Victoria Has Some Money From Her Birthday 322137

Victoria Has Some Money From Her Birthday And The Amount Is Modeled By

Victoria has some money from her birthday, and the amount is modeled by the function h(x) = 200. She read about a bank that has savings accounts that accrue interest according to the function s(x) = (1.05)^x - 1. Victoria is thinking about putting her money into the savings account to gain interest. Using complete sentences, explain to Victoria how she can combine her functions to create a new function, and explain what this new function means.

Paper For Above instruction

Victoria has a fixed initial amount of money from her birthday, represented by the function h(x) = 200, where x can be considered as the time period, such as years, since she received the money. The bank offers an interest rate modeled by the function s(x) = (1.05)^x - 1, which calculates the amount of interest accrued over time at a 5% annual interest rate. To understand how her total savings will grow over time after depositing her birthday money into this account, Victoria can combine her initial amount with the bank’s interest function.

The key is to create a new function that reflects the total amount of money she will have after a certain period, incorporating both her initial deposit and the interest earned. Since h(x) = 200 represents her starting principal, and s(x) = (1.05)^x - 1 represents the interest accrued, she can first compute the total growth factor of her initial amount by adding 1 to s(x), resulting in a function that multiplies her starting money by the total growth factor.

Specifically, the total amount of money after x years can be modeled by multiplying her initial deposit, 200, by the total growth factor, which is 1 plus the interest function:

Total amount, T(x) = h(x) * [1 + s(x)].

Substituting h(x) and s(x):

T(x) = 200 [1 + (1.05)^x - 1] = 200 (1.05)^x.

Thus, the combined function T(x) = 200 * (1.05)^x describes the total amount of money Victoria will have in her account after x years, considering her initial deposit and the accrued interest at a 5% annual rate.

This new function essentially models exponential growth: starting with her initial $200, her savings increase each year by 5%, compounding annually. The expression demonstrates how her money grows over time, illustrating the power of compound interest. Victoria can now use this function to predict her savings after any number of years, helping her make informed decisions about when to withdraw or add to her savings.

This approach highlights the importance of understanding how initial investments and interest rates work together to grow wealth over time. By combining her initial amount with the interest function, she gains a comprehensive picture of her potential savings growth, which can be vital for financial planning and goal setting.

References

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