1. Check it out! 1.2.1: Normal Distributions and the 68–95–99.7 Rule
1.2.1: Normal Distributions and the 68–95–99.7 Rule

2. The following diagram shows a cross section of a roof with support columns. The cross section is an isosceles triangle, with a base of 24 feet and a height of 9 feet. Five columns are placed at 4-foot intervals to support the roof. Each of the columns is perpendicular to the base of the cross section. Use this information to answer the questions that follow.
Common Core Georgia Performance Standard: MCC9–12.S.ID.4★
1.2.1: Normal Distributions and the 68–95–99.7 Rule

3. Find the area of the cross section of the roof. Use the formula .
What percent of the area of the cross section lies to the right of the centerline?
Find the heights of the support columns at the interval marking 8 feet and at the interval marking 16 feet.
What percent of the area of the cross section lies between the support columns at 8 feet and 16 feet?
1.2.1: Normal Distributions and the 68–95–99.7 Rule

4. Find the area of the cross section of the roof. Use the formula .
The cross section forms a triangle with a base of 24 feet and a height of 9 feet. The area of a triangle is equal to one-half the base times the height.
Formula for the area of a triangle
Substitute known values.
A = Multiply.
The area of the cross section is 108 ft2.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

5. What percent of the area of the cross section lies to the right of the centerline?
The cross section is symmetric to the centerline, so the area to the right of the centerline is equal to the area to the left of the centerline. Fifty percent of the area of the cross section lies to the right of the centerline.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

6. We have two options for finding each height. Method 1
Find the heights of the support columns at the interval marking 8 feet and at the interval marking 16 feet.
We have two options for finding each height.
Method 1
Determine the slope, which is , of one side of the roof.
The height of the cross section is 9 feet, and the length from the centerline of the cross section to the edge is 12 feet, as can be determined from the diagram.
The slope of one side of the roof is or .
1.2.1: Normal Distributions and the 68–95–99.7 Rule

7. Substitute known values. Multiply.
Since the support column at the interval marking 8 feet is a horizontal distance of 8 feet from the left edge of the roof (the run), the height of this column can be found by solving the slope equation for the rise.
Formula for slope
Substitute known values.
Multiply.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

8. The support column at this interval is 6 feet tall.
By symmetry, the height of the support column at the interval marking 16 feet is the same as the height at the interval marking 8 feet: 6 feet.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

9. Proportion using known values
Method 2
All of the right triangles formed by a support column, a portion of the base of the roof, and a portion of the sloped section of the roof are similar triangles. Thus, the height of the column at 8 feet is proportional to the height of the column at the centerline. Set up a proportion to determine the missing height.
Proportion using known values
Solve for x.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

10. The support columns at the interval marking 8 feet and at the interval marking 16 feet are each 6 feet tall.
Again by symmetry, the height of the support column at the interval marking 16 feet is the same as the height at the interval marking 8 feet: 6 feet.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

11. The area of a trapezoid can be found using the formula .
What percent of the area of the cross section lies between the support columns at 8 feet and 16 feet?
There are various methods for determining area. One such method is to find the area using the formula for a trapezoid. The region between the centerline and the support column at 8 feet forms a trapezoid, with base lengths of 6 feet and 9 feet and a height of 4 feet.
The area of a trapezoid can be found using the formula
1.2.1: Normal Distributions and the 68–95–99.7 Rule

12. Formula for the area of a trapezoid Substitute known values.
A = Simplify.
The area is 30 ft2.
By symmetry, the area of the cross section between the centerline and the support column at 16 feet is also 30 square feet. Thus, to find the area of the region between the support columns at the intervals marked as 8 feet and 16 feet, add 30 ft2 and 30 ft2.
= 60
The total area is 60 ft2.
1.2.1: Normal Distributions and the 68–95–99.7 Rule

13. To find this percent, divide 60 by 108.
In order to find the percent of the area between the support columns, calculate what percent 60 ft2 makes up of the total area of the cross section, 108 ft2.
To find this percent, divide 60 by 108.
The area between the support columns is of the total area.
Connection to the Lesson
Students will be finding proportions of values under the normal curve for specified intervals.
Students will be working with area as it relates to the probability in a normal distribution.
1.2.1: Normal Distributions and the 68–95–99.7 Rule