The diameter of the first circle in an infinite sequence of circles is 12 em. Each subsequent
circle’s diameter is two thirds of the diameter of the previous circle. Calculate
(1) the total circumference of all the circles.
(2) the total area of the all the circles.

Question

UpStudy AI · Accepted Answer

We start with a sequence of circles whose diameters form a geometric sequence. The first circle has a diameter of $$12$$ em, and each subsequent circle’s diameter is $$\frac{2}{3}$$ times that of the previous circle.

### Part (1): Total Circumference

The circumference $$C$$ of a circle with diameter $$d$$ is given by:
$$
C = \pi d
$$
Thus, the first circle’s circumference is:
$$
C_1 = \pi \times 12 = 12\pi
$$
The next circle has a diameter:
$$
12 \times \frac{2}{3} = 8 \quad \text{em}
$$
and its circumference is:
$$
C_2 = \pi \times 8 = 8\pi
$$

Since each diameter is multiplied by $$\frac{2}{3}$$, the circumferences form a geometric series with first term:
$$
a = 12\pi
$$
and common ratio:
$$
r = \frac{2}{3}
$$

For an infinite geometric series with $$|r| < 1$$, the sum $$S$$ is:
$$
S = \frac{a}{1 - r}
$$
Substituting the values, we have:
$$
S_{\text{circumference}} = \frac{12\pi}{1 - \frac{2}{3}} = \frac{12\pi}{\frac{1}{3}} = 36\pi
$$

### Part (2): Total Area

The area $$A$$ of a circle with diameter $$d$$ (or radius $$r = \frac{d}{2}$$) is given by:
$$
A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}
$$

For the first circle (diameter $$12$$ em):
$$
A_1 = \frac{\pi (12)^2}{4} = \frac{\pi \times 144}{4} = 36\pi
$$

For the next circle, the diameter is $$12 \times \frac{2}{3} = 8$$ em, so:
$$
A_2 = \frac{\pi (8)^2}{4} = \frac{\pi \times 64}{4} = 16\pi
$$

To find the common ratio for the areas, note that if the diameter is multiplied by $$\frac{2}{3}$$, then the area (involving $$d^2$$) is multiplied by:
$$
\left(\frac{2}{3}\right)^2 = \frac{4}{9}
$$

Thus, the areas also form a geometric series with first term:
$$
a = 36\pi
$$
and common ratio:
$$
r = \frac{4}{9}
$$

The sum of the infinite series is:
$$
S_{\text{area}} = \frac{36\pi}{1 - \frac{4}{9}} = \frac{36\pi}{\frac{5}{9}} = 36\pi \times \frac{9}{5} = \frac{324\pi}{5}
$$

### Final Answers

1. The total circumference of all the circles is:
   $$
   36\pi \quad \text{em}
   $$
2. The total area of all the circles is:
   $$
   \frac{324\pi}{5} \quad \text{square em}
   $$
The total circumference of all the circles is $$36\pi$$ em, and the total area is $$\frac{324\pi}{5}$$ square em.

The diameter of the first circle in an infinite sequence of circles is 12 em. Each subsequent
circle’s diameter is two thirds of the diameter of the previous circle. Calculate
(1) the total circumference of all the circles.
(2) the total area of the all the circles.

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The diameter of the first circle in an infinite sequence of circles is 12 em. Each subsequent circle’s diameter is two thirds of the diameter of the previous circle. Calculate (1) the total circumference of all the circles. (2) the total area of the all the circles.

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Latest Geometry Questions

The diameter of the first circle in an infinite sequence of circles is 12 em. Each subsequent
circle’s diameter is two thirds of the diameter of the previous circle. Calculate
(1) the total circumference of all the circles.
(2) the total area of the all the circles.