The U.S. Census Bureau estimates that the Hispanic...

The U.S. Census Bureau estimates that the Hispanic population in the United States will increase from 32.44 million in 2000 to 98.23 million in 2050 . (a) Find an exponential function that gives the Hispanic population in year $x,$ with $x=0$ corresponding to 2000 (b) What is the projected Hispanic population in 2010 and $2025 ?$ (c) In what year will the Hispanic population reach 55 million?

The U.S. Census Bureau estimates that the Hispanic population in the United States will increase from 32.44 million in 2000 to 98.23 million in 2050 .
(a) Find an exponential function that gives the Hispanic population in year $x,$ with $x=0$ corresponding to 2000
(b) What is the projected Hispanic population in 2010 and $2025 ?$
(c) In what year will the Hispanic population reach
55 million?

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Key Concepts

Solving Exponential Equations

Solving exponential equations requires applying logarithmic techniques to isolate the variable, especially when the variable is in the exponent. This is essential when determining the time at which an exponential process reaches a predefined threshold, such as predicting the year a population will hit a certain number.

Function Prediction

Function prediction refers to the use of a derived exponential function to estimate the value of the quantity at different points in time. Once the model is established, specific time values can be substituted into the function to predict the future state, such as a population size.

Exponential Growth Model

An exponential growth model describes a situation where a quantity increases by a constant percentage over equal time intervals. This model is typically expressed with the formula P(x) = P? × b? or equivalently P(x) = P? × e^(kx), where P? is the initial amount and b or k represents the growth factor or rate.

Parameter Estimation

Parameter estimation involves using known data points to determine the unknown constants in a mathematical model. In exponential growth problems, this typically means calculating the growth factor or rate by substituting two data points into the model and solving for the parameter, which then fully defines the model.

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