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)

Answer 1

To analyze the statements about the function \( a(t) = 500(0.8)^t \):

1. Option #1: "As t increases, a increases slowly at first and then quickly."
   This statement is false. The function \( a(t) \) is an exponential decay function since the base \( 0.8 \) is less than 1. As \( t \) increases, the amount of medicine \( a(t) \) decreases.

2. Option #2: "The y-intercept of the function is (0,500)."
   This statement is true. The y-intercept occurs when \( t = 0 \). Plugging in \( t = 0 \) into the function:
   \[ a(0) = 500(0.8)^0 = 500(1) = 500. \]
   Thus, the y-intercept is indeed \( (0, 500) \).

3. Option #3: "The range includes all real numbers."
   This statement is false. The range of \( a(t) \) is \( (0, 500] \), meaning it never reaches 0 and starts from 0 up to 500.

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