Problem 58 - Algebra 2

Question

Suppose a balloon is filled with 5000 \(\mathrm{cm}^{3}\) of helium. It then loses one fourth of its helium each day. a. Write the geometric sequence that shows the amount of helium in the balloon at the start of each day for five days. b. What is the common ratio of the sequence? c. How much helium will be left in the balloon at the start of the tenth day? d. Graph the sequence. Then sketch the graph. e. Critical Thinking How does the common ratio affect the shape of the graph?

Step-by-Step Solution

Answer

a. The geometric sequence representing the amount of helium from the first to the fifth day is: 5000, 3750, 2812.5, 2109.375, 1570.03125. b. The common ratio of this sequence is 3/4. c. On the tenth day, there will be approximately 886 \(cm^3\) helium left in the balloon. d. The sequence graph shows exponential decay. e. The graph shape (exponential decay) is determined by the common ratio which is less than 1.

Step 1: Define the Geometric Sequence

A geometric sequence is defined by its first term \(a\) and the common ratio \(r\). In this case, \(a\) is 5000 (the initial amount of helium in the balloon), and \(r\) is 3/4 \(1 - 1/4\), the remaining helium each day. The sequence for the amount of helium for the first five days is then 5000, 3750, 2812.5, 2109.375, and 1570.03125.

Step 2: Identify the Common Ratio

The common ratio \(r\) of a geometric sequence is the constant value we multiply to get from one term to the next. In this sequence, the common ratio is 3/4 \(1 - 1/4\).

Step 3: Find the Amount of Helium on the Tenth Day

To find the amount of helium on the tenth day, we use the formula for the nth term of a geometric sequence, which is \(a_n = a*(r)^{(n-1)}\). Here, \(n = 10\), \(a = 5000\), and \(r = 3/4\). Plugging these values, we get: \(a_{10} = 5000*(3/4)^9 = 885.735\). So, the balloon has approximately 886 \(cm^3\) of helium remaining at the start of the tenth day.

Step 4: Graph the Sequence

A geometric sequence can be graphed as a declining exponential curve in case of a common ratio less than 1, starting at 5000 and decreasing each day.

Step 5: Discuss the Effect of the Common Ratio on the Graph

When the common ratio is between 0 and 1, the graph will show exponential decay. The size of the ratio determines the rate of decline: the closer the ratio is to 1, the slower the decrease; the smaller the ratio, the more quickly the value of the sequence drops towards 0.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial component that determines how rapidly or slowly the sequence progresses. It is the factor by which each term in the sequence is multiplied to obtain the succeeding term. For example, in the given exercise, the initial volume of helium is 5000 cm³, and each day, the balloon retains three-fourths of its helium, leading to a common ratio of \(\frac{3}{4}\). This means every term in the sequence is calculated by multiplying the previous term by \(\frac{3}{4}\).

This constant multiplication allows us to generate subsequent terms with ease. Understanding the common ratio in a geometric sequence helps us predict and analyze the behavior of the sequence over time.

Exponential Decay

When a sequence demonstrates a common ratio less than 1, it leads to what is known as exponential decay. In this context, the term 'exponential' refers to the nature of the geometric sequence, where each term is a fixed fraction of its predecessor. Exponential decay is the gradual reduction of a quantity, here expressed through the geometric sequence.

In the balloon example, since it loses one-fourth of its helium every day, we observe an exponential decay pattern with a common ratio of \(\frac{3}{4}\).
The sequence starts at 5000 cm³ and shows a steady decline. Over time, the amount of helium decreases, approaching zero, but never quite reaching it. This illustrates how powerful exponential decay is in understanding various real-world phenomena, such as radioactivity or population declines.

Geometric Sequence Formula

The geometric sequence formula is a tool used to compute any term in the sequence without needing to list all the previous terms. The formula is given by:\[ a_n = a \, r^{n-1} \]Here, \(a_n\) represents the nth term, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number in the sequence.

In the balloon example, to find how much helium remains after ten days, we apply this formula with \(a = 5000\), \(r = \frac{3}{4}\), and \(n = 10\). Calculating this yields approximately 886 cm³ of helium. This illustrates how geometric sequences can mathematically model scenarios where a quantity decreases exponentially over time.

Mathematical Modeling

Geometric sequences are a powerful part of mathematical modeling, which involves using mathematical structures and formulas to represent real-world situations. The balloon problem is a straightforward example of how geometric sequences can model exponential decay effectively.

• Helps predict future states by calculating terms directly and accurately.
• Able to represent diverse scenarios like economics (depreciation), biology (population dynamics), and physics (radioactive decay).

In the case of the balloon losing helium, we see how mathematics precisely forecasts helium levels for each day by applying the geometric sequence formula. Understanding these models allows students to analyze complex systems efficiently and apply similar methods to more intricate problems.