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Sabrina is tracking the growth rates of a colony of ants and a bee hive. She has developed functions to represent the population growth of each type of insect, where [tex]\( y \)[/tex] represents the population and [tex]\( x \)[/tex] represents the number of weeks since Sabrina began her research.

Ant Colony Growth: [tex]\( y = 3.8^x \)[/tex]
Bee Hive Growth: [tex]\( y = 3.8x \)[/tex]

Which insect population is growing at a faster rate?

Sagot :

GinnyAnswer
Sure, let's analyze the growth rates of the ant colony and the bee hive.

Firstly, we'll understand the growth functions for both populations:

1. Ant Colony Growth:
The growth of the ant colony is given by the exponential function:
[tex]\[ y = 3.8^x \][/tex]
Here, [tex]\(y\)[/tex] is the population of ants, and [tex]\(x\)[/tex] is the number of weeks.

2. Bee Hive Growth:
The growth of the bee hive is given by the linear function:
[tex]\[ y = 3.8x \][/tex]
Again, [tex]\(y\)[/tex] is the population of bees, and [tex]\(x\)[/tex] is the number of weeks.

Now, let's look at their populations over a few weeks to understand the difference in growth rates:

1. Ant Population:
- For [tex]\(x = 0\)[/tex]: [tex]\(y = 3.8^0 = 1.0\)[/tex]
- For [tex]\(x = 1\)[/tex]: [tex]\(y = 3.8^1 = 3.8\)[/tex]
- For [tex]\(x = 2\)[/tex]: [tex]\(y = 3.8^2 = 14.44\)[/tex]
- For [tex]\(x = 3\)[/tex]: [tex]\(y = 3.8^3 = 54.872\)[/tex]
- For [tex]\(x = 4\)[/tex]: [tex]\(y = 3.8^4 = 208.514\)[/tex]
- For [tex]\(x = 5\)[/tex]: [tex]\(y = 3.8^5 = 792.352\)[/tex]

So, the sequence of ant colony populations over the weeks is: [tex]\([1.0, 3.8, 14.44, 54.872, 208.514, 792.352]\)[/tex].

2. Bee Population:
- For [tex]\(x = 0\)[/tex]: [tex]\(y = 3.8 \cdot 0 = 0.0\)[/tex]
- For [tex]\(x = 1\)[/tex]: [tex]\(y = 3.8 \cdot 1 = 3.8\)[/tex]
- For [tex]\(x = 2\)[/tex]: [tex]\(y = 3.8 \cdot 2 = 7.6\)[/tex]
- For [tex]\(x = 3\)[/tex]: [tex]\(y = 3.8 \cdot 3 = 11.4\)[/tex]
- For [tex]\(x = 4\)[/tex]: [tex]\(y = 3.8 \cdot 4 = 15.2\)[/tex]
- For [tex]\(x = 5\)[/tex]: [tex]\(y = 3.8 \cdot 5 = 19.0\)[/tex]

So, the sequence of bee populations over the weeks is: [tex]\([0.0, 3.8, 7.6, 11.4, 15.2, 19.0]\)[/tex].

Observing these values, we can see that the ant population grows much faster than the bee population as time progresses. Next, let's examine the rates of growth over the weeks:

1. Ant Growth Rate:
- Between [tex]\(x = 0\)[/tex] and [tex]\(x = 1\)[/tex]: Increase is [tex]\(3.8 - 1.0 = 2.8\)[/tex]
- Between [tex]\(x = 1\)[/tex] and [tex]\(x = 2\)[/tex]: Increase is [tex]\(14.44 - 3.8 = 10.64\)[/tex]
- Between [tex]\(x = 2\)[/tex] and [tex]\(x = 3\)[/tex]: Increase is [tex]\(54.872 - 14.44 = 40.432\)[/tex]
- Between [tex]\(x = 3\)[/tex] and [tex]\(x = 4\)[/tex]: Increase is [tex]\(208.514 - 54.872 = 153.642\)[/tex]
- Between [tex]\(x = 4\)[/tex] and [tex]\(x = 5\)[/tex]: Increase is [tex]\(792.352 - 208.514 = 583.838\)[/tex]

These correspond to the growth rates: [tex]\([2.8, 10.64, 40.432, 153.642, 583.838]\)[/tex].

2. Bee Growth Rate:
- Between [tex]\(each x\)[/tex] values: Increase is [tex]\(3.8\)[/tex] consistently.

These correspond to the growth rates: [tex]\([3.8, 3.8, 3.8, 3.8, 3.8]\)[/tex].

Clearly, the ant population is experiencing exponential growth, whereas the bee population is experiencing linear growth. This means that the ant population is growing at a much faster rate as compared to the bee population because the rate of growth for ants increases rapidly over time.

Thus, the ant colony's population is growing at a faster rate.
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