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Alanna goes to the pet store to buy fish for her new 15-gallon fish tank. Each large fish costs $8 and requires 2 gallons of water. Each small fish costs $3 and requires 0.5 gallon of water. She has $100 to spend on fish at the pet store. Which combinations of large and small fish can Alanna purchase for her fish tank?

User Chenell
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1 Answer

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Let's set up the cost constraint inequality.

x = number of large fish, some nonnegative integer

y = number of small fish, some nonnegative integer

8x = cost of just the large fish only

3y = cost of just the small fish only

8x+3y = total cost

The total cost must be $100 or smaller, so Alanna keeps to her budget.

We form the inequality 8x+3y ≤ 100 as the cost constraint.

Because x and y are nonnegative, this means x ≥ 0 and y ≥ 0

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Now let's form the water constraint inequality.

x = number of large fish

y = number of small fish

2x = amount of water needed for the large fish only

0.5y = amount of water needed for the small fish only

2x+0.5y = total amount of water needed

This total must be 15 gallons or fewer.

2x+0.5y ≤ 15

Let's multiply both sides by 2 to make that 0.5 turn into a whole number.

2x+0.5y ≤ 15

2*(2x+0.5y) ≤ 2*15

4x+y ≤ 30

This represents the water constraint.

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Here are the inequalities we found

8x+3y ≤ 100

4x+y ≤ 30

x ≥ 0

y ≥ 0

The first two items represent the cost constraint and water constraint in that order. The last two items make sure that x and y can't be negative.

Valid (x,y) solutions will be found in the shaded region of that system of inequalities. Points on the boundary are included in the shaded region due to the "or equal to" as part of each inequality sign.

Below is the graph of this system of inequalities. I used Desmos to make the graph. GeoGebra is another good choice. Let me know if you need to make the graph by hand rather than use technology.

The purple shaded region represents all possible (x,y) solutions.

For example, (5,5) is in that shaded region. I've marked it in the drawing below. This means x = 5 large fish and y = 5 small fish can be bought. Alanna will stick to her budget and she'll have enough water to go around.

On the other hand, (5,25) is NOT in the shaded region. This means Alanna cannot buy x = 5 large fish and y = 25 small fish (she'll either go over budget or she won't have enough water, or both happens).

Alanna goes to the pet store to buy fish for her new 15-gallon fish tank. Each large-example-1
User Dave Snyder
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