You can make 7 gal of liquid fertilizer by mixing 8 tsp of powdered fertilizer with water. Represent the relation between the teaspoons of powder used and the gallons of fertilizer made using a​ table, an​ equation, and a graph. Is the amount of fertilizer made a function of the amount of powder​ used?

Question

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Part 1
Which table below represents the​ relation?
A.
Number of​ tsp, x
Number of​ gal, y
0
0
8
7
16
14
24
21
B.
Number of​ tsp, x
Number of​ gal, y
0
0
7
8
14
16
21
24
C.
Number of​ gal, x
Number of​ tsp, y
0
21
8
14
16
7
24
0

Answer 1

To represent the relation between the teaspoons of powdered fertilizer used (x) and the gallons of fertilizer made (y), we can analyze the data you've provided. According to the information given, 8 teaspoons of powdered fertilizer yield 7 gallons of liquid fertilizer.

Step 1: Determine the relationship

From the information given, we can calculate how many gallons can be made with different amounts of teaspoons.

Using the ratio from the provided information (8 tsp produces 7 gal), we can find the gallons corresponding to any multiple of 8 tsp:

1. 0 tsp yields: \( 0 \) gal (since no powder means no liquid)
2. 8 tsp yields: \( 7 \) gal
3. 16 tsp yields: Since 16 tsp = 2 × 8 tsp, \( 7 \times 2 = 14 \) gal
4. 24 tsp yields: Since 24 tsp = 3 × 8 tsp, \( 7 \times 3 = 21 \) gal

So, the complete relationship can be summarized as:

• 0 tsp → 0 gal
• 8 tsp → 7 gal
• 16 tsp → 14 gal
• 24 tsp → 21 gal

Step 2: Create the table

Based on this analysis of the relationship, we can format a table (which corresponds to option A):

| Number of tsp, x | Number of gal, y | |-------------------|-------------------| | 0 | 0 | | 8 | 7 | | 16 | 14 | | 24 | 21 |

This corresponds to Table A from your options.

Step 3: Create the equation

Now we will derive the equation that describes this linear relationship.

The relationship can be interpreted as follows:

The number of gallons \( y \) can be expressed as a function of the number of teaspoons \( x \):

\[ y = \frac{7}{8}x \]

This equation tells us that for every 8 teaspoons of fertilizer (x), we produce 7 gallons of fertilizer (y).

Step 4: Graph the relationship

To graph this, you can plot the points (0,0), (8,7), (16,14), and (24,21) on a Cartesian coordinate system.

• The x-axis represents teaspoons of powdered fertilizer.
• The y-axis represents gallons of liquid fertilizer.

The points will form a straight line, starting from the origin and moving upward as \( x \) increases.

Step 5: Determine if it’s a function

Yes, the amount of fertilizer made is a function of the amount of powdered fertilizer used. In this case:

• For each input (amount of teaspoon powder), there is exactly one output (amount of gallons of fertilizer produced). This satisfies the definition of a function, which states that each input must correspond to exactly one output.

In summary:

1. The correct table representation is A.
2. The relationship can be described by the equation \( y = \frac{7}{8}x \).
3. The graph will be a straight line through the points derived.
4. The amount of fertilizer made is indeed a function of the amount of powder used.