Question

Help with Related rates?
A swimming pool is 12 meters long, 6m wide, 1m wide at the shallow end, and 3m deep at the deep end. Water is being pumped into the pool at 1/4 cubic meters per minute, and there is 1 meter of water at the deep end.
a. What percent of the pool is filled?
b. At what rate is the water level rising?

Answer 1

1 meter deep* at the shallow end

Answer 2

Wait one sec
http://m.imgur.com/w40HkBW
Does this help

Answer 3

Oh I see :O
by shallow they just mean the left edge of the pool
and it's a straight decline from there.

Answer 4

Yeah sorry

Answer 5

Lemme erase all my crap from before so we can rethink this :D

Answer 6

Ok, thanks so much.

Answer 7

the volume of the triangular prism is
\[V= \frac{1}{2} b *h *w\]

Answer 8

you know that the height must be 1/6 of the length since the height of the triangle is 2m and the length is 12

hence we could write this as
6h=l

Answer 9

the rate given is 1/4 m^3/min
which is volume per second
aka
dV/dt=.25 m^3/m

Answer 10

you know there is already 1 meter at the deep end or 1 meter deep

hence the length of the water must be 6m from the 6h=l relation above

hence you can calculate the total volume of water already existing in the pool and dub this number as the starting volume

Answer 11

now,
a is easy,

to find percentage, you need to calculate the initial volume of water
and then calculate the total volume of the pool

the total volume of the pool is the addition of a triangular prism along with a rectangular prism
these dimensions are already given

to get percentage, divide initial water volume by the total pool volume

Answer 12

any questions thus far?

Answer 13

No

Answer 14

Would the volume of it be 144?

Answer 15

volume of what? the initial volume of water or the volume of the pool?

Answer 16

Umm
The volume of the pool

Answer 17

Yes, the total volume of the pool would be 144 m^3

dont forget to include units

Answer 18

Right.

Answer 19

so what is the initial volume of water?

Answer 20

Im not sure how to find it.

Answer 21

you know the width of pool is 6 regardless of what you do, so we will ignore that for now

Answer 22

hence the dimensions of the triangular prism is 2x12

Answer 23

now, it states that the water is 1 meter deep

so what is the corresponding length of the triangle created by the water?

Answer 24

6

Answer 25

now, if you remember above, i stated, for the triangular portion of the pool

the length will always be 6 times the height

because any height you create will be a similar triangle to the big one

Answer 26

so yes, the length is 6m, the height is 1m

what is the corresponding volume created by the water?

Answer 27

6*6*1=36?

Answer 28

triangular prism

the area of the triangle times the width

(1/2 *6*1)*6

Answer 29

Oh
18

Answer 30

right,
you know the total volume of water and the total volume of the pool,

so what percentage of the pool is filled?

Answer 31

18/144
Is
.125
So 12.5%

Answer 32

yup

Answer 33

ok, now for part B

there will actually be 2 different answers for this, depending on whether the triangular prism is being filled or is already filled up

make sense?

Answer 34

or, let me ask,
will the water level of a triangular prism rise at the same rate of the water level of a rectangular prism?

Answer 35

ok, anyways to do the related rates problem

first identify what variables you are given and what variables you are solving for

dV/dt is given- this is the rate at which water is being pumped into the pool

dh/dt is the rate at which the water level is increasing

Answer 36

so, in order to get a equation with dV/dt, this means we would need to take the derivative of the volume equation with respect to time

so, for the triangular prism, we know
V= 1/2 *l *h*w

Answer 37

now, it is VERY importatnt to note which variables are constant
in this case, only w is constant at 6m and l is dependent on h

we know l=1/6 h
so we substitute that in
V=1/12 h*h*w

Answer 38

then, you take the derivative with respect to time to get an equation relating dV/dt and dh/dt

Answer 39

Sorry :/
Ive got to go
Thank you though, very much.