A drug is administered intravenously at a constant rate of r mg/hour and is excreted at a rate proportional to the quantity present, with constant of proportionality k>0. (Set up and) Solve a differential equation for the quantity, Q, in milligrams, of the drug in the body at time t hours. Assume there is no drug in the body initially. Your answer will contain r and k.
Q =
Graph Q against t. What is Q?, the limiting long-run value of Q?
Q?=
If r is doubled (to 2r), by what multiplicative factor is Q? increased?
Q? (for 2r) = Q? (for r)
Similarly, if r is doubled (to 2r), by what multiplicative factor is the time it takes to reach half the limiting value, 12Q?, changed?
t (to 12Q?), for 2r) = t (to 12Q?), for r)
If k is doubled (that is, we use 2k instead of k), by what multiplicative factor is Q? increased?
Q? (for 2k) = Q? (for k)
On the time to reach 12Q??
t (to 12Q?), for 2k) = t (to 12Q?), for k)

See explanation

Step-by-step explanation:

Solution:-

- We are told that a drug is administered to a patient at a rate of ( r ). The drug present in the patient at time t is Q. The drug also leaves the patient's body at a rate proportional to the amount of drug present at time t.

- We will set up the first order ODE for the rate of change of drug ( Q ) in the patient's body.

- We know that the rate of inflow is the rate at which the drug is administered that is ( r ) and the flow out is proportional to the amount currently present in the patient's body ( k*Q ). Where ( k ) is the constant of proportionality:

- Express the ODE in the standard form:

- The integrating factor ( u ) for the above ODE would be:

- Use the standard solution of ( Q ) using the integrating factor ( u ):

Where, c: the constant of integration.

- The initial value problem is such that there is no drug in the patient body initially. Hence, Q ( 0 ) = 0:

- The solution to the ODE is:

                 .... Answer

- We can use any graphing calculator to plot the amount of drug ( Q ) in the patient body. The limiting value of the drug in the long-run ( t -> ∞ ) can be determined as follows:

              Lim  ( t -> ∞ ) [ Q ( t ) ] = Lim  ( t -> ∞ ) [  

              Lim  ( t -> ∞ ) [ Q ( t ) ]  =  

- The long-run limiting value of drug in the body would be ( r / k ).

- If the rate of drug administrative rate is doubled then the amount of ( Q ) at any time t would be:

- The multiplicative factor is 2.

- To reach half the limiting value ( 0.5* r / k ) the amount of time taken for the double rate ( 2r ) of administration of drug would be:

- Similarly for the normal administration rate ( r ):

- The multiplicative factor ( M ) of time taken to reach half the limiting value is as follows:

- Similarly repeat the above calculation when the proportionality constant ( k ) is doubled.

12.5%

Step-by-step explanation:

In order to find the percentage the worker spent traveling, we can make us of the following ratio:

where number is how many hours he spent traveling, ammount is the total ammount of hours he reported and percent is the percentage we want to find. So we can solve the ratio for the percent, so we get:

now, we need to find the total amount of hours he spent traveling, this is gotten by adding the provided data:

6hrs+5hrs=11hrs

next we need to find the total amount of hours reported:

42hrs+46hrs=88hrs

so now we can substitute the data the problem provided us with:

which yields:

percent=12.5%