Maria took 500 mg of medicine for her headache. The drug depletes in the blood stream at a rate of 20% per hour. The situation can be modeled by the exponential equation a(t)=500(0.8)t

Question

, where a is the amount of medicine in the blood stream after t hours. Which one of the following statements is true about the graph of the function?

Option #1: As t increases, a increases slowly at first and then quickly.

Option #2: The y-intercept of the function is (0,500)
.

Option #3: The range includes all real numbers.

(1 point)
Option #
is the true statement.

Answer 1

Let's analyze each option to determine which statement about the graph of the function \( a(t) = 500(0.8)^t \) is true.

Option #1: As \( t \) increases, \( a \) increases slowly at first and then quickly.

• This statement is false. The function \( a(t) = 500(0.8)^t \) is an exponential decay function, meaning as \( t \) increases, \( a \) decreases. The rate of decrease slows down, but \( a \) does not increase.

Option #2: The y-intercept of the function is \( (0, 500) \).

• This statement is true. When \( t = 0 \), we have \( a(0) = 500(0.8)^0 = 500(1) = 500 \). Therefore, the y-intercept is indeed \( (0, 500) \).

Option #3: The range includes all real numbers.

• This statement is false. The range of the function \( a(t) = 500(0.8)^t \) is limited to positive values. As \( t \) approaches infinity, \( a(t) \) approaches 0, but never becomes negative. Therefore, the range is \( (0, 500] \).

Given this analysis, the correct answer is:

Option #2 is the true statement.