The number of personal bankruptcy filings by fiscal years...

The number of personal bankruptcy filings by fiscal years ending September 30 between 2010 and 2012 was declining at the rate of $$R(t)=0.077 t+0.0825 \quad(0 \leq t \leq 2)$$ million cases/year, $t$ years after September $30,2010 .$ The number of filings as of September 30 2010, stood at approximately 1.538 million cases. a. Estimate the change in the number of personal bankruptcy cases filed between September 30 2010, and September 30, 2012. b. What was the approximate number of personal bankruptcy cases filed in $2012 ?$ Hint: If $N(t)$ denotes the number of bankruptcy filings in year $t,$ then $N^{\prime}(t)=-R(t)$. Source: Administrative Office of the U.S. Courts.

The number of personal bankruptcy filings by fiscal years ending September 30 between 2010 and 2012 was declining at the rate of $$R(t)=0.077 t+0.0825 \quad(0 \leq t \leq 2)$$ million cases/year, $t$ years after September $30,2010 .$ The number of filings as of September 30 2010, stood at approximately 1.538 million cases.
a. Estimate the change in the number of personal bankruptcy cases filed between September 30
2010, and September 30, 2012.
b. What was the approximate number of personal bankruptcy cases filed in $2012 ?$
Hint: If $N(t)$ denotes the number of bankruptcy filings in year $t,$ then $N^{\prime}(t)=-R(t)$.
Source: Administrative Office of the U.S. Courts.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Definite Integration

Definite integration is the process of summing a rate of change over a specific interval to calculate the total change in a quantity. When given a rate function, integrating it over an interval yields the net accumulation or decrease of the quantity during that time period, which is essential for computing changes in cumulative amounts like the number of bankruptcy filings.

Initial Value Problems (IVP)

An initial value problem involves a differential equation along with an initial condition that specifies the state of the system at a particular time. This concept allows for determining the unique function that models the evolution of a quantity over time. By knowing the starting amount and the rate of change, one can integrate and apply the initial value to find the total number at any given time.

Rate of Change Functions

This concept involves understanding how a quantity changes over time. In calculus, the rate at which a variable changes with respect to another is represented by its derivative. In practical scenarios, such as changes in filing rates, the derivative indicates how many cases are increasing or decreasing per unit time, which can be used to predict future behavior when integrated.

Transcript

Please give Ace some feedback