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 #2
avatar+9460 
+1

We could do it pretty quickly by applying the exponent rules....but if that's confusing or if you're not super comfortable using exponent rules, we can go through the first part like this:

 

 

First let's just look at  (pq3)3  :

 

     \({(pq^3)^3}\ =\ (pq^3)(pq^3)(pq^3)\)

 

     Imagine replacing every q3 with qqq. How many p's would there be? 3. How many q's would there be? 9. So...

 

     \( {(pq^3)^3}\ =\ p^3q^9\)

 

 

Next let's look at  (4p2q)2  :

 

     \((4p^2q)^2\ =\ (4p^2q)(4p^2q)\)

 

     In this case, how many p's are there? 4. How many q's are there? 2. How many 4's are there? 2. So...

 

     \({(4p^2q)^2}\ =\ 16p^4q^2\)

 

 

Now let's look at  (2pq2)3  :

 

     \((2pq^2)^3\ =\ (2pq^2) (2pq^2) (2pq^2)\)

 

     Again let's ask how many p's, q's , and 2's there are to get that...

 

     \( {(2pq^2)^3}\ =\ 8p^3q^6\)

 

 

And so...

 

\(\dfrac{(pq^3)^3(4p^2q)^2}{(2pq^2)^3}\ =\ \dfrac{(p^3q^9)(16p^4q^2)}{8p^3q^6}\)

 

There are 3 p's in the first set of parenthesees in the numerator and 4 p's in the second. That makes 7 p's total in the numerator. There are 9 q's in the first set of parenthesees in the numerator and 2 q's in the second. That makes 11 q's total in the numerator. I'm not going to write explanations for these next steps... so feel free to ask if anything confuses you!!

 

\(\dfrac{(pq^3)^3(4p^2q)^2}{(2pq^2)^3}\ =\ \dfrac{(p^3q^9)(16p^4q^2)}{8p^3q^6}\\~\\ \phantom{\dfrac{(pq^3)^3(4p^2q)^2}{(2pq^2)^3}}\ =\ \dfrac{16p^7q^{11}}{8p^3q^6} \\~\\ \phantom{\dfrac{(pq^3)^3(4p^2q)^2}{(2pq^2)^3}}\ =\ \dfrac{16}{8}\cdot\dfrac{p^7}{p^3}\cdot\dfrac{q^{11}}{q^6}\\~\\ \phantom{\dfrac{(pq^3)^3(4p^2q)^2}{(2pq^2)^3}}\ =\ 2\cdot p^4\cdot q^5\\~\\ \phantom{\dfrac{(pq^3)^3(4p^2q)^2}{(2pq^2)^3}}\ =\ 2p^4q^5\)

 

Anytime you don't know how to handle an exponent, it can help to rewrite it as repeated multiplication....I used to always do this until I became super familiar with exponent rules...and I still have to do it sometimes!!

 

Hope this helps!

Nov 18, 2020
 #1
avatar+9460 
0
Nov 18, 2020
 #5
avatar+9460 
0
Nov 18, 2020
 #1
avatar+9460 
+2

We can draw a right triangle where the hypotenuse is the slant height and the legs are the radius and the height of the cone. Then if we can find the slant height and the radius of the base of the cone, we can use the Pythagorean theorem to find the height.

 

The slant height of the cone is the same as the radius of the sector, which is 18

 

Now we need to find the radius of the base of the cone.

 

The circumference of the base is the same as the length of the sector, which we can find like this:

 

\(\frac{\text{arc length of sector}}{\text{circumference of circle}}=\frac{288^\circ}{360^\circ} \\~\\ \frac{\text{arc length of sector}}{2\pi(\text{radius of circle})}=\frac{288^\circ}{360^\circ} \\~\\ \frac{\text{arc length of sector}}{2\pi(18)}=\frac{288^\circ}{360^\circ} \\~\\ \text{arc length of sector}=\frac{288^\circ}{360^\circ} \cdot 2\pi(18)\\~\\ \text{arc length of sector}=28.8\pi\)   (The circle here is the circle from which the sector is taken)

 

And so

 

circumference of base   =   28.8 pi

 

radius of base   =   circumference of base  /  (2 pi)   =   28.8 pi / (2 pi)   =   14.4

 

Now that we know the radius of the base of the cone and we know the slant height of the cone, we can use the Pythagorean theorem to find the height of the cone.

 

(radius of base)2 + (height)2   =   (slant height)2

                                                                               Plug in what we know...

14.42  +  ( height )2   =   182

                                                        Subtract  14.42  from both sides

( height )2   =   182 - 14.42

                                                        Simplify the right side

( height )2   =   116.64

                                                        Take the positive square root of both sides

height   =   10.8

Nov 15, 2020