QuestionLet be the balance in a savings account after years. Complete parts (a) through ( ) below.
(Kound to the nearest cent as needed.)
(d) What differential equation is satisfied by ?
The differential equation that is satisfied by is .
(e) Use the results from parts (c) and (d) to determine how fast the balance is growing after 11 years.
The balance is growing at approximately per year after 11 years.
(Round to the nearest cent as needed.)
Studdy Solution
STEP 1
1. The function represents the balance in a savings account over time.
2. The problem involves finding the rate of change of the balance at a specific time.
3. The differential equation given is .
STEP 2
1. Understand the given differential equation.
2. Calculate the derivative using the differential equation.
3. Evaluate at to find the rate of growth of the balance.
STEP 3
The given differential equation is . This means that the rate of change of the balance, , is proportional to the current balance with a proportionality constant of 0.03.
STEP 4
To find , we use the differential equation directly:
Since , substitute this into the equation:
STEP 5
Evaluate at to determine how fast the balance is growing after 11 years:
Calculate the exponent:
Now, compute using a calculator:
Finally, calculate :
Therefore, the balance is growing at approximately \$125.28 per year after 11 years.
The balance is growing at approximately per year after 11 years.
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