지수함수와 로그함수 전체 문제

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

Write an exponential model representing the amount of the drug remaining in the patient’s system after $t$ hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after $24$ hours. Round to the nearest hundredth of a gram.

For the following exercises, use this scenario: A soup with an internal temperature of $350^\circ$ Fahrenheit was taken off the stove to cool in a $\text{71°F}$ room. After fifteen minutes, the internal temperature of the soup was $\text{175°F}.$

Use Newton’s Law of Cooling to write a formula that models this situation.

For the following exercises, use this scenario: A soup with an internal temperature of $350^\circ$ Fahrenheit was taken off the stove to cool in a $\text{71°F}$ room. After fifteen minutes, the internal temperature of the soup was $\text{175°F}.$

How many minutes will it take the soup to cool to $\text{85°F?}$

What is the carrying capacity for a population modeled by the logistic equation $P(t)=\frac{250,000}{1+499{e}^{-0.45t}}?$ What is the initial population for the model?

The population of a culture of bacteria is modeled by the logistic equation $\,P(t)=\frac{14,250}{1+29{e}^{-0.62t}},$ where $t$ is in days. To the nearest tenth, how many days will it take the culture to reach $75\%$ of its carrying capacity?

The population of a pod of bottlenose dolphins is modeled by the function $A(t)=8{(1.17)}^{t},$ where $t$ is given in years. To the nearest whole number, what will the pod population be after $3$ 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 rock concert with a sound intensity of $4.7\cdot {10}^{-1}$ watts per square meter?

A bottle of soda with a temperature of $71^\circ$ Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of $\text{35° F}.$ After ten minutes, the internal temperature of the soda was $\text{63° F}.$ Use Newton’s Law of Cooling to write a formula that models this situation. To the nearest degree, what will the temperature of the soda be after one hour?

The population of a wildlife habitat is modeled by the equation $P(t)=\frac{360}{1+6.2{e}^{-0.35t}},$ where $t$ is given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity?

The population of a lake of fish is modeled by the logistic equation $P(t)=\frac{16,120}{1+25{e}^{-0.75t}},$ where $t$ is time in years. To the nearest hundredth, how many years will it take the lake to reach $80\%$ of its carrying capacity?