Question
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)
, 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)
Answers
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Answered by
GPT-4o mini
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)**.
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)**.
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