what is the logic in this operation

[eddy2017]

Hi, this is a similar problem to another one i posted yesterday. I tried to follow the pattern and then decided at the end to set up a proportion to determine the amount time needed to pump the water out...

A basement has a 24-foot by 32-foot rectangular floor. The basement is flooded with water to a depth of 18 inches. Three pumps are used to pump the water out of the basement. Each pump will pump 8 gallons of water per minute. If a cubic foot of water contains 7.5 gallons, how many minutes will it take to pump all of the water out of the basement using the three pumps?

following lev88 top down approach
Data

Measurements= A = L+W

24 * 32=768 ft^2



Note that 18 in = 1.5 ft 18in/12in=1.5 ft

h=1.5ft

FORMULA

Time = (size of job)/rate.

Find volume( that is the size of the job)

So....the volume of the basement

V = 24 x 32 x 1.5 = 1152 ft^3

1152 ft^3 can hold 8617.558 gallons of water
to pump all these gallons of water out I have 3 pumps pumping out 8 gallons per minute.

Pump #1

Pumpin’ 8 gallons every minute in 60 minutes (1h) pumps 480 gallons/h

Pump#2

480 gallons/h

Pump#3

480 gallons/ h

480 gallons * 3 pumps

=1440 gallons every hour(60 minutes)

here i decide to go with a proportion.
dimensions

TiME WATER

in 60 minutes-----------1440 gallons are pumped
in x(t) ------------------8617.558

359 minutes /60
= 359 minutes

359min /60min= approximately 6 hours
that was my work. I just want you to confirm what i did, if it is correct.

=======================
another solution from another guy was this one:
Note that 18 in = 1.5 ft
So....the cubic feet of water in the basement = 24 x 32 x 1.5 = 1152 ft^3 ( same was as mine to find the volume)

Each pump pumps out 8 gal of water / minute (correct)

So....the number of cubic feet they pump out each minute = 8 / 7.5 = 16/15 ft^3 min (?????) here!

So...three of them pump out 3 (16/15) = 48/15 = 16/5 ft^3 min

So.....the number of minutes to pump out the basement is 1152 / (16/5) = 1152 * 5 / 16 = 360 minutes = 6 hours


I do not understand why he/she equals 8/7.5 to 16/15 ft^3 ?????? why, what is the logic here?

thank you all
eddy

 

[Dr.Peterson]

I do not understand why he/she equals 8/7.5 to 16/15 ft^3 ?????? why, what is the logic here?

Rather than just use a calculator to divide 8 by 7.5, they want to use fractions, to keep things exact. To eliminate the decimal, they chose to double the numerator and denominator: `8/7.5=(8*2)/(7.5*2)=16/15.............................`(fixed typo)

 

[eddy2017]

Rather than just use a calculator to divide 8 by 7.5, they want to use fractions, to keep things exact. To eliminate the decimal, they chose to double the numerator and denominator: `8/7.5=(8*2)/(7.5*2)=16/15`

= 16/15
I got it Doc, thanks a lot.
One question, does it always work that way when you want to get rid of the decimal form?. can i multiply both numerator and denominator by 2?.

 

[Dr.Peterson]

= 16/15
I got it Doc, thanks a lot.
One question, does it always work that way when you want to get rid of the decimal form?. can i multiply both numerator and denominator by 2?.

Did you try some other examples to see what happens? That's how you learn math well: not just asking, but trying.

If the decimal is .5, then doubling does the trick. Otherwise, in the worst case, you have to multiply by a power of ten. For example, to simplify 3.4/5.27, I'd multiply by 100 and get 340/527. Then I'd see if that can be simplified.

 

[eddy2017]

Did you try some other examples to see what happens? That's how you learn math well: not just asking, but trying.

If the decimal is .5, then doubling does the trick. Otherwise, in the worst case, you have to multiply by a power of ten. For example, to simplify 3.4/5.27, I'd multiply by 100 and get 340/527. Then I'd see if that can be simplified.

