Right Cone

[vampirewitchreine]

15. The volume of a right cone is $\displaystyle 144 \pi$ and the area of its base is $\displaystyle 36\pi$. Find the radius, height and slant height of the cone.



What I currently have is the radius.

Since the area of a circle is: $\displaystyle 2\pi r^2$
I did:
$\displaystyle \frac {36 \pi}{2 \pi}= \frac {2 \pi r^2}{2 \pi}$
$\displaystyle \sqrt {18} = \sqrt {r^2}$
$\displaystyle 4.24 ~\approx r$


What I've started to work out as I've been typing:

And the area of a cone is: $\displaystyle \frac {1}{3} Bh \Rightarrow \frac{1}{3} \pi r^2 h$
So I have
$\displaystyle 114 \pi = \frac{1}{3} \pi r^2 h$
$\displaystyle 114 \pi = \frac{1}{3} \pi (4.24)^2 h$
$\displaystyle 114 \pi = \frac{1}{3} \pi (17.98) h$
(And now I'm at a stand still.... could someone please help me work from here? If you could help me find the height, I can see if I can find the slant on my own.)

 

[Deleted member 4993]

15. The volume of a right cone is $\displaystyle 144 \pi$ and the area of its base is $\displaystyle 36\pi$. Find the radius, height and slant height of the cone.



What I currently have is the radius.

Since the area of a circle is: $\displaystyle 2\pi r^2$ <<<< No area of a circle = π * r²

Now fix rest of the problem
I did:
$\displaystyle \frac {36 \pi}{2 \pi}= \frac {2 \pi r^2}{2 \pi}$
$\displaystyle \sqrt {18} = \sqrt {r^2}$
$\displaystyle 4.24 ~\approx r$


What I've started to work out as I've been typing:

And the area of a cone is: $\displaystyle \frac {1}{3} Bh \Rightarrow \frac{1}{3} \pi r^2 h$ <<< B is given to you B = 36 * π

144 * π = 1/3 * (36 * π) * h

h = (144 * π)/[1/3 * (36 * π)] = 12

Why make life harder than it needs to be!!!

So I have
$\displaystyle 114 \pi = \frac{1}{3} \pi r^2 h$
$\displaystyle 114 \pi = \frac{1}{3} \pi (4.24)^2 h$
$\displaystyle 114 \pi = \frac{1}{3} \pi (17.98) h$
(And now I'm at a stand still.... could someone please help me work from here? If you could help me find the height, I can see if I can find the slant on my own.)

.

 

[soroban]

Hello, vampirewitchreine!

15. The volume of a right cone is $\displaystyle 144 \pi$ and the area of its base is $\displaystyle 36\pi$.
Find the radius, height and slant height of the cone.

Since the area of a circle is: $\displaystyle 2\pi r^2$ . No!

We know that: .$\displaystyle V \:=\:\frac{1}{3}Bh$

So we have: .$\displaystyle 144\pi \:=\:\frac{1}{3}(36\pi)h \quad\Rightarrow\quad \boxed{h \,=\,12}$

The area of the circular base is: /$\displaystyle B \,=\,\pi r^2$
. . Hence: .$\displaystyle 36\pi \,=\,\pi r^2 \quad\Rightarrow\quad \boxed{r\,=\,6}$

 

[vampirewitchreine]

Okay, so I messed up the formula and mashed up Circumference with the Area.

My corrected equations did end up with the exact same results as soroban (thank you)

$\displaystyle \frac {36 \pi}{\pi}= \frac {\pi r^2}{\pi}$
$\displaystyle \sqrt {36} = \sqrt {r^2}$
$\displaystyle r = 6$


$\displaystyle 114 \pi = \frac{1}{3} \pi r^2 h$
$\displaystyle 114 \pi = \frac{1}{3} (36 \pi)h$
$\displaystyle 114 \pi = (12 \pi) h$
$\displaystyle \frac {144 \pi}{12 \pi} = \frac {12 \pi h}{12 \pi}$
$\displaystyle 12 = h$



So my slant height comes from Pythagorean Theorem:
$\displaystyle s^2 = 6^2 + 12^2$
$\displaystyle s^2 = 36 + 144$
$\displaystyle \sqrt {s^2} = \sqrt {180}$
$\displaystyle s ~\approx 13.42$



And Dennis..... isn't that suppose to be You're? Not Yer? Thought text talk was a no-no in the forum





On another note. Lets say that I have the Area of a square pyramid and the height. Would I divide out the height first then multiply both sides by 3 and get what the area of the base is? (And square root that answer to get what one length is?)

 

[Mrspi]

Okay, so I messed up the formula and mashed up Circumference with the Area.

My corrected equations did end up with the exact same results as soroban (thank you)

$\displaystyle \frac {36 \pi}{\pi}= \frac {\pi r^2}{\pi}$
$\displaystyle \sqrt {36} = \sqrt {r^2}$
$\displaystyle r = 6$


$\displaystyle 114 \pi = \frac{1}{3} \pi r^2 h$
$\displaystyle 114 \pi = \frac{1}{3} (36 \pi)h$
$\displaystyle 114 \pi = (12 \pi) h$
$\displaystyle \frac {144 \pi}{12 \pi} = \frac {12 \pi h}{12 \pi}$
$\displaystyle 12 = h$



So my slant height comes from Pythagorean Theorem:
$\displaystyle s^2 = 6^2 + 12^2$
$\displaystyle s^2 = 36 + 144$
$\displaystyle \sqrt {s^2} = \sqrt {180}$
$\displaystyle s ~\approx 13.42$



And Dennis..... isn't that suppose to be You're? Not Yer? Thought text talk was a no-no in the forum





On another note. Lets say that I have the Area of a square pyramid and the height. Would I divide out the height first then multiply both sides by 3 and get what the area of the base is? (And square root that answer to get what one length is?)

You have the AREA of a square pyramid....

What would be the basic formula you would start with?

From what I see, you seem to be confusing AREA with VOLUME.

Please clarify...

 

[vampirewitchreine]

You have the AREA of a square pyramid....

What would be the basic formula you would start with?

Area for a square pyramid is $\displaystyle \frac {1}{2}ps+B$

From what I see, you seem to be confusing AREA with VOLUME.

Please clarify...

>.< I seem to be doing this a lot lately...... So now I need to find the slant as well as the area of the base.....


The exact problem (from the book) is:
A pyramid (assuming that it's a square pyramid due to that's what this section is on) has a height of 15 ft and a volume (my bad, it was volume..... I posted wrong.) of 450 ft³. What is the area of the base of the pyramid?

Correct Hazel: just wanted to c if u'd notice :idea:

So most students just glance over it and don't think anything of it?

Oh, yeah.... and see.. you'd