Question

a patient in a hospital is given an injection of medicine. the amount of medicine (in milligrams) left in the patient’s bloodstream after ( t ) hours can be modeled with an exponential function. the graph of this function is shown below.
use the model to answer the parts to the right.
(a) what is the initial amount of medicine in the patient’s bloodstream? (square) mg
the initial amount of medicine is represented by the (\boldsymbol{\text{select}}) of the graph.
(b) for the first 20 hours, as time increases, the amount of medicine (\boldsymbol{\text{select}})
(c) give the equation of the asymptote. ( y = square )
choose the statement that best describes the meaning of the asymptote.
(\bigcirc) the amount of medicine in the patient’s bloodstream does not fall below 0 mg.
(\bigcirc) the amount of medicine in the patient’s bloodstream does not rise above 100 mg.
(\bigcirc) the patient can go home after 40 hours.

Explanation:

Part (a)

Step 1: Identify the initial point

The initial amount of medicine is when \( t = 0 \) (time is 0 hours). On the graph of the exponential function, the \( y \)-intercept (where \( t = 0 \)) gives the initial amount. From the graph, when \( t = 0 \), the amount of medicine ( \( y \)-value) is 100 mg. The initial amount is represented by the \( y \)-intercept of the graph.

Step 1: Analyze the graph's trend

The graph of the exponential function for the amount of medicine over time is a decreasing curve (since it starts at 100 and goes down as time \( t \) increases). So, for the first 20 hours, as time increases, the amount of medicine decreases.

Step 1: Determine the asymptote

For an exponential decay function (which models the decrease of medicine over time), the horizontal asymptote is \( y = 0 \) (since the amount of medicine can't be negative, it approaches 0 as time goes to infinity).

Step 2: Interpret the asymptote

The statement "The amount of medicine in the patient’s bloodstream does not fall below 0 mg" is correct because the asymptote \( y = 0 \) means the function approaches 0 but never goes below it. The other options are incorrect: the initial amount is 100, but the function can go below 100 (it does, as it's decreasing), and the asymptote has nothing to do with when the patient can go home.

Answer:

100; \( y \)-intercept

Part (b)