Thanks, got it. it isn't worth the effort if there is a calculator to be had. But it is sure good to know that. thanks again. and yes, trying is the best way to learn. I am doing it believe it or not. every day, every spare moment, I have a math exercise in front of me, and you can see me in your mind's eye noodling over it. And you all inspire me, you all have been an inspiration to me. I have boatloads of respect for knowledge.

 

[topsquark]

I have boatloads of respect for knowledge.

Then listen to Dr.Peterson about this... approximate expressions like decimals are for the Physical Sciences... exact expressions like fractions or radicals are for Math. Give not using calculators for your Math problems some more thought.

-Dan

 

[eddy2017]

Then listen to Dr.Peterson about this... approximate expressions like decimals are for the Physical Sciences... exact expressions like fractions or radicals are for Math. Give not using calculators for your Math problems some more thought.

-Dan

Thanks for the info and advice. I am a newbie. sometimes i talk out of my ignorance. thanks again. advice taken. Not in vain math has been termed an exact science. thankS for taking the time to remind me of that.

 

[JeffM]

Actually, even in applied work, you need to be careful of calculators because of something called “error build up.” If you are doing lots of calculations then even small errors can accumulate until they are significant.

 

[Dr.Peterson]

Hi, this is a similar problem to another one i posted yesterday. I tried to follow the pattern and then decided at the end to set up a proportion to determine the amount time needed to pump the water out...

A basement has a 24-foot by 32-foot rectangular floor. The basement is flooded with water to a depth of 18 inches. Three pumps are used to pump the water out of the basement. Each pump will pump 8 gallons of water per minute. If a cubic foot of water contains 7.5 gallons, how many minutes will it take to pump all of the water out of the basement using the three pumps?

Just to keep the discussion going, here is what I would have done:
depth: 18 inches = 18/12 = 1.5 feet
volume: 24*32*1.5 = 1152 cubic feet of water
convert to gallons: 1152 ft³*7.5gal/ft³ = 8640 gal

flow rate: 3 pumps, each 8 gal/min = 24 gal/min

time: 8640 gal/(24 gal/min) = 360 min

The only decimal needed was exact, so there's no inaccuracy to worry about. (Not that that affected what I did.)

Such nice, neat numbers are a sign that this isn't a real problem!

EDIT: And, yes, the use of the approximation 7.5 without mention that it is accurate only to the nearest tenth is one of those (falsely) nice numbers!

 

[Deleted member 4993]

Just to keep the discussion going, here is what I would have done:
depth: 18 inches = 18/12 = 1.5 feet
volume: 24*32*1.5 = 1152 cubic feet of water
convert to gallons: 1152 ft³*7.5gal/ft³ = 8640 gal

flow rate: 3 pumps, each 8 gal/min = 24 gal/min

time: 8640 gal/(24 gal/min) = 360 min

The only decimal needed was exact, so there's no inaccuracy to worry about. (Not that that affected what I did.)

Such nice, neat numbers are a sign that this isn't a real problem!

Just to spoil things:

1 cft = 7.48052 gallons ± few drops

 

[eddy2017]

GREAT!. Thanks!!!.

 

[eddy2017]

Just to keep the discussion going, here is what I would have done:
depth: 18 inches = 18/12 = 1.5 feet
volume: 24*32*1.5 = 1152 cubic feet of water
convert to gallons: 1152 ft³*7.5gal/ft³ = 8640 gal

flow rate: 3 pumps, each 8 gal/min = 24 gal/min

time: 8640 gal/(24 gal/min) = 360 min

The only decimal needed was exact, so there's no inaccuracy to worry about. (Not that that affected what I did.)

Such nice, neat numbers are a sign that this isn't a real problem!

EDIT: And, yes, the use of the approximation 7.5 without mention that it is accurate only to the nearest tenth is one of those (falsely) nice numbers!

Neat!!!
But if truth is to be honored, that was the way i did it. Remember that i was comparing my top down approach with whomever ever did it the other way, which i thought a little bit cumbersome.