Question

1. if a quantity triples every 3 years, which of the following best represents its function?

a. $f(t) = 100 \times 3^{\frac{t}{3}}$
b. $f(t) = 100 \times (1.3)^t$
c. $f(t) = 100 \times (3)^t$
d. $f(t) = 100 \times (0.3)^t$

Explanation:

Step1: Recall exponential growth formula

The general form of exponential growth is \( f(t) = P \times a^{\frac{t}{k}} \), where \( P \) is the initial quantity, \( a \) is the growth factor, \( t \) is time, and \( k \) is the time it takes for the quantity to multiply by \( a \). Here, the quantity triples (so \( a = 3 \)) every 3 years (so \( k = 3 \)), and the initial quantity \( P = 100 \).

Step2: Substitute values into the formula

Substituting \( P = 100 \), \( a = 3 \), and \( k = 3 \) into the formula, we get \( f(t)=100\times3^{\frac{t}{3}} \) (which is equivalent to \( 100\times3^{\frac{t}{3}} = 100\times3^{\frac{1}{3}t}=100\times(3^{\frac{1}{3}})^t \), but the form in option a is \( 100\times3^{\frac{t}{3}} \), which matches our derivation. Let's check other options:

• Option b: \( 100\times(1.3)^t \) has a growth factor of 1.3, not related to tripling every 3 years.
• Option c: \( 100\times(3)^t \) would mean tripling every year, not every 3 years.
• Option d: \( 100\times(0.3)^t \) is a decay function (since 0.3 < 1), not growth.

Answer:

a. \( f(t) = 100 \times 3^{\frac{t}{3}} \)