Question

sabrina is tracking the growth rates of a colony of ants and of a bee hive. from her research, she has developed a function to represent the population growth of each type of insect, where ( y ) represents the population and ( x ) represents the number of weeks since sabrina began her research.
ant colony growth: ( y = 3.8^x )
bee hive growth: ( y = 3.8x )
which insect population is growing at a faster rate?
the ant colony is growing at a faster rate. it will add 3.8 each week while the bee hive will subtract 3.8 each week.
the bee hive is growing at a faster rate. it will multiply by 3.8 each week while the ant colony will add 3.8 each week.
the ant colony is growing at a faster rate. it will multiply by 3.8 each week while the bee hive will add 3.8 each week

Explanation:

Step1: Analyze Ant Colony Growth

The ant colony growth function is \( y = 3.8^x \), which is an exponential function. For exponential growth, the base \( 3.8>1 \), so the population multiplies by \( 3.8 \) each week. For example, when \( x = 1 \), \( y=3.8^1 = 3.8 \); when \( x = 2 \), \( y = 3.8^2=3.8\times3.8 = 14.44 \), showing a multiplicative growth.

Step2: Analyze Bee Hive Growth

The bee hive growth function is \( y = 3.8x \), which is a linear function. For linear growth, the population increases by \( 3.8 \) each week (additive growth). For example, when \( x = 1 \), \( y = 3.8\times1=3.8 \); when \( x = 2 \), \( y=3.8\times2 = 7.6 \).

Step3: Compare Growth Rates

Exponential growth (ant colony) with base \( 3.8 \) will outpace linear growth (bee hive) with slope \( 3.8 \) as \( x \) increases. The ant colony multiplies by \( 3.8 \) each week, while the bee hive adds \( 3.8 \) each week. So the ant colony grows faster.

Answer:

The ant colony is growing at a faster rate. It will multiply by 3.8 each week while the bee hive will add 3.8 each week.