Question

suppose you randomly choose an integer n between 1 to 5, and then draw a circle with a radius of n centimetres. What is the expected area of this circle to the nearest hundredths of a square centimetres?

Answer 1

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.

1. For n=1: A = π(1)² = π cm²
2. For n=2: A = π(2)² = 4π cm²
3. For n=3: A = π(3)² = 9π cm²
4. For n=4: A = π(4)² = 16π cm²
5. 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.

Answer 2

The set of possible integers is 1, 2, 3, 4, and 5.

The area of a circle with radius of n centimeters = π(n)^2.

So, the set of possible values are:

n area = π(n^2)

1 π

2 4π

3 9π

4 16π

5 25π

And the expected value of the area may be determined as the mean (average) of the five possible areas:

expected value of the area = [ π+ 4π + 9π + 16π + 25π] / 5 = 11π ≈ 34.5575, which rounded to the nearest hundreth is 34.56

Answer: 34.56