A number of circles are given and the area of the smallest circles is 4π.The area of the next circle is 9/4 times of the previous circle.if there are 6 circles,determine the length of the diameter of the last circle.

Let's solve this problem step by step:

Step 1: Calculate the area of the first circle.
The area of the smallest circle is given as 4π.

Step 2: Calculate the radius of the first circle.
We can use the area formula of a circle to find the radius.
Area of a circle = π * r^2
4π = π * r^2
Dividing both sides by π, we get:
4 = r^2
Taking the square root of both sides, we get:
r = 2

Step 3: Calculate the area of the second circle.
The area of the second circle is 9/4 times the area of the first circle.
Area of the second circle = (9/4) * 4π = 9π

Step 4: Calculate the radius of the second circle.
Using the same formula as before, we get:
9π = π * r^2
Dividing both sides by π, we get:
9 = r^2
Taking the square root of both sides, we get:
r = 3

Step 5: Repeat steps 3 and 4 for the next four circles.
We can use the same formula to find the areas and radii of the next four circles.
Area of the third circle = (9/4) * 9π = 81/4 * π
Radius of the third circle = √(81/4) = 9/2

Area of the fourth circle = (9/4) * (81/4) * π = 729/16 * π
Radius of the fourth circle = √(729/16) = 27/4

Area of the fifth circle = (9/4) * (729/16) * π = 6561/64 * π
Radius of the fifth circle = √(6561/64) = 81/8

Area of the sixth (last) circle = (9/4) * (6561/64) * π = 59049/256 * π
Radius of the sixth (last) circle = √(59049/256) = 243/16

Step 6: Calculate the diameter of the last circle.
The diameter is twice the radius, so:
Diameter of the last circle = 2 * (243/16) = 486/16 = 30.375

Therefore, the length of the diameter of the last circle is 30.375 units.

To solve this problem, we need to find the area and diameter of each circle by applying the given information.

1. First circle:
Given: Area = 4π
Formula: Area = πr², where r is the radius of the circle.
Therefore, we can rearrange the formula to solve for the radius: r = √(Area/π) = √(4π/π) = 2.

Since the diameter is double the radius, the diameter of the first circle is 2 × 2 = 4.

2. Second circle:
Given: The area of the next circle is 9/4 times the area of the previous circle.
Formula: Area = πr²
The area of the second circle is (9/4) times the area of the first circle: (9/4) × 4π = 9π.

Again, solve for the radius: r = √(Area/π) = √(9π/π) = 3.

The diameter of the second circle is 2 × 3 = 6.

3. Third circle:
Given: The area of the next circle is 9/4 times the area of the previous circle.
The area of the third circle is (9/4) times the area of the second circle: (9/4) × 9π = 81/4 π.

The radius of the third circle is r = √(Area/π) = √((81π/4)/π) = √(81/4) = 9/2.

The diameter of the third circle is 2 × (9/2) = 9.

4. Fourth circle:
Using the same method, the radius of the fourth circle is r = (9/2) × (9/2) = 81/4.

The diameter of the fourth circle is 2 × (81/4) = 81/2.

5. Fifth circle:
The radius of the fifth circle is r = (81/2) × (81/2) = 6561/4.

The diameter of the fifth circle is 2 × (6561/4) = 6561/2.

6. Sixth circle:
The radius of the sixth circle is r = (6561/2) × (6561/2) = 43046721/4.

The diameter of the sixth circle is 2 × (43046721/4) = 43046721/2.

Therefore, the length of the diameter of the last circle is 43046721/2.