Find the unknowns and assign them variables and explain their meaning. Set up a system of equations to model this scenario. How long will it take for the bucket to have the exact same amount of water?

Question

You have two 5-gallon buckets. One is filled with water, but has a slow leak, leaking out water at 7 ounces per minute. The other is empty but is being used to catch water from a leaky faucet at rate of 4 ounces per minute. Assume they both start at the same time. Hint: There are 128 ounces in a gallon

NOT ENOUGH SPACE FOR ME TO PUT THE REST OF THE QUESTIONS SO SOME OF THEM ARE HERE: \/

How much water will there be(in gallons)?

Answer 1

First, since the water’s rate of change is measured in OUNCES per minutes, we must change our gallons to ounces.

1 gallon = 128 oz

5 gallons = (128)(5) oz = 640 oz

First bucket starts with 5 gallons/ 640 oz of water and is leaking out at a rate of 7 oz/minute.

This can be expressed as;

V₁ = 640 - 7t

where t = minutes and V₁ = volume of water in the first, leaky bucket.

Second bucket starts with 0 gallons/0 oz of water and is gaining water at a rate of 4 oz/minute

This can be expressed as;

V₂ = 0 + 4t

where t = minutes again and V₂ = volume of water in the second bucket that is gaining water.

Since we want the two volumes of water to be the same, we can set up the following equation;

V₁ = V₂

Then, substitute

640 - 7t = 0 + 4t

Solve for t

640 - 7t = 4t

+7t +7t

640 = 11t

640 = 11t

11 11

640 = t

11

t ~ 58.18

So, since t represents minutes, it will take 58.18 minutes for each bucket to have the same volume.

To find that volume, plug 58.18 or 640/11 into either equation (since the volumes will be the same).

(Note: I personally plug in the fraction since it is more precise.)

V₂ = 4(640/11)

= 2560/11 oz

~ 232.73 oz

So, we found that after about 58.18 minutes, the two buckets will have the same volume of 232.73 OUNCES.

However, the question asks for the answer in gallons so we must convert it back.

To do this, we can set up the ratio;

1 gallon = 128 oz

X gallons 232.73 oz

Cross multiply to get;

232.73 = 128x

232.73 = 128x

128 128

x ~ 1.82 gallons

Finally, we found that after about 58.18 minutes, the two buckets will have the same volume of about 1.82 gallons.

Answer 2

m represents the number of minutes

640 - 7m = 4m (5 gal = 640 oz)

Can you solve for m and answer?