Questions and Answers
If the annual depreciation rate is 11%, which equation can be used to determine the approximate current value of a car purchased for $25,000?
What is the approximate pH of a solution with hydrogen ion concentration of $2 imes 10^{-9}$?
8.7
How old is a sample of Fermium-257 that contains 60% of its original amount if its half-life is about 100 days?
What is the approximate difference in hydrogen ion concentration between a solution with pH 11.2 and another with pH 2.4?
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After how many years will an environment no longer be able to support a population that increases exponentially?
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How long was Jacques's money in the account if he deposited $1,900 and it grew to $3,875.79 with 4% interest compounded semiannually?
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By what approximate factor was the intensity of the first earthquake greater than the second earthquake, given magnitudes of 8.6 and 8.2?
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How much money is in Isabel's account after 4 years if she deposited $6,000 at 1.5% interest compounded monthly?
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Which equation models the concentration of hydrogen ions (H+) in a solution with pH 7.2?
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How much of a 10-mg sample of a radioactive isotope remains after 16 days if its half-life is 8 days?
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Study Notes
Exponential and Logarithmic Equations in Real-World Applications
- The depreciation of a vehicle can be modeled with the equation (y = A(1 - r)^t), used to calculate the current value based on purchase price, depreciation rate, and time.
- An example case: A car purchased for $25,000 with an annual depreciation rate of 11% after 6 years.
pH and Hydrogen Ion Concentration
- pH is determined by the formula (pH = -\log[H^+]), where ([H^+]) denotes the concentration of hydrogen ions.
- A solution with a hydrogen ion concentration of (2 \times 10^{-9}) has an approximate pH of 8.7.
- The difference in hydrogen ion concentration between basic (pH 11.2) and acidic solutions (pH 2.4) can be quantified, revealing a concentration difference of (4.0 \times 10^{-3}).
Exponential Decay and Half-Life
- The remaining quantity of a substance can be calculated using (P(t) = A(1/2)^{t/h}), where (h) represents the half-life.
- In a Fermium-257 sample retaining 60% of its original amount with a 100-day half-life, the age of the sample cannot be concluded to be 60 days.
Population Growth and Carrying Capacity
- Population growth in a defined environment follows an exponential trend, with a maximum carrying capacity of 600 organisms.
- Analyzing population increases over time helps determine when the environment will no longer sustain the growth.
Compound Interest and Growth Over Time
- Compound interest is modeled with the formula (V(t) = P(1 + r/n)^{nt}), where variables include principal amount, interest rate, and compounding frequency.
- Example: A deposit of $1,900 growing to $3,875.79 at 4% interest compounded semiannually suggests a duration of approximately 18 years.
- For a deposit of $6,000 at an annual interest rate of 1.5% compounded monthly, the total after 4 years amounts to $6,370.78.
Earthquake Magnitudes and Intensity
- Magnitude of earthquakes is expressed through logarithmic intensity comparisons using (M = \log(i/i_0)).
- The intensity factor between two earthquakes, one with a magnitude of 8.6 and another of 8.2, indicates the first earthquake was approximately 2.51 times more intense.
Modeling Hydrogen Ion Concentration from pH
- The hydrogen ion concentration ([H^+]) can be derived from pH values, such as pH 7.2 yielding ([H^+] = 10^{-7.2}).
Radioactive Decay Calculation
- In radioactive decay, the quantity remaining after a set period is calculated similarly to other exponential decay processes.
- For a 10-mg sample with a half-life of 8 days, only 2.5 mg remains after 16 days.
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Description
Test your knowledge on modeling with exponential and logarithmic equations in Common Core Algebra II. This quiz covers key concepts including depreciation calculations and the application of exponential functions. Prepare to solve real-world problems using these mathematical models.
