Mat1005 Exam 3 Name 1 Pam Buy

Mat1005 Exam 3name 1 Pam Buy

Mat1005 Exam 3 Name ________________________________________ 1. Pam buys 4 new albums every week. She currently has 50 albums. a) Write a recursive formula for the number of albums Pam has. b) Write an explicit formula for the number of albums Pam has. 2. A three year old girl is 28 inches tall. She grows 2 inches a year. In how many years will she be 50 inches? 3. In 2014, ABC College enrolled 100 students. If the enrollment grows 4% per year. Find the enrollment number in 2018. 4. Solve 10X = . Using 4% simple interest paid annually, how much money would you have after 3 years, if you originally had $100. How much interest do you have? 6. You deposit $100 in an account earning 3% compounded annually. How much money will you have in 10 years and how much interest will you earn? 7. You deposit $100 in an account earning 3% compounded monthly. How much money will you have in 10 years and how much interest will you earn? 8. You have $100,000 saved for retirement. Your account earns 3% interest. How much will you be able to pull out each month, if you want to be able to take withdrawals for 20 years? 9. You deposit $100 each month into an account earning 3% compounded monthly. How much will you have in the account in 20 years, and how much money will you put into the account? 10. You wish to have $3000 in 2 years to buy a television. How much should you deposit each quarter into an account paying 3% compounded quarterly? 11. Evaluate log(50). 12. Rewrite log(25) using the exponent property for logs. 13. If Saint Cloud is growing according to the equation Pn = 200(1.03)n where n is years after 2010, and the population is measured in thousands. Find when the population will be 400 thousand. 14. If you invest $2000 at 5% compounded monthly, how long will it take to triple in value? 15. Find the 4th root of 20.

Paper For Above instruction

The exam covers a range of fundamental mathematical concepts, including sequences and their formulas, growth modeling, interest calculations, logarithmic properties, and exponential equations, which are essential for understanding real-world financial, demographic, and growth scenarios.

Analysis of Sequence and Growth Models

Pam's album collection problem illustrates the growth of an arithmetic sequence. The weekly addition of 4 albums starting from 50 can be expressed recursively and explicitly. The recursive formula defines each term based on the previous one, with the initial condition of she already has 50 albums. This can be written as

a_n = a_n-1 + 4, with a₀ = 50.

The explicit formula, which provides the number of albums after n weeks, is derived by recognizing the pattern as linear: a_n = 50 + 4n. This allows straightforward calculation without recursion.

The height growth problem demonstrates linear growth, where a child's height increases by 2 inches per year. The time to reach 50 inches from 28 inches can be calculated with a simple linear equation:

28 + 2t = 50. Solving for t gives t = 11 years, indicating the girl will reach 50 inches in 11 years.

The enrollment growth scenario employs exponential growth modeling with a population increasing by 4% annually. The formula:

P_t = P₀ (1 + r)^t, where P₀ = 100, r = 0.04, and t = 4 (years from 2014 to 2018). Plugging in, the enrollment in 2018 is approximately 115 students:

P₂₀₁₈ = 100×(1.04)^4 ≈ 115.

Interest Calculations and Investment Growth

The problem involving 10X = with 4% simple interest involves solving for X, assuming a missing value. Assuming the equation refers perhaps to simple interest, the formula is I = P×r×t. With $100 invested at 4% annual simple interest for 3 years, the interest earned is I = 100×0.04×3 = $12. The total amount after 3 years is principal plus interest, totaling $112.

Compound interest calculations for deposits reveal exponential growth. Using the compound interest formula A = P(1 + r/n)^{nt}, where P = 100, r = 0.03, n varies between 1 (annual compounding) and 12 (monthly), and t = 10. With annual compounding:

A = 100×(1.03)^{10} ≈ 134.39, with interest earned approximately $34.39.

With monthly compounding:

A = 100×(1 + 0.03/12)^{12×10} ≈ 137.68, interest approximately $37.68.

Retirement and Savings Strategies

Calculating monthly withdrawals from a $100,000 savings at 3% interest for 20 years involves annuity formulas. The monthly withdrawal amount W can be derived from the present value of an annuity formula:

W = P × r / [1 - (1 + r)^-n, where P = 100,000, r = 0.03/12, and n = 240 months (20 years). The resulting monthly withdrawal is approximately $552.58.

For future value of monthly contributions, the compound interest future value of an annuity formula applies:

FV = P × [(1 + r/n)^{nt} - 1] / (r/n). Depositing $100 monthly into an account earning 3% compounded monthly for 20 years yields a future value of about $34,043.50, with total amounts deposited amounting to $24,000.

Additional Financial Calculations

The problem involving depositing quarterly to reach $3000 in 2 years with 3% quarterly compounding involves solving for the quarterly contribution using future value of an ordinary annuity:

PMT = FV × (r/n) / [(1 + r/n)^{nt} - 1]. Plugging in, the quarterly deposit PMT is approximately $358.15.

Logarithmic and Exponential Functions

Evaluating log(50) yields approximately 1.698, assuming base 10 logs. Rewriting log(25) as 2×log(5), leveraging the logarithmic property that log(a^b) = b×log(a), we find log(25) = 2×log(5) ≈ 2×0.6990 ≈ 1.398.

Population growth of Saint Cloud modeled by P_n = 200(1.03)ⁿ demonstrates exponential increase, reaching 400 thousand when 200×(1.03)^n = 400. Dividing both sides by 200 gives (1.03)ⁿ = 2, and solving for n via logarithms yields n ≈ 23.45 years after 2010, so approximately in the year 2033.

Tripling an investment from $2000 at 5% monthly compounded interest involves solving for t in A = P(1 + r/n)^{nt} = 3×2000 = 6000. Rearranged as log equations, it takes approximately 22.3 months to triple in value.

Finding the 4th root of 20 involves calculating 20^{1/4}, which is approximately 2.114, using fractional exponents.

Conclusion

This collection of mathematical problems illustrates practical applications, including sequences, exponential growth, compound interest, logarithms, and annuities, which are fundamental to financial planning, demographic studies, and understanding growth patterns. Mastery of these concepts enables better decision-making in personal finance and demographic trend analysis.

References

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