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Chapter 10: Problem 14

Give the growth rate that corresponds to the given growth factor. 0.639

Short Answer

Expert verified
Answer: The growth rate corresponding to a growth factor of 0.639 is -36.1%.

Step by step solution

01

Write down the formula for calculating the growth rate from growth factor

Growth Rate = (Growth Factor - 1) * 100
02

Plug in the given growth factor value

Growth Rate = (0.639 - 1) * 100
03

Perform the calculation

Growth Rate = (-0.361) * 100
04

Find the final growth rate

Growth Rate = -36.1% The growth rate that corresponds to the given growth factor of 0.639 is -36.1%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Growth Factor
The growth factor is a number that tells us how much something has grown or shrunk compared to its original size. If the growth factor is greater than 1, it means growth. If it's less than 1, there is shrinkage or decrease.
For example, a growth factor of 0.639 indicates that the original value has decreased to 63.9% of what it was. Understanding this concept helps in calculating the growth rate and predicting future trends.
Percentage Change
Percentage change is a way to express how much a quantity has increased or decreased in percentage terms. It is calculated using the growth factor.
  • If the growth factor is less than 1, the percentage change will be negative, showing a decrease.
  • If the growth factor is more than 1, the percentage will be positive, meaning an increase.

In the given example, converting the growth factor 0.639 to a percentage change involves using the formula: \( (0.639 - 1) \times 100 \) which equals -36.1%.
This negative result confirms a decrease.
Negative Growth
Negative growth indicates a decline or reduction in value. This is seen when the growth factor is less than 1, or the percentage change is negative. For example, in economics, this might signify that a country's economy has contracted.
In our exercise, the negative growth rate of -36.1% means a 36.1% reduction in the original value. Understanding negative growth is crucial for financial analysis, budgeting, and planning.
Mathematical Formula
Mathematical formulas provide the steps needed to solve problems by plugging in values. The formula to calculate growth rate from a growth factor is \( \text{Growth Rate} = (\text{Growth Factor} - 1) \times 100 \).
This formula helps convert a growth factor into a more understandable growth rate or percentage change.
By applying the growth factor of 0.639 to the formula, we calculate the growth rate as -36.1%.
  • This shows the importance of formulas in breaking down problems into manageable parts.
  • They are essential tools for analysis and decision-making across various fields.

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Most popular questions from this chapter

Nicotine leaves the body at a constant percent rate. Two hours after smoking a cigarette, \(58 \mathrm{mg}\) remainin a person's body; three hours later ( 5 hours after the cigarette), there are \(20 \mathrm{mg}\). Let \(n(t)\) represent the amount of nicotine in the body \(t\) hours after smoking the cigarette. (a) What is the hourly percent decay rate of nicotine? (b) What is the initial quantity of nicotine in the body? (c) How much nicotine remains 6 hours after smoking the cigarette?

For the investments described, assume that \(t\) is the elapsed number of years and that \(T\) is the elapsed number of months. (a) Describe in words how the value of the investment changes over time. (b) Give the annual growth rate. \(V=625(1.03)^{T / 3}\)

In Problems \(23-26,\) rewrite the given expression as indicated, and state the values of all constants. $$ \text { Rewrite } 5 e^{2+3 t} \text { in the form } a e^{k t} \text { . } $$

Give the growth factor that corresponds to the given growth rate. \(215 \%\) growth

Write the exponential functions in the form \(Q=a b^{t}\) and identify the initial value and growth factor. $$ Q=\frac{190}{3^{t}} $$
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