Question

The conical hat shown here is a traditional sun-protective straw hat originating in eastern Asia. What is the approximate outer surface area of one of these hats that has a radius of 9 inches and a slant height of inches? Keep in mind that there is an opening in the hat where the base surface would normally be. Show the steps of your solution and explain your work.

Answer 1

Slant height is 12 in *****

Answer 2

@AnimalAin

Answer 3

how many inches is the slant height?

Answer 4

do you know the formula for surface area of a cone?

Answer 5

sa cone : pie x r x s + pie x radius squared.

Answer 6

slant height is 12 inches .

Answer 7

you first need to find the height of the cone using the Pythagorean Theorem

Answer 8

\[a ^{2}+b ^{2}=c ^{2}\]
a=9, b=height, c=12

Answer 9

solve for b

Answer 10

7.9 ?

Answer 11

7.94 just to be safe-oops you didn't need to find the height haha that's for the volume..sorry!
Now plug everything into the SA formula
\[\pi(9)(12)+\pi(9)\]

Answer 12

\[\pi(9)^{2}\]

Answer 13

First find it in terms of pi, it will be easier

Answer 14

so i didnt need to do the pythagoream theorm ?

Answer 15

sa = 367.5

Answer 16

leave it as \[189\pi\] for now
we have to subtract the area of the base now

Answer 17

\[189\pi-\pi(9)^{2}\]

Answer 18

108 pie ?

Answer 19

@J.L. ??

Answer 20

Yeah, you're right

Answer 21

since a hat doesn't have a circular base, it's open

Answer 22

so its 367.5 for the outer surface area?

Answer 23

http://www.google.com/search?q=108pi&ie=utf-8&oe=utf-8&aq=t

Answer 24

Your answer should be 339.292

Answer 25

Consider it approximated by a triangle that is 12 tall and has a base of 18 pi. That might not get you a ten-digit correct answer, but will give a satisfactory approximation.

Answer 26

By the way, if you say the cut and flattened hat forms a sector of a 12 inch radius circle that is three fourths of the circle. The area of that sector is 108 pi, exactly the same as my approximation above.

Answer 27

???