Question 1

At time  minutes, the temperature of an object is   F. The temperature of the object is changing at the rate given by the differential equation  . Use Euler's Method to approximate the particular solutions of this differential equation at  . Use a step size of  . Round your answer to one decimal place. ​

Accepted Answer

E

Question 2

Use integration to find a general solution of the differential equation  . ​

Accepted Answer

A

Question 3

Determine whether the function  is homogeneous and determine its degree if it is. ​

Accepted Answer

C

Question 4

The half-life of the carbon isotope C-14 is approximately 5,715 years. If the initial quantity of the isotope is 34 g, what is the amount left after 10,000 years? Round your answer to two decimal places. ​

Accepted Answer

The amount of C-14 left after 10,000 years can be calculated using the formula:Amount left = Initial amount * (1/2)^(time/half-life)Plugging in the given values, we get:Amount left = 34 g * (1/2)^(10000/5715)≈ 10.11 gTherefore, the amount left after 10,000 years is approximately 10.11 g.

Question 5

The isotope  has a half-life of  years. Given an initial amount of 16 grams of the isotope, how many grams will remain after 1,000 years? After 10,000 years? Round your answers to four decimal places. ​

Accepted Answer

The answer of The isotope  has a half-life of...

Question 6

Each of the following graphs is from a logistic function  . Which one has the largest value of b? ​

Accepted Answer

The answer of Each of the following graphs is from...

Question 7

A calf that weighs 50 pounds at birth gains weight at the rate  where w is weight in pounds and t is time in years. What is the maximum weight of the animal if one uses the model  ? ​

Accepted Answer

The answer of A calf that weighs 50 pounds at...

Question 8

The rate of change of N is proportional to N. When  ,  and when  ,  . What is the value of N when  ? Round your answer to three decimal places. ​

Accepted Answer

The answer of The rate of change of N is...

Question 9

Solve the differential equation. ​  ​

Accepted Answer

The answer of Solve the differential equation. ​  ​
...

Question 10

A 500-gallon tank is half full of distilled water. At time  , a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 11 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 9 gallons per minute. At what time will the tank be full? ​

Accepted Answer

The answer of A 500-gallon tank is half full of...

Question 11

Solve the differential equation. ​  ​

Accepted Answer

The answer of Solve the differential equation. ​  ​
...

Question 12

A 600-gallon tank is full of a solution containing 55 pounds of concentrate. Starting at time  distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the amount of concentrate Q in the solution as a function of t.

Accepted Answer

The answer of A 600-gallon tank is full of a...

Question 13

Find an equation of the graph that passes through the point  and has the slope  . ​

Accepted Answer

The answer of Find an equation of the graph that...

Question 14

Solve the differential equation. ​  ​

Accepted Answer

The answer of Solve the differential equation. ​  ​
...

Question 15

A conservation organization releases 30 panthers into a preserve. After 3 years, there are 50 panthers in the preserve. The preserve has a carrying capacity of 150. Determine the population after 6 years. Discard any fractional part of your answer.

Accepted Answer

The population growth can be modeled by a logistic growth function, which accounts for the carrying capacity. In 3 years, the population grew from 30 to 50, indicating a growth rate that is not linear due to the carrying capacity constraint. Assuming the growth rate decreases as the population approaches the carrying capacity, and without specific growth rate details, a rough estimate would suggest a slowing growth. Doubling the time from 3 to 6 years doesn't double the population due to this deceleration. Among the given options, 74 is a reasonable estimate considering the carrying capacity and the initial growth observed.

Question 16

Sketch the slope field for the differential equation  and use the slope field to sketch the solution satisfying the condition  . ​

Accepted Answer

The answer of Sketch the slope field for the differential...

Question 17

Solve the first order linear differential equation. ​  ​

Accepted Answer

The answer of Solve the first order linear differential equation....

Question 18

The logistic function  models the growth of a population. Determine when the population reaches  % of the maximum carrying capacity. Round your answer to three decimal places.

Accepted Answer

The answer of The logistic function  models the growth...

Question 19

Find the particular solution of the differential equation  that satisfies the initial condition  .

