지수함수와 로그함수 전체 문제

For the following exercises, use this scenario: A pot of warm soup with an internal temperature of $100^\circ$ Fahrenheit was taken off the stove to cool in a $\text{69° F}$ room. After fifteen minutes, the internal temperature of the soup was $\text{95° F}.$

To the nearest degree, what will the temperature be after $2$ and a half hours?

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of $\text{165°F}$ and is allowed to cool in a $\text{75°F}$ room. After half an hour, the internal temperature of the turkey is $\text{145°F}.$

Write a formula that models this situation.

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of $\text{165°F}$ and is allowed to cool in a $\text{75°F}$ room. After half an hour, the internal temperature of the turkey is $\text{145°F}.$

To the nearest degree, what will the temperature be after 50 minutes?

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of $\text{165°F}$ and is allowed to cool in a $\text{75°F}$ room. After half an hour, the internal temperature of the turkey is $\text{145°F}.$

To the nearest minute, how long will it take the turkey to cool to $\text{110° F?}$

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

Recall the formula for calculating the magnitude of an earthquake, $M=\frac{2}{3}\log \left(\frac{S}{{S}_{0}}\right).$ One earthquake has magnitude $3.9$ on the MMS scale. If a second earthquake has $750$ times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.

For the following exercise, choose the correct answer choice.

A doctor injects a patient with 13 milligrams of radioactive dye that decays exponentially. After 12 minutes, there are 4.75 milligrams of dye remaining in the patient’s system. Which is an appropriate model for this situation?

• ⓐ $f(t)=13{(0.0805)}^{t}$
• ⓑ $f(t)=13{e}^{0.9195t}$
• ⓒ $f(t)=13{e}^{(-0.0839t)}$
• ⓓ $f(t)=\frac{4.75}{1+13{e}^{-0.83925t}}$

To the nearest whole number, what is the initial value of a population modeled by the logistic equation $P(t)=\frac{175}{1+6.995{e}^{-0.68t}}?$ What is the carrying capacity?

For the following exercises, use this scenario: The population $P$ of a koi pond over $x$ months is modeled by the function $P(x)=\frac{68}{1+16{e}^{-0.28x}}.$

What was the initial population of koi?

For the following exercises, use this scenario: The population $P$ of a koi pond over $x$ months is modeled by the function $P(x)=\frac{68}{1+16{e}^{-0.28x}}.$

How many koi will the pond have after one and a half years?

For the following exercises, use this scenario: The population $P$ of a koi pond over $x$ months is modeled by the function $P(x)=\frac{68}{1+16{e}^{-0.28x}}.$

How many months will it take before there are $20$ koi in the pond?

For the following exercises, use this scenario: The population $P$ of a koi pond over $x$ months is modeled by the function $P(x)=\frac{68}{1+16{e}^{-0.28x}}.$

Use the intersect feature to approximate the number of months it will take before the population of the pond reaches half its carrying capacity.

For the following exercises, use this scenario: The population $P$ of an endangered species habitat for wolves is modeled by the function $P(x)=\frac{558}{1+54.8{e}^{-0.462x}},$ where $x$ is given in years.

What was the initial population of wolves transported to the habitat?

For the following exercises, use this scenario: The population $P$ of an endangered species habitat for wolves is modeled by the function $P(x)=\frac{558}{1+54.8{e}^{-0.462x}},$ where $x$ is given in years.

How many wolves will the habitat have after $3$ years?

For the following exercises, use this scenario: The population $P$ of an endangered species habitat for wolves is modeled by the function $P(x)=\frac{558}{1+54.8{e}^{-0.462x}},$ where $x$ is given in years.

How many years will it take before there are $100$ wolves in the habitat?

For the following exercises, use this scenario: The population $P$ of an endangered species habitat for wolves is modeled by the function $P(x)=\frac{558}{1+54.8{e}^{-0.462x}},$ where $x$ is given in years.

Use the intersect feature to approximate the number of years it will take before the population of the habitat reaches half its carrying capacity.

Recall that the general form of a logistic equation for a population is given by $P(t)=\frac{c}{1+a{e}^{-bt}},$ such that the initial population at time $t=0$ is $P(0)={P}_{0}.$ Show algebraically that $\frac{c-P(t)}{P(t)}=\frac{c-{P}_{0}}{{P}_{0}}{e}^{-bt}.$

The population of a herd of deer is represented by the function $A(t)=205{(1.13)}^{t},$ where $t$ is given in years. To the nearest whole number, what will the herd population be after $6$ years?

The formula for measuring sound intensity in decibels $D$ is defined by the equation $D=10\log \left(\frac{I}{{I}_{0}}\right),$ where $I$ is the intensity of the sound in watts per square meter and ${I}_{0}={10}^{-12}$ is the lowest level of sound that the average person can hear. How many decibels are emitted from a large orchestra with a sound intensity of $6.3\cdot {10}^{-3}$ watts per square meter?

The population of a city is modeled by the equation $P(t)=256,114{e}^{0.25t}$ where $t$ is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?

For the following exercises, use this scenario: A doctor prescribes $300$ milligrams of a therapeutic drug that decays by about $17\%$ each hour.

To the nearest minute, what is the half-life of the drug?