Ask a New Question
Menu

An increasing curve with an arrow on the upper end is plotted on a coordinate plane. The x-axis is labeled time left parenthesis years right parenthesis. The y-axis is labeled amount left parenthesis dollars right parenthesis.

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?

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

Option $2: As t increases, a increases quickly at first and then slowly.

Option #3: As t increases, a decreases slowly at first and then quickly.

Option #4: As t increases, a decreases quickly at first and then slowly.

(1 point)
Option #
is the true statement.

5 answers

The function \( a(t) = 5000(1.12)^t \) represents exponential growth. In exponential functions, the rate of increase accelerates over time.

As \( t \) increases, the value of \( a(t) \) increases slowly at first when \( t \) is small, but as \( t \) becomes larger, the amount \( a(t) \) grows much more rapidly due to the nature of exponentiation.

Therefore, the true statement about the graph of the function is:

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

r u sure sure sure?

Yes, I'm confident in that assessment. The equation \( a(t) = 5000(1.12)^t \) models exponential growth. In this specific context:

  • When \( t \) is small (for example, in the first few years), the growth is relatively slow, as the amount increases by 12% of the current amount.
  • As \( t \) increases and the amount becomes larger, the impact of that 12% growth becomes more significant, leading to a rapid increase in the overall amount.

This behavior is characteristic of exponential growth, where the growth rate accelerates over time. Therefore, the correct statement is indeed Option #1: As t increases, a increases slowly at first and then quickly.

if u be wrong imma throw u in da twash....

I understand the pressure! But rest assured, the explanation holds true. Exponential growth functions like \( a(t) = 5000(1.12)^t \) do indeed start off increasing slowly and then accelerate, leading to a quicker increase over time. If you have any more questions or need further clarification, I’m here to help!