The function f(t)=1900(1.075)ᵗ/60 represents the change in quantity over t seconds. What does the constant 1.075 reveal about the rate of change of the quantity?
The Correct Answer and Explanation is :
The function ( f(t) = 1900(1.075)^{t/60} ) describes a quantity that changes over time, where ( t ) is measured in seconds. In this function, the constant ( 1.075 ) is crucial because it represents the growth factor of the quantity per unit of time. Specifically, it indicates that the quantity is increasing by 7.5% for every 60 seconds that pass.
Explanation
- Understanding the Constant: The base of the exponent, ( 1.075 ), signifies the rate of growth. When a base greater than 1 is used in an exponential function, it indicates growth; conversely, a base less than 1 would signify decay. In this case, since ( 1.075 > 1 ), it confirms that the quantity is indeed growing over time.
- Rate of Change: The specific value of ( 1.075 ) shows that for each interval of 60 seconds, the quantity increases to 107.5% of its previous value. This can be interpreted as a 7.5% increase. To visualize this, if at ( t = 0 ), the quantity is 1900, then after 60 seconds (when ( t = 60 )), the new quantity becomes ( 1900 \times 1.075 = 2042.5 ). This illustrates the compounding nature of the growth.
- Continuous Growth: The exponential nature of the function indicates that as time progresses, the quantity will not only continue to grow, but the rate of growth itself will accelerate. This is due to the compounding effect inherent in exponential functions, where the amount being added to the original quantity increases over time because it is calculated based on the ever-increasing value.
- Application in Real World: This type of growth model is commonly found in various real-world applications, such as population growth, finance (interest compounding), and certain biological processes. Understanding the implications of a growth factor like ( 1.075 ) can help in forecasting future quantities based on current data, aiding in decision-making processes across various fields.
In conclusion, the constant ( 1.075 ) in the function ( f(t) ) reveals a consistent growth rate of 7.5% for every 60 seconds, highlighting an exponential increase in the quantity over time.