Number of Doctors The numbers of medical doctors D (in...

Number of Doctors The numbers of medical doctors $D$ (in thousands) in the United States from 2000 through 2009 can be modeled by $D=\sqrt{664,267.0+31,003.65 t}, \quad 0 \leq t \leq 9$ where $t$ represents the year, with $t=0$ corresponding to $2000 .$ (Source: American Medical Association) (a) In which year did the number of medical doctors reach $925,000 ?$ (b) Use the model to predict when the number of medical doctors will reach $1,100,000 .$ Is this prediction reasonable? Explain.

Number of Doctors The numbers of medical doctors $D$ (in thousands) in the United States from 2000 through 2009 can be modeled by $D=\sqrt{664,267.0+31,003.65 t}, \quad 0 \leq t \leq 9$
where $t$ represents the year, with $t=0$ corresponding to $2000 .$ (Source: American Medical Association)
(a) In which year did the number of medical doctors reach $925,000 ?$
(b) Use the model to predict when the number of medical doctors will reach $1,100,000 .$ Is this prediction reasonable? Explain.

Key Concepts

Extrapolation

Extrapolation refers to the process of using a model to predict values outside the range of observed data. It is critical to assess the reasonableness of predictions made through extrapolation, since the accuracy of the model beyond the provided interval may be compromised by additional factors or changes in trends not accounted for in the original data.

Solving Equations

Solving equations in this context involves isolating the variable in the model equation by performing inverse operations, such as squaring both sides of the equation to eliminate the square root. It is important to pay attention to the domain restrictions and to use algebraic manipulation carefully to determine the value corresponding to a specific output.

Mathematical Modeling

Mathematical modeling involves creating algebraic expressions or equations to represent real-world phenomena. In this context, a formula is constructed to model the number of doctors over time, by relating the variable representing time to the output quantity through a square root function. This approach helps in making predictions and understanding trends within the given data range.

Square Root Functions

Square root functions are used to express relationships where the output is proportional to the square root of a linear expression. They have a domain restriction due to the radical, implying that the expression under the square root must be non-negative. This type of function often models growth processes or phenomena where increases become progressively smaller as the independent variable grows.

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