Drag each response to the correct location on the table. Each response can be used more than once, but not all responses will be used.
Consider the two exponential equations shown. Identify the attributes for each equation to complete the table.
8. 9%
9. 40
10. 250
11. Growth
12. 89 %
13. 111%
14. Decay
15. 11%
| | |
| :---------------------------- | :----------------------------- |
| [tex]$40=250(0.89)^x$[/tex] | [tex]$250=40(1.11)^T$[/tex] |
| Initial Value: | Initial Value: |
| Growth or Decay: | Growth or Decay: |
| Growth/Decay Rate: | Growth/Decay Rate: |
Answer (2)
The initial value for the first equation 40 = 250 ( 0.89 ) x is 250, representing decay with a decay rate of 11%. The initial value for the second equation 250 = 40 ( 1.11 ) T is 40, representing growth with a growth rate of 11%. Therefore, the completed table shows the respective values for both equations.
;
The initial value of the first equation 40 = 250 ( 0.89 ) x is 250, and it represents decay.
The decay rate of the first equation is 1 − 0.89 = 0.11 , or 11%.
The initial value of the second equation 250 = 40 ( 1.11 ) T is 40, and it represents growth.
The growth rate of the second equation is 1.11 − 1 = 0.11 , or 11%. The completed table is then filled with these values, and the final answers are: Initial Value: 250, Growth or Decay: Decay, Growth/Decay Rate: 11% for the first equation and Initial Value: 40, Growth or Decay: Growth, Growth/Decay Rate: 11% for the second equation.
Explanation
Analyzing the Problem We are given two exponential equations and asked to identify the initial value, whether they represent growth or decay, and the growth/decay rate for each. Let's analyze each equation separately.
Analyzing the First Equation For the first equation, 40 = 250 ( 0.89 ) x , the initial value is the coefficient of the exponential term. In this case, the initial value is 250. Since the base of the exponential term (0.89) is between 0 and 1, this equation represents exponential decay. The decay rate is calculated as 1 − 0.89 = 0.11 , which is 11%.
Analyzing the Second Equation For the second equation, 250 = 40 ( 1.11 ) T , the initial value is the coefficient of the exponential term, which is 40. Since the base of the exponential term (1.11) is greater than 1, this equation represents exponential growth. The growth rate is calculated as 1.11 − 1 = 0.11 , which is 11%.
Completing the Table Now, let's fill in the table with the attributes we found:
40 = 250 ( 0.89 ) x
250 = 40 ( 1.11 ) T
Initial Value:
250
40
Growth or Decay:
Decay
Growth
Growth/Decay Rate:
11%
11%
Final Answer Therefore, the initial value for the first equation is 250, it represents decay with a rate of 11%. The initial value for the second equation is 40, it represents growth with a rate of 11%.
Examples
Exponential equations are used in various real-world scenarios. For example, they can model population growth, radioactive decay, and the spread of diseases. Understanding the initial value, growth/decay, and growth/decay rate helps in predicting future values and making informed decisions. For instance, if a population grows exponentially with an initial size of 1000 and a growth rate of 5% per year, we can use an exponential equation to estimate the population size after a certain number of years. Similarly, in finance, compound interest can be modeled using exponential equations to calculate the future value of an investment.
