## Podcast Beta

## Questions and Answers

How long will it take for $900 to grow to $14,700 at an interest rate of 5.7% if the interest is compounded continuously? Round the number of years to the nearest hundredth.

7.89

According to the formula p(t) = 10,472e^(0.004t), in how many years will the population reach 15,708? Round to the nearest tenth of a year.

15.2

Select the equation that describes the graph shown.

Which of the following is the same as 3log(4x) for x>0?

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Using the formula p(t) = 9779e^(0.004t), what is the population after 10 years?

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How many watches must Bob repair to have the lowest cost if the cost is given by c(x) = 4x^(2) - 312x + 49?

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How long will it take a sample of radioactive substance to decay to half of its original amount according to A(t) = 500e^(-0.153t)? Round to the nearest hundredth year.

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Find the required annual interest rate for $5200 to grow to $6500 if interest is compounded quarterly for 4 years.

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How many years will it take for the population of cars to grow from 69 million to 95 million at a growth rate of 4.1% annually?

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What is the length of the longest side of a rectangular plot with area 126 ydÂ² and perimeter of 46 yd?

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What is the rate on an investment that doubles $3171 in 14 years if interest is compounded quarterly?

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Solve the equation (x + 5)(x - 4) = 5 using the quadratic formula.

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Write 4^(1/2) = 2 in logarithmic form.

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Solve the equation sqrt(x + 3) = x - 3.

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Solve the equation (9x/(x - 9)) - (4/x) = 36/(x^(2) - 9x).

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Solve the equation | 7x + 5 | - 4 = -2.

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Solve the equation e^(x - 2) = (1/e^(6))^(x + 3).

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## Study Notes

### Continuous Compound Interest

- Use the formula A = Pe^(rt) to calculate the time for an investment to grow.
- For $900 to grow to $14,700 at a 5.7% interest rate, solve for t using continuous compounding.

### Exponential Growth Model

- Population growth can be modeled with p(t) = 10,472e^(0.004t).
- To determine when the population reaches 15,708, isolate t and solve.

### Parabola Equations

- Identify the equation of a parabola by examining shifts in the graph.
- The form y = (x + h)Â² + k indicates horizontal shift by h and vertical shift by k.

### Logarithmic Properties

- The expression 3log(4x) can be rewritten by raising the inside to the power of 3, yielding log(64xÂ³).
- Understanding properties of logarithms is essential for algebraic manipulation.

### Exponential Growth Calculation

- For the population modeled by p(t) = 9779e^(0.004t), calculate population after 10 years by substituting t = 10.

### Cost Optimization

- The cost function c(x) = 4xÂ² - 312x + 49 can be analyzed to find the minimum cost by calculating vertex form using y = -b/(2a).

### Radioactive Decay

- The decay of a substance is modeled by A(t) = 500e^(-0.153t).
- Determine the time for the sample to decay to half its original amount using the decay function.

### Interest Rate Calculation

- For compounding interest, use r = n((A/P)^(1/nt) - 1) to find the interest rate needed for investments to grow.

### Population Growth Rate

- The formula N = Ie^(kt) models population growth, where k represents the growth constant.
- Solve for the time required to increase from 69 million to 95 million cars, given a 4.1% annual growth rate.

### Area and Perimeter of Rectangles

- The area of a rectangular plot is given as 126 ydÂ² with a perimeter of 46 yd.
- Solve for side lengths by testing values within the established constraints.

### Investment Doubling Time

- To find the rate when $3171 doubles in 14 years, apply the quarterly compounding interest formula.

### Quadratic Equations

- Simplify the quadratic equation to the standard form axÂ² + bx + c.
- Use the quadratic formula x = (-b Â± âˆš(bÂ² - 4ac))/(2a) to find solutions.

### Logarithmic Conversion

- Convert exponential expressions to logarithmic form, using the identity a^b = c corresponds to log_a(c) = b.

### Solving Radical Equations

- To solve âˆš(x + 3) = x - 3, square both sides and apply quadratic solving methods.

### Rational Equation Solutions

- For equations like (9x/(x - 9)) - (4/x) = 36/(xÂ² - 9x), combine and simplify fractions then factor.

### Absolute Value Equations

- For equations involving absolute values, isolate and simplify to determine possible values of x.

### Exponential Equations

- Solve equations of the form e^(x - 2) = (1/e^(6))^(x + 3) by applying properties of exponents for simplification.

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## Description

This quiz reviews key concepts from Pre-Calculus Algebra, focusing on exponential growth and continuous compounding. You will solve problems involving population growth and interest calculations. Ensure you are prepared for your final exam with these essential flashcards.