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Question

A decreasing curve with an arrow at the end is plotted on a coordinate plane. The y-axis is labeled amount left parenthesis milligrams right parenthesis. The x-axis is labeled time left parenthesis hours right parenthesis.

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
, 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.
Remaining Attempts : 3
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Answer 1

Let's evaluate the statements based on the provided information about Maria's medicine depletion model \( a(t) = 500(0.8)^t \).

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

   • This statement is false. The function \( a(t) = 500(0.8)^t \) is an exponentially decreasing function. As \( t \) increases, \( a(t) \) actually decreases slowly at first and then more rapidly, not the other way around.

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

   • This statement is true. The y-intercept occurs when \( t = 0 \): \[ a(0) = 500(0.8)^0 = 500(1) = 500 \] So 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 only includes positive amounts of medicine up to 500 mg, not all real numbers.

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