Question

Question

Use the image to answer the question.

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
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.
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Answer 1

To analyze the growth of Lincoln's college savings account, we have the equation:

\[ a(t) = 5,000(1.12)^t \]

This equation represents exponential growth since the amount \( a(t) \) is a function of time \( t \) raised to an exponent (in this case, \( 1.12 \)).

Behavior of Exponential Growth:

1. At first (for small values of \( t \)), the growth is relatively slow because the base of the exponent (1.12) hasn't compounded much yet.
2. As \( t \) increases, the value of \( (1.12)^t \) grows significantly faster, leading to more rapid increases in \( a(t) \).

Therefore, with this understanding, the curve will start slow and then steepen as time progresses.

Based on the options given:

• Option #1: As \( t \) increases, \( a \) increases slowly at first and then quickly. (This is the correct statement)
• Option #2: As \( t \) increases, \( a \) increases quickly at first and then slowly. (This is incorrect)
• Option #3: As \( t \) increases, \( a \) decreases slowly at first and then quickly. (This is incorrect)
• Option #4: As \( t \) increases, \( a \) decreases quickly at first and then slowly. (This is incorrect)

Thus, the correct answer is:

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