Question

question
the scatter plot and line of best fit below show the length of 11 people’s femur (the long leg bone in the thigh) and their height in centimeters. what is the meaning of the x - value on the line when y = 160?

image of scatter plot with x - axis: femur length (centimeters), y - axis: height (centimeters)

answer
the length of an actual person with a femur length of x centimeters is...
the length of an actual person with a femur length of x centimeters is 95.6 centimeters.
an expected height of 160 centimeters when the femur has a length of x centimeters.
an expected height of 95.6 centimeters when the femur has a length of x centimeters.

Explanation:

To solve for the \( x \)-value (femur length) when \( y = 160 \) (height) using the line of best fit, we first need to determine the equation of the line. From the scatter plot, we can observe two points on the line of best fit, for example, when \( x = 40 \), \( y = 140 \) and when \( x = 50 \), \( y = 160 \) (by analyzing the trend).

Step 1: Calculate the slope (\( m \))

The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \( (40, 140) \) and \( (50, 160) \):
\[
m = \frac{160 - 140}{50 - 40} = \frac{20}{10} = 2
\]

Step 2: Determine the equation of the line

Using the point-slope form \( y - y_1 = m(x - x_1) \) with the point \( (40, 140) \) and \( m = 2 \):
\[
y - 140 = 2(x - 40)
\]
Simplify to slope-intercept form (\( y = mx + b \)):
\[
y - 140 = 2x - 80 \\
y = 2x + 60
\]

Step 3: Solve for \( x \) when \( y = 160 \)

Substitute \( y = 160 \) into the equation \( y = 2x + 60 \):
\[
160 = 2x + 60
\]
Subtract 60 from both sides:
\[
100 = 2x
\]
Divide both sides by 2:
\[
x = 50
\]

Answer:

The missing \( x \)-value (femur length) when \( y = 160 \) (height) is \(\boxed{50}\) (in centimeters).