
    • 2 years ago
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    Then plug the value for surface area that they provided,\[\large 402=2\pi\left(\dfrac{1}{3}h\right)^2+2\pi\left(\dfrac{1}{3}h\right)h\]

  6. zepdrix
    • 2 years ago
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    From here it's just a few algebra steps to solve for h. Lemme know if you're confused by any of that, or need to see more steps.

  7. julianahe
    • 2 years ago
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    can u waalk me throught the whole thing? im having trouble solving

  8. zepdrix
    • 2 years ago
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    \[\large 402=2\pi \color{orangered}{r}^2+2\pi \color{orangered}{r} h \qquad \rightarrow \qquad 402=2\pi\color{orangered}{\left(\dfrac{1}{3}h\right)}^2+2\pi\color{orangered}{\left(\dfrac{1}{3}h\right)}h\]So we found or relationship with r and h and made the replacement. Now let's try to solve for h.\[\large 402=2\pi\left(\dfrac{1}{3}h\right)^2+\color{royalblue}{2\pi\left(\dfrac{1}{3}h\right)h}\]Let's work on the blue part first. \[\large 402=2\pi\left(\dfrac{1}{3}h\right)^2+\color{royalblue}{\dfrac{2\pi}{3}h^2}\]

  9. zepdrix
    • 2 years ago
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    \[\large 402=\color{#CC0033}{2\pi\left(\dfrac{1}{3}h\right)^2}+\dfrac{2\pi}{3}h^2\]Now to simplify the red term, make sure you square both the h and the 1/3.\[\large 402=\color{#CC0033}{2\pi\left(\dfrac{1}{9}h^2\right)}+\dfrac{2\pi}{3}h^2\]Which simplifies to,\[\large 402=\color{#CC0033}{\dfrac{2\pi}{9}h^2}+\dfrac{2\pi}{3}h^2\]

  10. zepdrix
    • 2 years ago
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    We have a couple of fractions, let's get a common denominator, multiplying the second term by \(\dfrac{3}{3}\).\[\large 402=\dfrac{2\pi}{9}h^2+\color{#996666}{\dfrac{3}{3}}\cdot\dfrac{2\pi}{3}h^2\]Giving us,\[\large 402=\dfrac{2\pi}{9}h^2+\dfrac{6\pi}{9}h^2\]Adding these terms together gives us,\[\large 402=\dfrac{8\pi}{9}h^2\]

  11. zepdrix
    • 2 years ago
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    To get rid of the fraction on the right, we'll multiply both sides by it's reciprocal.\[\large \color{#662FFF}{\left(\dfrac{9}{8\pi}\right)}402=\dfrac{8\pi}{9}h^2\color{#662FFF}{\left(\dfrac{9}{8\pi}\right)}\]We can cancel the fractions on the right,\[\large \color{#662FFF}{\left(\dfrac{9}{8\pi}\right)}402=\cancel{\dfrac{8\pi}{9}}h^2\cancel{\color{#662FFF}{\left(\dfrac{9}{8\pi}\right)}}\]Giving us,\[\large h^2=\dfrac{9\cdot402}{8\pi}\]

  12. zepdrix
    • 2 years ago
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    Punch that number into your calculator, and take the square root, and voila!!

  13. zepdrix
    • 2 years ago
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    Sorry if that was too slow, I like to add the colors. I think they make it a little easier to read.

  14. julianahe
    • 2 years ago
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    Ok thank you!!

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