Q. 93 - Precalculus Enhanced with Graphing Utilities

Question

Population as a Function of Age The function

$P\left(a\right)=0.015a^2-4.962a+290.580$

represents the population P (in millions) of Americans that are a years of age or older.

(a) Identify the dependent and independent variables.

(b) Evaluate $P\left(20\right)$. Provide a verbal explanation of the meaning of $P\left(20\right)$.

(c) Evaluate $P\left(0\right)$. Provide a verbal explanation of the meaning of $P\left(0\right)$.

Step-by-Step Solution

Answer

(a) The dependent variable is $P$ and independent variable is $a$.

(b) The value of $P\left(20\right)=197.34$, there are $197.34$ million Americans 20 years of age or older.

(c) The value of $P\left(0\right)=290.580$, there are $290.580$ million Americans 0 years of age or older.

Step 1. Given Information

The function $P\left(a\right)=0.015a^2-4.962a+290.580$represents the population P (in millions) of Americans that are a years of age or older.

(a) Identify the dependent and independent variables.

(b) Evaluate $P\left(20\right)$. Provide a verbal explanation of the meaning of $P\left(20\right)$.

(c) Evaluate $P\left(0\right)$. Provide a verbal explanation of the meaning of $P\left(0\right)$.

Part (a) Step 1. We have to identify the dependent and independent variables.

The given function is $P\left(a\right)=0.015a^2-4.962a+290.580$.

As we know, the dependent variable is characterized as the variable whose quality depends on the estimation of another variable in its condition. 

In this given function width="14" height="19" style="max-width: none;" $\mathit{P}$ depends on the estimation of another variable in its condition.

So the dependent variable is $P$.

Part (a) Step 2. As we know, an independent variable describes a variable whose values are independent of changes.

In this given function variable $\mathit{a}$are independent of changes.

So the independent variable is $a$.

Part (b) Step 1. Evaluating the value of P ( 20 ) . Putting 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250LXN0eWxlOm5vcm1hbDt9PC9zdHlsZT48L2RlZnM+PHRleHQgZm9udC1mYW1pbHk9ImFlMmVmNTI0ZmJmM2Q5ZmU2MTFkNWE4ZTkwZmVmZGMiIGZvbnQtc2l6ZT0iMTYiIGZvbnQtc3R5bGU9Iml0YWxpYyIgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iNC41IiB5PSIxNiI+YTwvdGV4dD48dGV4dCBmb250LWZhbWlseT0ibWF0aDE3ZjM5ZjgzMTdmYmRiMTk4OGVmNGM2MjhlYiIgZm9udC1zaXplPSIxNiIgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iMTguNSIgeT0iMTYiPj08L3RleHQ+PHRleHQgZm9udC1mYW1pbHk9IkFyaWFsIiBmb250LXNpemU9IjE2IiB0ZXh0LWFuY2hvcj0ibWlkZGxlIiB4PSIzNi41IiB5PSIxNiI+MjA8L3RleHQ+PC9zdmc+" role="math" localid="1646321542217" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/8509aa5b-37f1-4a15-abf7-a5b361e0b100.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220303%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220303T154239Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=b928f65db799cbe8d4a71d882230dd7e80c4220652d8d0226a58541f576d068f" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/8509aa5b-37f1-4a15-abf7-a5b361e0b100.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220303%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220303T154117Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=319c6fbadb90cdc75ad03a7d224b28915a364559a83f054a8944d12c33904b96" a = 20 on the given function.

$$\begin{gathered} P\left(20\right)=0.015{\left(20\right)}^2-4.962\times 20+290.580 \\ P\left(20\right)=0.015\times 400-4.962\times 20+290.580 \\ P\left(20\right)=6-99.24+290.580 \\ P\left(20\right)=197.34 \end{gathered}$$

Part (b) Step 2. A verbal explanation of the meaning of P ( 20 ) .

The meaning of $P\left(20\right)$ is " There are $197.34$ million Americans $20$ years of age or older."

Part (c) Step 1. Evaluating the value of P ( 0 ) . Putting x = 0 on the given function.

$$\begin{gathered} P\left(0\right)=0.015{\left(0\right)}^2-4.962\times 0+290.580 \\ P\left(0\right)=0-0+290.580 \\ P\left(0\right)=290.580 \end{gathered}$$

Part (b) Step 2. A verbal explanation of the meaning of P ( 0 ) .

The meaning of $P\left(0\right)$ is " There are $290.580$ million Americans $0$ years of age or older."