Question

chapter 8 practice quiz

1. with the information below, answer the following:

estimate the area of idaho using only one polygon
estimate the area of idaho using 2 polygons.
the actual area of idaho is about 83,564 miles squared. which estimate is more accurate? how could you increase the accuracy of your estimate?

Explanation:

Step1: Estimate with one polygon

Assume the state can be approximated as a rectangle. If we consider the length along one - side as approximately $300$ miles and the other side as approximately $300$ miles (by looking at the rough dimensions from the given information). The area of a rectangle is $A = l\times w$. So, $A_1=300\times300 = 90000$ square miles.

Step2: Estimate with two polygons

We can divide the shape into two rectangles. Let's say one rectangle has dimensions $150\times300$ and the other has dimensions $150\times300$. The area of the first rectangle $A_{r1}=150\times300 = 45000$ square miles, and the area of the second rectangle $A_{r2}=150\times300 = 45000$ square miles. Then the total area $A_2=A_{r1}+A_{r2}=45000 + 45000=90000$ square miles.

Step3: Analyze accuracy

The estimate using one or two polygons gives an area of $90000$ square miles, while the actual area is $83564$ square miles. To increase the accuracy, we can use more polygons to better fit the irregular shape of Idaho. For example, using smaller rectangles or other polygons like triangles to fill in the gaps more precisely.

Answer:

• Estimate with one polygon: 90000 square miles
• Estimate with two polygons: 90000 square miles
• Which estimate is better: Neither is clearly better as they are the same in this simple - case approximation.
• How to increase accuracy: Use more and smaller polygons to fit the shape more precisely.