Question

A pile of earth standing on flat ground by an abandoned mine has height 15 meters. The ground is the xy-plane; the origin is directly below the top of the pile and the z-axis is upward. The cross-section at height z is x^2+y^2=15-z for 0 12 years ago

Answer 1

Can you please draw the x,y, and z axis ?

Answer 2

i dont have the drawing of it

Answer 3

Wouldn't it just be when z=0?

Answer 4

Yea all points along that circle when z=0

Answer 5

x^2+y^2=15-z

z=0 gives

x^2+y^2=15
So it's a circle, of radius sqrt15

Answer 6

oh wow i feel stupid! theres other questions that go with this that i need help with but my book is at home so ill post the other questions later tonight! thank you dear! @agent0smith

Answer 7

What is the area of the base of the pile? @agent0smith

Answer 8

So it's a circle, of radius sqrt15

Use that to find area Meg :P

Answer 9

i have no common sense tonight lol thank youuuu

Answer 10

imagine the pile sliced horizontally. What is the area, A(z), of the slice at height z? What are its units?

Answer 11

Hmm. Will have to think about that.

Answer 12

i cant figure it out...did u?:/

Answer 13

Oh it's just horizontal circles!

Answer 14

x^2+y^2=15-z

the radius is sqrt(15-z)

So the area is just using that, and a circle's area!

Answer 15

i dont get what you mean:/

Answer 16

Radius of x^2+y^2=15
is squrt15 right?

So the radius of
x^2+y^2=15-z
is sqrt(15-z)

Answer 17

ooooh i see it now!

Answer 18

:)

Answer 19

so what does it mean by how many units?

Answer 20

I knew you would, you smart girl :D

Answer 21

They're all in m, so m^2

Answer 22

where did m come from? lol

Answer 23

Now you know the radius, use it for... What is the area, A(z), of the slice at height z?

Answer 24

Area of a circle

Answer 25

pi(15-z)

Answer 26

:)

Answer 27

okay so now if delta(z) is the thickness of the slice at height z, what is represented by A(z) delta(z)?

Answer 28

thanks for helping me with this problem btw

Answer 29

Anytime!

Answer 30

Area times thickness will give what?

Think of a cylinder. What does the area of its base times height give you?

Answer 31

how thick the cylinder is?

Answer 32

No, for any prism or cylinder, area of base times height gives its...?

Answer 33

The "thickness" of the slice is the same thing as its height.

Answer 34

How do you find the volume of a cylinder?

Answer 35

volume.

Answer 36

So the area of a slice times its thickness gives the slice's...?

Answer 37

volume

Answer 38

:)

Answer 39

okay ive got one more question for this question lol

evaluate and interpret the integral from 0 to 15 of A(z)dz. what are its units?

Answer 40

That'd be the volume of the whole hill thingy. in m^3