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?

1. julianahe

plz helppp!! i

2. klimenkov

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

3. julianahe

SA= 2πr^2 + 2πrh

4. zepdrix

|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

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

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

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

8. zepdrix

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

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

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

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

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

13. zepdrix

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

14. julianahe

Ok thank you!!