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Math Homework: Investment and Exponential Equations

This homework assignment involves solving investment and exponential equations, calculating compound interest, and understanding exponential growth and decay. Topics include investing money, calculating future values, solving for unknowns, and graphing systems of equations.

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Math Homework: Investment and Exponential Equations

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  1. total: Homework, pencil, notebook, textbook, red pen, calculator Have out: U3D9 Bellwork: 1. Edgar invests $100 in a bank account earning 8% annual interest compounded monthly. Find how much will be in the account after 5 years. 2. Solve for x. Be sure to give the exact and approximate answers. a) b)

  2. 1. Edgar invests $100 in a bank account earning 8% annual interest compounded monthly. Find how much will be in the account after 5 years. total: +1 +1 +1 +1 A(t) = amount (value) A0 = 100 r = 0.08 n = 12 t = 5 A(5) ≈ $148.98 +2 2. Solve for x. Be sure to give the exact and approximate answers. +1 a) b) +1 +1 +2 +1 +1 +1

  3. FX – 102 Suppose your state legislature passes a law that allows the fees for the State University system to increase by 5% per year without approval by the public or the legislature (larger increases have to be approved.) Suppose the annual fees for the university are currently $1000. The Smiths just had a baby girl and wanted to plan for her college education. They figured that, in 18 years, their daughter would be attending one of the state universities. Since 5% of $1000 is $50, they figured the increase in fees in 18 years will be 18 times $50, or $900; therefore, when she starts college the annual fees will be $1900. Explain to the Smiths why they may need to save more than they are anticipating.

  4. FX – 102 The Smiths used an “arithmetic” (or linear) equation rather than an exponential equation. What the Smiths did: What the Smiths should have done: Initial value = $1000 Initial value = $1000 multiplier = 100% + 5% difference = $1000 (0.05) = 105% = $50 = 1.05 V(t) = 50t + 1000 V(t) = 1000(1.05)t V(18) = 50(18) + 1000 V(18) = 1000(1.05)18 V(18) = 900 + 1000 V(18)  $2407 V(18) = $1900 The actual cost of the tuition will be about $2407.

  5. If you can purchase an item that costs $10.00 now and inflation continues at 4% per year (compounded yearly), when will the cost double? Show how you would compute this cost. FX – 103 V(t) = 10(1.04)t Initial value = $10 multiplier = 100% + 4% 20 = 10(1.04)t = 104% 10 10 = 1.04 2 = (1.04)t t  18 years Be sure to set up the equation!!! Then use a calculator to estimate the year. The item will double in cost in about 18 years.

  6. Bill just graduated from college and started work with a base pay of $32,000. Each year Bill receives a 5% raise. FX – 104 a) Write an expression using an exponent for Bill’s income in 3 years. I(3) = 32,000(1.05)3 Let’s answer part (d) first. b) What is Bill’s annual income in 5 years? 32,000(1.05)5  $40841.01 c) When will Bill’s income exceed $70,000? t  17 years 70,000 = 32,000(1.05)t d) What is the multiplier? What was the initial value (principal)? Write a function, I(t), to express Bill’s income I at any time t. multiplier = 100% + 5% Initial value = $32,000 I(t) = 32,000(1.05)t = 105% = 1.05 e) After 30 years of service, Bill retires. He receives an annual pension equal to his last year’s salary. What is Bill’s annual pension? (Note: Bill receives a total of 29 raises.) I(29) = 32,000(1.05)29  $131,716.34

  7. Find the annual rate of growth on an account that was worth $1000 in 1996 and was worth $1400 in 1999. (1.4) = (m3) m = (1.4) FX – 106 Initial value = $1000 year t V(t) multiplier = m 1996 0 1000 time = 3 1997 1 V(t) = k(m)t 1400 = 1000(m)3 1998 2 1000 1000 1999 3 1400 1.4 = m3 111.8688942% –100.0000000% exact 11.8688942% approx m  1.118688942… The annual rate of growth is about 11.87%.

  8. Find the monthly rate of decay on a radioactive sample that weighed 100 grams in May and weighs 50 grams in November. (0.5) = (m6) m = (0.5) FX – 107 Initial value = 100 Month t V(t) multiplier = m May 0 100 time = 6 V(t) = k(m)t 50 = 100m6 June 1 100 100 July 2 0.5 = m6 … … … Nov 6 50 exact 100.0000000% m  0.890898718… – 89.0898718% approx 10.9101282% The monthly rate of decay is about 10.91%.

  9. Finish today's assignment: FX 102 - 104, 106, 107, 109 - 112, 114, 115 & Worksheet

  10. Graph the system of equations. In the first graph be sure to include a value between 9 and 1 and a value between -1 and 0. y 10 x –5 5 –10 FX – 108 a) How many times do these functions cross? They cross 3 times. b) What are the coordinates of their intersections? (–1, –1), (0, 0), (1, 1) c) Solve the equation x3 = x. x = 0 x3 = x –x –x x = 1 x – 1 = 0 x3 – x = 0 x(x2 – 1) = 0 x = –1 x + 1 = 0 x(x – 1)(x + 1) = 0 d) How are the solutions in part (c) related to the points of intersection in part (b)? The solutions in part (c) are the x–values of the points of intersection.

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