Lincoln’s parents invested $5,000 in a college savings account when he was born. If the account has a growth rate of 12%, the situation can be modeled by the equation a(t)=5,000(1.12)t , where a is the amount after t years. Which one of the following statements is true about the graph of the function? (1 point) Responses As t increases, a decreases slowly at first and then quickly. As t increases, a decreases quickly at first and then slowly. As t increases, a increases slowly at first and then quickly.

Question

Lincoln’s parents invested $5,000 in a college savings account when he was born. If the account has a growth rate of 12%, the situation can be modeled by the equation a(t)=5,000(1.12)t , where a is the amount after t years. Which one of the following statements is true about the graph of the function? (1 point) Responses As t increases, a decreases slowly at first and then quickly. As t increases, a decreases quickly at first and then slowly.  As t increases, a increases slowly at first and then quickly.

GPT-4o mini · Answer

The equation given is $ a(t) = 5000(1.12)^t $, which models exponential growth. In this function, as time $ t $ increases, the amount $ a(t) $ will increase. To analyze the growth: - When $ t $ is small (for the first few years), the value of $ 1.12^t $ will still be relatively close to 1, making the increase in $ a(t) $ appear small. - However, as $ t $ increases, the term $ (1.12)^t $ grows significantly larger, leading to a more rapid increase in $ a(t) $. Therefore, the correct interpretation is: **As t increases, a increases slowly at first and then quickly.**