Q. 20 - Precalculus Enhanced with Graphing Utilities

Question

Height versus Head Circumference A pediatrician wanted to find a linear model that relates a child’s height, H, to head circumference, C. She randomly selects nine children from her practice, measures their height and head circumference, and obtains the data shown. Let H represent the independent variable and C the dependent variable.

(a) Use a graphing utility to draw a scatter diagram.

(b) Use a graphing utility to find the line of best fit that models the relation between height and head circumference. Express the model using function notation.

(c) Interpret the slope.

(d) Predict the head circumference of a child that is 26 inches tall.

(e) What is the height of a child whose head circumference is 17.4 inches?

Step-by-Step Solution

Answer

(a). The scatter diagram is

(b). The line of best fit is y = 0.734x + 7.3268

(c). The slope of the best fit line is 0.734.

(d). The head circumference of a child that is 26 inches tall is 17.0352.

(e). The height of a child whose head circumference is 17.4 inches is 26.98 approximately. 

Part (a) Step 1. Given information

The given data is 

Part (a) Step 1. Scatter plot.

The scatter plot diagram is by using the given data as,

Part (b) Step 1. Best fit line

The line of best fit is y = 0.3734x + 7.3268.

Part (c) Step 1. Write the Slope of the obtained model.

The obtained model is y = 0.3734x + 7.3268 . Compare the equation by y=mx+c. Where m is slope.

 The slope is 0.3734.

Part (d) Step 1. Solution

The head circumference of a child that is 26 inches tall is

Put 26 for x  in the linear model.

$\begin{array}{rcl}C& =& 0.3734\times 26+7.3268\\ & =& 9.7084+7.3268\\ & =& 17.0352\end{array}$

Part (e) Step 1. Find the height when the head circumference is 17.4 inches.

Substitute 17.4 for y in the obtained model.

$\begin{array}{rcl}17.4& =& 0.373x+7.3268\\ 0.373x & =& 10.0732\\ x& =& 26.98 \mathrm{inches}\end{array}$

Thus, the obtained value is 26.98 inches.