Question

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?

Answer 1

plz helppp!! i

Answer 2

I will help you if you'll write a formula for the surface area for a cylinder.

Answer 3

SA= 2πr^2 + 2πrh

Answer 4

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.

\[\large A=2\pi r^2+2\pi r h \qquad \rightarrow \qquad A=2\pi\left(\dfrac{1}{3}h\right)^2+2\pi\left(\dfrac{1}{3}h\right)h\]

Answer 5

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

Answer 6

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.

Answer 7

can u waalk me throught the whole thing? im having trouble solving

Answer 8

\[\large 402=2\pi \color{orangered}{r}^2+2\pi \color{orangered}{r} h \qquad \rightarrow \qquad 402=2\pi\color{orangered}{\left(\dfrac{1}{3}h\right)}^2+2\pi\color{orangered}{\left(\dfrac{1}{3}h\right)}h\]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.
\[\large 402=2\pi\left(\dfrac{1}{3}h\right)^2+\color{royalblue}{\dfrac{2\pi}{3}h^2}\]

Answer 9

\[\large 402=\color{#CC0033}{2\pi\left(\dfrac{1}{3}h\right)^2}+\dfrac{2\pi}{3}h^2\]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,\[\large 402=\color{#CC0033}{\dfrac{2\pi}{9}h^2}+\dfrac{2\pi}{3}h^2\]

Answer 10

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,\[\large 402=\dfrac{8\pi}{9}h^2\]

Answer 11

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,\[\large h^2=\dfrac{9\cdot402}{8\pi}\]

Answer 12

Punch that number into your calculator, and take the square root, and voila!!

Answer 13

Sorry if that was too slow, I like to add the colors. I think they make it a little easier to read.

Answer 14

Ok thank you!!