Final answer:
To find the expected area of a circle when randomly selecting an integer n between 1 to 5 as the radius, calculate the area of each possible circle and find the average. The expected area, rounded to the hundredths, is approximately 34.54 cm².
Step-by-step explanation:
When you randomly choose an integer n between 1 to 5, and then draw a circle with a radius of n centimetres, the expected area is calculated by taking the average of the areas for all possible radii.
The area of a circle is given by the formula A = πr². Since the integer n can be 1, 2, 3, 4, or 5, we calculate the area for each of these radii and then find the average.
- For n=1: A = π(1)² = π cm²
- For n=2: A = π(2)² = 4π cm²
- For n=3: A = π(3)² = 9π cm²
- For n=4: A = π(4)² = 16π cm²
- For n=5: A = π(5)² = 25π cm²
Add all the areas and divide by the number of possible radii:
Expected Area = (π + 4π + 9π + 16π + 25π) / 5
Expected Area = (55π) / 5
Expected Area = 11π cm²
Now, use the value of π up to two decimal places (3.14) to get the expected area in hundredths of a square centimetre:
Expected Area ≈ 11 × 3.14 cm²
Expected Area ≈ 34.54 cm²
The expected area of the circle, to the nearest hundredths of a square centimetre, is approximately 34.54 cm².