Problem 33 - Algebra 1: Concepts and Skills

Question

An ocean sunfish, the mola mola, is about 0.006 foot long when it hatches. By the time it reaches adulthood, the largest of the mola mola will have tripled its length about 7 times. What is the growth factor for the length of a mola mola?

Step-by-Step Solution

Answer

Therefore, the growth factor for the length of a mola mola is 2187.

Step 1: Understand the Problem

The problem gives the initial length of the mola mola when it hatches, which is 0.006 foot. It also states that this length triples 7 times as the mola mola grows to adulthood. The task is to find out the growth factor of the mola mola.

Step 2: Specify the growth

The mola mola triples its length 7 times. So, to calculate the growth factor, 3 should be multiplied by itself 7 times.

Step 3: Calculate the Growth Factor

In this step, simply calculate \(3^7 = 2187\). As the mola mola triples its length 7 times, the growth factor for the length of a mola mola is 2187.

Key Concepts

Growth Factor

When we talk about growth factor in algebra, we refer to the number by which a quantity multiplies over a period. The concept is commonly used to describe how quickly something is growing exponentially. In the context of the ocean sunfish, or mola mola, we see that the initial length of a hatchling, which is 0.006 foot, is multiplied several times as it grows. Specifically, the exercise states that the length triples 7 times by adulthood.

To clarify the growth process, imagine the fish's length not merely doubling, but tripling with each growth spurt. After the first growth, its length becomes 0.018 foot. With the next spurt, that length triples again, and so on, until this process has occurred seven times. Determining the growth factor involves raising 3 (the number it is increasing by) to the power of 7 (the number of times it triples). This reveals the growth factor as an impressive 2187, highlighting the remarkable growth power of the ocean sunfish.

Exponents

Exponents are shorthand for repeated multiplication of the same factor. They play a critical role in understanding exponential growth and are represented as a smaller number raised to the right of a base number. For instance, in the term 3⁷, 3 is the base, and 7 is the exponent, indicating that 3 should be multiplied by itself 6 more times - for a total of seven multiplications. So, this expression simplifies to 3 * 3 * 3 * 3 * 3 * 3 * 3, which equals 2187.

Exponents are not just limited to whole numbers; they can also be fractions, which represent roots, or even negative numbers, which indicate a reciprocal function. But in the context of exponential growth, like the one we see with the mola mola, the exponent typically is a whole number that shows how many times the growth has been compounded.

Geometric Sequences

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In this context, the length of the mola mola at each stage of its growth forms a geometric sequence.

The hatchling length is the first term of the sequence, and the common ratio is the factor by which its length grows, which is 3. Each term can be found by multiplying the previous term by this ratio, thus forming a sequence that might start with 0.006, 0.018, 0.054, and so forth, continuing for seven stages. Geometric sequences are essential for modeling situations where there is consistent percent change, whether that is growth, as with the mola mola, or decay in other scenarios.