To find the amount of interest earned each year using the points provided on your graph, we can interpret the points as follows:
- The first point (3, 21) indicates that after 3 years, the total amount (principal + interest) in the account is $21.
- The second point (8, 56) indicates that after 8 years, the total amount in the account is $56.
To find the amount of interest earned each year (let's call it \( r \)), we can use the simple interest formula:
\[ A = P + Prt \]
where:
- \( A \) is the total amount after time \( t \),
- \( P \) is the principal amount (initial deposit),
- \( r \) is the interest rate per year, and
- \( t \) is the time in years.
We can also express the relationship for the two points in terms of the interest earned.
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After 3 years: \[ A = 350 + 3r = 21 \] \( \Rightarrow \) \( 3r = 21 - 350 \) \( \Rightarrow \) \( 3r = -329 \) \( \Rightarrow \) \( r = -\frac{329}{3} \text{ (not possible since interest cannot be negative)} \)
-
After 8 years: \[ A = 350 + 8r = 56 \] \( \Rightarrow \) \( 8r = 56 - 350 \) \( \Rightarrow \) \( 8r = -294 \) \( \Rightarrow \) \( r = -\frac{294}{8} \text{ (also not possible)} \)
Since both points lead to a nonsensical negative interest rate, let's instead calculate the annual increase in the total amount between these two points to find a consistent impression for the annual interest earned.
To calculate the increase in amount from year 3 to year 8:
\[ A_{t=8} - A_{t=3} = 56 - 21 = 35 \] This occurs over a period of: \[ 8 - 3 = 5 \text{ years} \]
Now, we can find the interest earned per year based on the increase:
\[ \text{Annual interest} = \frac{35}{5} = 7 \]
Hence, the amount of interest earned each year is $7.
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