Note: Use the information below to answer each part of the question, and show all necessary work. You can type or upload your work in the space provided.%0D%0A%0D%0AAn adult takes 600 mg of ibuoprofen. Each hour, the amount of ibuprofen in the person's system decreases by about 29%.%0D%0A%0D%0A %0D%0A%0D%0Aa) Write an exponential function to model this situation. (2 pts)%0D%0A%0D%0Ab) About how much ibuprofen would be in the person's system after 5 hours? (2 pts)%0D%0A%0D%0Ac) After about how many hours would there be 30mg of ibuprofen left in their system? (3 pts)
a) Let's denote the amount of ibuprofen in the person's system at time t as A(t). Since the amount of ibuprofen decreases by about 29% each hour, we can write the exponential function as:
A(t) = 600 * (0.71)^t
b) To find the amount of ibuprofen in the person's system after 5 hours, we substitute t = 5 into the exponential function:
A(5) = 600 * (0.71)^5
A(5) ≈ 600 * 0.1681
A(5) ≈ 100.86 mg
Therefore, there would be approximately 100.86 mg of ibuprofen in the person's system after 5 hours.
c) To find the time at which there would be 30 mg of ibuprofen left in their system, we set A(t) = 30 and solve for t:
30 = 600 * (0.71)^t
Dividing both sides of the equation by 600, we get:
0.05 = 0.71^t
Taking the natural logarithm of both sides, we have:
ln(0.05) = ln(0.71^t)
Using the power rule of logarithms, we can move the exponent down:
ln(0.05) = t * ln(0.71)
Now, we can solve for t by dividing both sides by ln(0.71):
t ≈ ln(0.05) / ln(0.71)
t ≈ -2.9957 / -0.3425
t ≈ 8.7483
Therefore, after about 8.7483 hours, there would be approximately 30 mg of ibuprofen left in their system.