Let's evaluate each statement about the function \( a(t) = 500(1.07)^t \):
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Statement #1: As \( t \) increases, \( a \) increases slowly at first, then quickly.
- This is true. Exponential growth starts slowly for small \( t \) values and then accelerates rapidly as \( t \) increases.
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Statement #2: The y-intercept is greater than 0.
- This is true. When \( t = 0 \), \( a(0) = 500(1.07)^0 = 500 \), which is greater than 0.
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Statement #3: There is a vertical asymptote at \( t=0 \).
- This is false. Vertical asymptotes occur typically in rational functions or other cases, but here \( a(t) \) is defined at \( t = 0 \) and does not approach infinity as \( t \) approaches 0.
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Statement #4: As \( t \) increases, \( a \) increases at a decreasing rate.
- This is false. In exponential growth, the rate of increase is always increasing, not decreasing.
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Statement #5: The t-values include all real numbers greater than or equal to 0.
- This is true. The function is defined for all \( t \geq 0 \).
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Statement #6: The graph increases without bound as \( t \) approaches positive infinity.
- This is true. As \( t \) increases, \( a(t) \) grows indefinitely.
Based on the analysis:
- True statements are 1, 2, 5, and 6.
Therefore, the correct response is:
The true statements are 1, 2, 5, and 6.