Question

If a person takes a prescribed dose of 10 milligrams of Valium, the amount of Valium in that person's bloodstream at any time can be modeled with the exponential decay function A ( t ) = 10 e − 0.0188 t where t is in hours. a . How much Valium remains in the person's bloodstream 12 hours after taking a 10 -mg dose? Round to the nearest tenth of a milligram. mg b . How long will it take 10 mg to decay to 5 mg in a person's bloodstream? Round to two decimal places. hours c . At what rate is the amount of Valium in a person's bloodstream decaying 6 hours after a 10 -mg dose is taken. Round the rate to three decimal places.

Answer 1

(a)8.0mg

(b)36.87 hours

(c)-0.168

Explanation:

The amount of Valium in that person's bloodstream at any time can be modeled with the exponential decay function
`A ( t ) = 10 e^{ -0.0188 t` (t in hours)

(a)After 12 Hours

`A ( 12) = 10 e^( -0.0188*12)\\=7.98\\\approx 8.0 mg $(to the nearest tenth of a milligram)`

(b)If A(t)=5mg

Then:

`5= 10 e^( -0.0188 t)\\$Divide both sides by 10\\0.5=e^( -0.0188 t)\\$Take the natural logarithm of both sides\\ln (0.5)=-0.0188 t\\t=ln (0.5) / (-0.0188)\\$t=36.87 hours (to two decimal places.)`

(c)

`A ( t ) = 10 e^( -0.0188 t)\\A'(t)=10(-0.0188)e^( -0.0188 t)\\A'(t)=-0.188e^( -0.0188 t)\\$At 6 hours, the rate at which the amount of Valium is decaying therefore is:\\A'(6)=-0.188e^( -0.0188*6)\\$=-0.168 ( to three decimal places)`