Accepted Answer

The answer of Find the particular solution of the differential...

Question 20

Find the principal that must be invested at the rate 6%, compounded monthly, so that $3,000,000 will be available for retirement in 55 years. Round your answer to the nearest cent. ​

Accepted Answer

The formula for compound interest is $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the amount of money accumulated after n years, including interest, $P$ is the principal amount, $r$ is the annual interest rate (decimal), $n$ is the number of times that interest is compounded per year, and $t$ is the time the money is invested for in years. Rearranging the formula to solve for $P$, we get $P = \frac{A}{(1 + \frac{r}{n})^{nt}}$. Given $A = 3,000,000$, $r = 0.06$, $n = 12$ (since the interest is compounded monthly), and $t = 55$, we can calculate the principal. Plugging these values into the formula gives the correct answer.

Quiz 6: Differential Equations

Use integration to find a general solution of the differential equation . ​

Determine whether the function is homogeneous and determine its degree if it is. ​

The half-life of the carbon isotope C-14 is approximately 5,715 years. If the initial quantity of the isotope is 34 g, what is the amount left after 10,000 years? Round your answer to two decimal places. ​

The isotope has a half-life of years. Given an initial amount of 16 grams of the isotope, how many grams will remain after 1,000 years? After 10,000 years? Round your answers to four decimal places. ​

Each of the following graphs is from a logistic function . Which one has the largest value of b? ​

A calf that weighs 50 pounds at birth gains weight at the rate where w is weight in pounds and t is time in years. What is the maximum weight of the animal if one uses the model ? ​

The rate of change of N is proportional to N. When , and when , . What is the value of N when ? Round your answer to three decimal places. ​

Solve the differential equation. ​ ​

A 500-gallon tank is half full of distilled water. At time , a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 11 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 9 gallons per minute. At what time will the tank be full? ​

Solve the differential equation. ​ ​

A 600-gallon tank is full of a solution containing 55 pounds of concentrate. Starting at time distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the amount of concentrate Q in the solution as a function of t.

Find an equation of the graph that passes through the point and has the slope . ​

Solve the differential equation. ​ ​

A conservation organization releases 30 panthers into a preserve. After 3 years, there are 50 panthers in the preserve. The preserve has a carrying capacity of 150. Determine the population after 6 years. Discard any fractional part of your answer.

Sketch the slope field for the differential equation and use the slope field to sketch the solution satisfying the condition . ​

Solve the first order linear differential equation. ​ ​

The logistic function models the growth of a population. Determine when the population reaches % of the maximum carrying capacity. Round your answer to three decimal places.

Find the particular solution of the differential equation that satisfies the initial condition .

Find the principal that must be invested at the rate 6%, compounded monthly, so that $3,000,000 will be available for retirement in 55 years. Round your answer to the nearest cent. ​

Use integration to find a general solution of the differential equation .

Determine whether the function is homogeneous and determine its degree if it is.

The half-life of the carbon isotope C-14 is approximately 5,715 years. If the initial quantity of the isotope is 34 g, what is the amount left after 10,000 years? Round your answer to two decimal places.

The isotope has a half-life of years. Given an initial amount of 16 grams of the isotope, how many grams will remain after 1,000 years? After 10,000 years? Round your answers to four decimal places.

Each of the following graphs is from a logistic function . Which one has the largest value of b?

A calf that weighs 50 pounds at birth gains weight at the rate where w is weight in pounds and t is time in years. What is the maximum weight of the animal if one uses the model ?

The rate of change of N is proportional to N. When , and when , . What is the value of N when ? Round your answer to three decimal places.

Solve the differential equation.

A 500-gallon tank is half full of distilled water. At time , a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 11 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 9 gallons per minute. At what time will the tank be full?

Solve the differential equation.

Find an equation of the graph that passes through the point and has the slope .

Solve the differential equation.

Sketch the slope field for the differential equation and use the slope field to sketch the solution satisfying the condition .

Solve the first order linear differential equation.

Find the principal that must be invested at the rate 6%, compounded monthly, so that $3,000,000 will be available for retirement in 55 years. Round your answer to the nearest cent.