Question

objective 2 - understand the continuous form
objective 1 - use growth/decay rates/factors
guided practice activity #3 - continuous vs. annual rates
for each scatterplot:

• identify the continuous rate and use it in a sentence to quantify the change of the output
• compute the associated rate per unit input (3 decimal places for the rate, r) and write

down the equation (5 decimal places for the factor, 1 + r):
( p = p _ { 0 } cdot ( 1 + r ) ^ { t } )

1. mutual fund

amount in the fund

Explanation:

Step1: Identify continuous rate

The continuous growth model is \( y = P_0 e^{kt} \), here \( y = 1200e^{0.1655x} \), so continuous rate \( k = 0.1655 \). The amount in the mutual fund grows continuously at a rate of 0.1655 per year.

Step2: Find annual rate \( r \)

Relate continuous growth \( y = P_0 e^{kt} \) and annual growth \( P = P_0(1 + r)^t \). So \( e^{k}=1 + r \). Substitute \( k = 0.1655 \), \( e^{0.1655}\approx1 + r \). Calculate \( e^{0.1655}\approx1.179 \) (using calculator), so \( r\approx0.179 \) (3 decimal places).

Step3: Write annual growth equation

\( P_0 = 1200 \), \( r = 0.179 \), so equation is \( P = 1200\cdot(1 + 0.179)^t=1200\cdot1.179^t \) (1 + r = 1.17900 when rounded to 5 decimal places as \( e^{0.1655}\approx1.17900 \)).

Answer:

• Continuous rate: The amount in the mutual fund grows continuously at a rate of 0.1655 per year.
• Annual rate \( r \approx 0.179 \)
• Annual growth equation: \( P = 1200\cdot1.179^t \) (or \( P = 1200\cdot(1 + 0.179)^t \))