Chapter 8.6, Problem 26E

a.

To determine

To make:

A scatterplot of the data pairs $(x,y)$ . Draw the line that appears to best fit the data points.

Answer to Problem 26E

The line of best fit would look like:

  

Explanation of Solution

Given:

Since 1912, scientists have created five maps of the glaciers on top of Mount Kilimanjaro in Africa. The maps indicate that the glaciers are shrinking as shown by the table.

  

Calculation:

Let x be the number of years since 1912. Lety be the area of the glaciers (in square kilometers).

Our ordered pairs would be $(1,12.1),(41,6.7),(64,4.2),(77,3.3),(88,2.2)$ .

Upon plotting our given values on coordinate plane and drawing a line of best fit, we will get:

  

b.

To determine

To write:

An equation for the line of best fit.

Answer to Problem 26E

The equation of line of best fit would be $y=-0.094x+10.572$ .

Explanation of Solution

Given:

Since 1912, scientists have created five maps of the glaciers on top of Mount Kilimanjaro in Africa. The maps indicate that the glaciers are shrinking as shown by the table.

  

Calculation:

Let us find slope of line passing through points (77,3.3) and (41,6.7) as:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{6.7-3.3}{41-77}\\ m=\frac{3.4}{-36}\\ m=-0.094\end{array}$

Now we will use point-slope form to write our equation as:

  $\begin{array}{l}y-y_1=m(x-x_1)\\ y-3.3=-0.094(x-77)\\ y-3.3=-0.094x+7.2722\\ y-3.3+3.3=-0.094x+7.2722+3.3\\ y=-0.094x+10.572\end{array}$

Therefore, the equation of line of best fit would be $y=-0.094x+10.572$ .

c.

To determine

To estimate:

The year when the glaciers will disappear.

Answer to Problem 26E

The glaciers will disappear in 2024.

Explanation of Solution

Given:

Since 1912, scientists have created five maps of the glaciers on top of Mount Kilimanjaro in Africa. The maps indicate that the glaciers are shrinking as shown by the table.

  

Calculation:

To predict the year, when the glaciers will disappear, we will substitute $y=0$ in equation of line of best fit and solve for x as:

  $\begin{array}{l}y=-0.094x+10.572\\ 0=-0.094x+10.572\\ -0.094x+10.572=0\\ -0.094x=-10.572\\ \frac{-0.094x}{-0.094}=\frac{-10.572}{-0.094}\\ x=112.46\\ x\approx 112\end{array}$

Asx represents the number of years since 1912, so $x=112$ will represent year $1912+112=2024$ . Therefore, the glaciers will disappear in 2024.