Let's evaluate each statement based on the given function \( a(t) = 500(1.07)^t \):
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Statement #1: As t increases, a increases slowly at first, then quickly.
- True. This is characteristic of an exponential growth function - it grows slowly at first and then becomes steeper as \( t \) increases.
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Statement #2: The y-intercept is greater than 0.
- True. When \( t = 0 \), \( a(0) = 500(1.07)^0 = 500 \). The y-intercept is 500, which is indeed greater than 0.
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Statement #3: There is a vertical asymptote at t=0.
- False. The function does not have a vertical asymptote at \( t=0 \). The function is defined at \( t=0 \) with a value of 500.
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Statement #4: As t increases, a increases at a decreasing rate.
- False. In an exponential growth function, while the rate of growth does increase (the function itself grows faster), the amount by which it increases does not decrease; it increases at an increasing rate.
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Statement #5: The t-values include all real numbers greater than or equal to 0.
- True. The function is only defined for \( t \geq 0 \) since it represents time in years.
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Statement #6: The graph increases without bound as t approaches positive infinity.
- True. As \( t \) approaches positive infinity, \( a(t) \) indeed increases without bound.
Now, let's summarize the true statements:
- True statements: 1, 2, 5, and 6.
Thus, the set of true statements are 1, 2, 5, and 6.
None of the provided responses match this exactly, but if we select the closest comprehension from the options, the correct choice regarding the true statements would be: The true statements are 1, 2, 5, and 6.