A kennel is to be enclosed with a 20m of fencing. The length of the kennel is to be an integer length and is to be chosen randomly. What is the expected area of the kennel?
To find the expected area of the kennel, we need to consider all the possible lengths of the kennel.
Let's define the length of the kennel as "x". Given that the total length of fencing available is 20m, we can write the equation:
2x + 4 = 20
Simplifying this equation, we get:
2x = 16
x = 8
So, the kennel can have a length of 8 meters.
To calculate the expected area, we need to consider all the possible lengths (from 1 to 8) and calculate the corresponding areas.
Using the formula for the area of the rectangle, which is length x width, the area of the kennel can be calculated as:
Area = x * (20 - 2x)
Let's calculate the areas for each possible length from 1 to 8:
For x = 1, Area = 1 * (20 - 2) = 1 * 18 = 18
For x = 2, Area = 2 * (20 - 4) = 2 * 16 = 32
For x = 3, Area = 3 * (20 - 6) = 3 * 14 = 42
For x = 4, Area = 4 * (20 - 8) = 4 * 12 = 48
For x = 5, Area = 5 * (20 - 10) = 5 * 10 = 50
For x = 6, Area = 6 * (20 - 12) = 6 * 8 = 48
For x = 7, Area = 7 * (20 - 14) = 7 * 6 = 42
For x = 8, Area = 8 * (20 - 16) = 8 * 4 = 32
Now, let's calculate the probabilities for each possible length:
The length can be 1 with a probability of 1/8.
The length can be 2 with a probability of 1/8.
The length can be 3 with a probability of 1/8.
The length can be 4 with a probability of 1/8.
The length can be 5 with a probability of 1/8.
The length can be 6 with a probability of 1/8.
The length can be 7 with a probability of 1/8.
The length can be 8 with a probability of 1/8.
To calculate the expected area, we multiply each area by its corresponding probability and sum them up:
Expected Area = (18 * 1/8) + (32 * 1/8) + (42 * 1/8) + (48 * 1/8) + (50 * 1/8) + (48 * 1/8) + (42 * 1/8) + (32 * 1/8)
Expected Area = 3.875
Therefore, the expected area of the kennel is approximately 3.875 square meters.
To find the expected area of the kennel, we need to consider all possible lengths for the kennel and calculate the corresponding areas for each length. Then we can calculate the average or expected area.
Let's start by considering the possible lengths for the kennel. Since it is given that the length should be an integer, we need to find all the integer values that satisfy the given conditions.
Given that the perimeter of the kennel is 20m, we can write an equation to represent this:
2 * length + 2 * width = 20
Since the width is not given, we can assume it to be a constant value or take it as a variable. Let's assume the width as "w".
So the equation becomes:
2 * length + 2 * w = 20
We need to rearrange the equation to express length in terms of w:
length = (20 - 2 * w) / 2
length = 10 - w
From this equation, we can see that the length will vary depending on the width "w". To calculate the possible lengths, we need to consider integer values for "w" between 1 and 9, as any value greater than 10 would result in a negative length or a non-integral value.
Now let's calculate the area for each possible length and find the expected area.
For w = 1:
length = 10 - 1 = 9
area = length * width = 9 * 1 = 9
For w = 2:
length = 10 - 2 = 8
area = length * width = 8 * 2 = 16
Similarly, we can calculate the areas for w = 3, 4, 5, 6, 7, 8, and 9.
w = 3:
length = 10 - 3 = 7
area = 7 * 3 = 21
w = 4:
length = 10 - 4 = 6
area = 6 * 4 = 24
w = 5:
length = 10 - 5 = 5
area = 5 * 5 = 25
w = 6:
length = 10 - 6 = 4
area = 4 * 6 = 24
w = 7:
length = 10 - 7 = 3
area = 3 * 7 = 21
w = 8:
length = 10 - 8 = 2
area = 2 * 8 = 16
w = 9:
length = 10 - 9 = 1
area = 1 * 9 = 9
To find the expected area, we calculate the average area by summing up all the calculated areas and dividing by the total number of possible lengths (which is 9 in this case).
Expected area = (9 + 16 + 21 + 24 + 25 + 24 + 21 + 16 + 9) / 9
Expected area = 165 / 9
Expected area ≈ 18.33 square meters
Therefore, the expected area of the kennel is approximately 18.33 square meters.
area=l*w=
but 2L+2w=20 or w=10-L
area= L(10-L)
but L is random. 1,2,3,4,5 but less than 10
expected area, if L can be 1,2,3,4,5,6,7,8,9, then 5 is the mean, or average, or expected after many trials.
Area= 25