Unit 5, Lesson 7
Interpreting and Using Exponential Functions (Optional)
Algebra 2
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The half-life of carbon-14 is about 5,730 years. A fossil had 6 picograms of carbon-14 at one point in time. (A picogram is a trillionth of a gram, or \(1 \times 10^{\text-12}\) gram.) Which expression describes the amount of carbon-14, in picograms, \(t\) years after it was measured to be 6 picograms.
\(6 \boldcdot \left(\frac12\right)^{\frac{t}{5,730}}\)
\(6 \boldcdot \left(\frac12\right)^{5,730t}\)
\(6 \boldcdot (5,\!730)^{\frac12 t}\)
\(\frac12 \boldcdot (6)^{\frac{t}{5,730}}\)
The half-life of carbon-14 is about 5,730 years. A tree fossil was estimated to have about 4.2 picograms of carbon-14 when it died. (A picogram is a trillionth of a gram.) The fossil now has about 0.5 picogram of carbon-14. About how many years ago did the tree die? Show your reasoning.
Nickel-63 is a radioactive substance with a half-life of about 100 years. An artifact had 9.8 milligrams of nickel-63 when it was first measured. Write an equation to represent the mass of nickel-63, in milligrams, as a function of:
Tyler says that the function \(f(x) = 5^x\) is exponential and so it grows by equal factors over equal intervals. He says that factor must be \(\sqrt[10]{5}\) for an interval of \(\frac{1}{10}\) because ten of those intervals makes an interval of length 1. Do you agree with Tyler? Explain your reasoning.
The population in a city is modeled by the equation \(p(d) = 100,\!000 \boldcdot (1+ 0.3)^d\), where \(d\) is the number of decades since 1970.
The function \(f\) is exponential. Its graph contains the points \((0,5)\) and \((1.5,10)\).
Select all expressions that are equal to \(8^{\frac23}\).
\(\sqrt[3]{8^2}\)
\(\sqrt[3]{8}^2\)
\(\sqrt{8^3}\)
\(2^2\)
\(2^3\)
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