Solving Circle Circumference In Arithmetic Sequence

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Hey math enthusiasts! Today, we're diving into a fun geometry problem involving circles and arithmetic sequences. Get ready to flex those math muscles! We'll be figuring out the circumference of a circle in a sequence where the areas of the circles follow a specific pattern. Let's break down the problem and find out how to solve it together.

Understanding the Problem: Circles in Arithmetic Progression

Alright, guys, let's get our heads around the situation. We have five circles, and their areas are arranged in an arithmetic sequence. This means the difference between the areas of consecutive circles is constant. Think of it like climbing stairs; each step (or circle area difference) is the same height.

We're given two key pieces of information: the area of the largest circle (1386 cm²) and the area of the smallest circle (154 cm²). Our mission? To calculate the circumference of the circle in the third position. Remember, the circumference is the distance around the circle – like the length of a hula hoop! We will use π=227\pi = \frac{22}{7} for the calculations. Now, let's decode this puzzle step by step.

Firstly, let's denote the areas of the five circles as A1, A2, A3, A4, and A5, where A1 is the smallest area and A5 is the largest. Since they form an arithmetic sequence, the difference between consecutive terms is constant. Mathematically, this can be represented as A2 - A1 = A3 - A2 = A4 - A3 = A5 - A4 = d, where d is the common difference.

We are given A1 = 154 cm² and A5 = 1386 cm². We can express A5 in terms of A1 and the common difference d. Because there are four intervals of d from A1 to A5, we have A5 = A1 + 4d. Substituting the given values, we get 1386 = 154 + 4d.

Solving for d, we find that 4d = 1386 - 154 = 1232, which gives us d = 308 cm². This means that the area of each successive circle increases by 308 cm².

Now we can calculate the area of the third circle, A3. We know that A3 = A1 + 2d. Substituting the known values gives us A3 = 154 + 2(308) = 154 + 616 = 770 cm².

To find the circumference of the third circle, we need to find its radius. We know that the area of a circle is given by the formula A = πr², where r is the radius. For the third circle, we have 770 = (22/7)r². Rearranging the formula, we get r² = (770 * 7) / 22 = 245. Therefore, r = √245 = 7√5 cm.

Finally, the circumference C of a circle is given by the formula C = 2πr. Substituting the values, we get C = 2 * (22/7) * 7√5 = 44√5 cm. Therefore, the circumference of the circle in the third position is 44√5 cm.

Decoding the Arithmetic Sequence: Area and Radius

Alright, let's get into the nitty-gritty of this problem. The core concept here is understanding how the areas of the circles relate to each other within an arithmetic sequence. Because the areas form an arithmetic sequence, the difference between consecutive areas is constant. This constant difference is the key to unlocking this problem.

We know the area of the largest circle, A5, and the area of the smallest circle, A1. Using these two values, we can find the common difference, d. The area of any circle in the sequence can be expressed as An = A1 + (n-1)d, where n is the position of the circle in the sequence. So, the area of the third circle, A3, can be found by using this formula.

Now, let's talk about the relationship between area and radius. The area of a circle is πr², where r is the radius and π (pi) is approximately 3.14159, but we're given that π=227\pi = \frac{22}{7}. Knowing the area, we can find the radius by rearranging the formula to r = √(A/π).

Once we know the radius, we can calculate the circumference, which is 2πr. This is the final step, and it gives us our answer. Remember, the circumference is the distance around the circle.

So, to recap, we'll be using the arithmetic sequence properties to find the common difference, calculate the area of the third circle, find its radius, and finally, calculate its circumference. Sounds like a plan, right?

Now, let's break down the steps in detail. First, we'll find the common difference (d) using the areas of the smallest and largest circles. We know A1 and A5, and we can express A5 in terms of A1 and d. Then we solve for d.

Next, we will calculate A3 using the formula A3 = A1 + 2d. With A3 in hand, we can find the radius of the third circle using the formula r = √(A3/π). Finally, we calculate the circumference using the formula C = 2πr.

Step-by-Step Solution: Unraveling the Circumference

Okay, let's get our hands dirty with the calculations. We're going to break this down into manageable steps so you can follow along easily. Ready? Let's go!

  1. Finding the Common Difference (d): We know A1 = 154 cm² and A5 = 1386 cm². Since the areas are in an arithmetic sequence, A5 = A1 + 4d. Plugging in the values, we get 1386 = 154 + 4d. Solving for d, we have:

    • 4d = 1386 - 154 = 1232
    • d = 1232 / 4 = 308 cm²

  2. Calculating the Area of the Third Circle (A3): Now that we have d, we can find A3. Remember, A3 = A1 + 2d. So:

    • A3 = 154 + 2(308) = 154 + 616 = 770 cm²

  3. Determining the Radius of the Third Circle (r): The area of a circle is A = πr². We have A3 = 770 cm² and π = 22/7. Let's solve for r:

    • 770 = (22/7)r²
    • r² = (770 * 7) / 22 = 245
    • r = √245 = 7√5 cm

  4. Calculating the Circumference of the Third Circle (C): Finally, the circumference is C = 2πr. We have r = 7√5 cm and π = 22/7. So:

    • C = 2 * (22/7) * 7√5
    • C = 44√5 cm

Therefore, the circumference of the circle in the third position is 44√5 cm. Bam! We solved it!

Practical Applications and Further Exploration

This problem isn't just about math; it has real-world applications! The concepts of arithmetic sequences and circle properties are used in various fields. For example, architects might use these principles when designing circular structures, or engineers could use them to calculate the optimal size and spacing of pipes in a system.

Want to explore further? Try these:

  • Change the values: What if the largest circle had an area of 2000 cm²? How would that change the calculations?
  • Different sequences: Instead of an arithmetic sequence, what if the areas formed a geometric sequence? How would the solution change?
  • Real-world examples: Look for circular objects around you and try to estimate their circumferences. Measure the areas and see how close your estimations are.

Math is all around us, guys. The more we practice and apply it, the more we'll understand and appreciate its beauty!

In conclusion, we've successfully navigated this problem using our knowledge of arithmetic sequences, circle areas, and circumferences. Keep practicing, keep exploring, and keep that math spark alive! Until next time, happy calculating!