Question

The table below shows the cost of downloading songs from a website. Number of Songs Total Cost 14 $8.96 16 $10.24 18 $11.52 If c represents the total cost in dollars and cents for any number of songs downloaded, s, write a proportional equation for c in terms of s that matches the S, context.

Answer 1

The proportional equation that matches the context is: c ≈ 0.64s

From the table, we can see that the cost increases proportionally with the number of songs downloaded. This means we can represent the relationship with a proportional equation of the form:

c = ks

where:

c is the total cost in dollars and cents (dependent variable)

s is the number of songs downloaded (independent variable)

k is the constant of proportionality

To find the constant of proportionality, we can use any of the data points from the table. Let's use the first data point, where 14 songs cost $8.96:

8.96 = k * 14

Solve for k:

k = 8.96 / 14

k ≈ 0.64

Therefore, the proportional equation that matches the context is:

c ≈ 0.64s

This equation tells us that the cost in dollars and cents for downloading any number of songs (s) can be found by multiplying the number of songs by the constant of proportionality, which is approximately 0.64.

Answer 2

Given:

c = total cost.

s = number of songs.

To solve this question, follow the steps below.

Step 01: Find 2 points (s, c).

You can choose 2 points from the table.

Let's choose (14, 8.96) and (16, 10.24).

Step 02: Substitute one point in the general equation.

Given the general equation:

`c=ms+b`

Substituting the first point.

`\begin{gathered} 8.96=14m+b \\ 8.96-14m=b \\ b=8.96-14m \end{gathered}`

Isolating b by subtracting 8.96 m from both sides:

`\begin{gathered} 14-8.96m=8.96m+b-8.96m \\ 14-8.96m=b \\ b=14-8.96m \end{gathered}`

Step 03: Substitute b in the general equation.

`\begin{gathered} c=ms+b \\ c=ms+8.96-14m \end{gathered}`

Step 04: Substitute the second point in the equation above.

`\begin{gathered} c=ms+8.96-14m \\ 10.24=16m+8.96-14m \\ 10.24=2m+8.96 \end{gathered}`

Step 05: Solve the equation above for m.

To do it, first, subtract 8.96 from both sides.

`\begin{gathered} 10.24-8.96=2m+8.96-8.96 \\ 1.28=2m \end{gathered}`

Now, divide both sides by 2.

`\begin{gathered} (1.28)/(2)=(2)/(2)m \\ m=0.64 \end{gathered}`

Step 06: Substitute m in the equation from step 02 to find b.

`\begin{gathered} b=8.96-14m \\ b=8.96-14\cdot0.64 \\ b=8.96-8.96 \\ b=0 \end{gathered}`

Step 07: Write the equation.

`\begin{gathered} c=0.64s+0 \\ c=0.64s \end{gathered}`

Answer:

c = 0.64s