A fractal is created: A circle is drawn with radius 8 cm. Another circle is drawn with half the radius of the previous circle. The new circle is tangent to the previous circle. Suppose this pattern continues through five steps. What is the sum of the areas of the circles? Express your answer as an exact fraction.
area = π(8)^2 + π(4)^2 + π(2)^2 + π(1)^2 + π(.5)^2
= 64π + 16π + 4π + π + (1/4)π
this is a GS where a = 64π and r = 1/4
but , rather than using the formula in this case it is just as easy to just add them up
total = 85.25π
check:
area = 64π( 1 - (1/4)^5)/(3/4) = 85.25π
Let's break down the problem step by step:
Step 1: The first circle has a radius of 8 cm. Therefore, its area is A1 = π * (8 cm)^2 = 64π cm².
Step 2: The second circle has half the radius of the first circle. Therefore, its radius is 8 cm / 2 = 4 cm. Its area is A2 = π * (4 cm)^2 = 16π cm².
Step 3: The third circle has half the radius of the second circle. Therefore, its radius is 4 cm / 2 = 2 cm. Its area is A3 = π * (2 cm)^2 = 4π cm².
Step 4: The fourth circle has half the radius of the third circle. Therefore, its radius is 2 cm / 2 = 1 cm. Its area is A4 = π * (1 cm)^2 = π cm².
Step 5: The fifth circle has half the radius of the fourth circle. Therefore, its radius is 1 cm / 2 = 1/2 cm. Its area is A5 = π * (1/2 cm)^2 = π/4 cm².
To find the sum of the areas of all the circles, we add the areas together:
A_total = A1 + A2 + A3 + A4 + A5
= 64π cm² + 16π cm² + 4π cm² + π cm² + π/4 cm²
= (64 + 16 + 4 + 1 + 1/4)π cm²
= 85.25π cm²
So, the sum of the areas of the circles is 85.25π cm².
To find the sum of the areas of the circles, we need to calculate the area of each individual circle in the pattern and then add them all together.
Step 1: The first circle has a radius of 8 cm, so its area is π(8 cm)^2 = 64π cm^2.
Step 2: The second circle has half the radius of the first circle, so its radius is 8 cm * (1/2) = 4 cm. The area of the second circle is then π(4 cm)^2 = 16π cm^2.
Step 3: The third circle has half the radius of the second circle, so its radius is 4 cm * (1/2) = 2 cm. The area of the third circle is then π(2 cm)^2 = 4π cm^2.
Step 4: The fourth circle has half the radius of the third circle, so its radius is 2 cm * (1/2) = 1 cm. The area of the fourth circle is then π(1 cm)^2 = π cm^2.
Step 5: The fifth circle has half the radius of the fourth circle, so its radius is 1 cm * (1/2) = 0.5 cm. The area of the fifth circle is then π(0.5 cm)^2 = 0.25π cm^2.
Now let's add up the areas of all five circles:
64π cm^2 + 16π cm^2 + 4π cm^2 + π cm^2 + 0.25π cm^2 = 85.25π cm^2
Therefore, the sum of the areas of the circles is 85.25π cm^2.