지수함수와 로그함수 전체 문제

For the following exercises, use this scenario: A tumor is injected with $0.5$ grams of Iodine-125, which has a decay rate of $1.15\%$ per day.

Write an exponential model representing the amount of Iodine-125 remaining in the tumor after $t$ days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days. Round to the nearest tenth of a gram.

For the following exercises, use this scenario: A tumor is injected with $0.5$ grams of Iodine-125, which has a decay rate of $1.15\%$ per day.

The half-life of Radium-226 is $1590$ years. What is the annual decay rate? Express the decimal result to four decimal places and the percentage to two decimal places.

For the following exercises, use this scenario: A tumor is injected with $0.5$ grams of Iodine-125, which has a decay rate of $1.15\%$ per day.

A wooden artifact from an archeological dig contains 60 percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is $5730$ years.)

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.

Plot each set of approximate values of intensity of sounds on a logarithmic scale: Whisper: ${10}^{-10} \frac{W}{{m}^{2}},$ Vacuum: ${10}^{-4}\frac{W}{{m}^{2}},$ Jet: ${10}^{2} \frac{W}{{m}^{2}}$

For the following exercises, use this scenario: The equation $N(t)=\frac{500}{1+49{e}^{-0.7t}}$ models the number of people in a town who have heard a rumor after t days.

How many people started the rumor?

For the following exercises, use this scenario: The equation $N(t)=\frac{500}{1+49{e}^{-0.7t}}$ models the number of people in a town who have heard a rumor after t days.

To the nearest whole number, how many people will have heard the rumor after 3 days?

For the following exercises, use this scenario: The equation $N(t)=\frac{500}{1+49{e}^{-0.7t}}$ models the number of people in a town who have heard a rumor after t days.

As $t$ increases without bound, what value does $N(t)$ approach? Interpret your answer.

What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.

What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense?

What is regression analysis? Describe the process of performing regression analysis on a graphing utility.

What might a scatterplot of data points look like if it were best described by a logarithmic model?

Rewrite the exponential model $A(t)=1550{(1.085)}^{x}$ as an equivalent model with base $e.$ Express the exponent to four significant digits.

A logarithmic model is given by the equation $h(p)=67.682-5.792\ln (p).$ To the nearest hundredth, for what value of $p$ does $h(p)=62?$

A logistic model is given by the equation $P(t)=\frac{90}{1+5{e}^{-0.42t}}.$ To the nearest hundredth, for what value of t does $P(t)=45?$

Verify the conjecture made in the previous exercise. Round all numbers to six decimal places when necessary.

Find the inverse function ${f}^{-1}(x)$ for the logistic function $f(x)=\frac{c}{1+a{e}^{-bx}}.$ Show all steps.

Determine whether the function $y=156{(0.825)}^{t}$ represents exponential growth, exponential decay, or neither. Explain

Find an exponential equation that passes through the points $(2, 2.25)$ and $(5,60.75).$

A retirement account is opened with an initial deposit of $8,500 and earns $8.12\%$ interest compounded monthly. What will the account be worth in $20$ years?

Hsu-Mei wants to save $5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with $7.5\%$ APR, compounded daily, in order to reach her goal in $3$ years?