A circle's circumference is s feet, and its area is t square feet. Given that s = 3t, what's the circle's radius?
A. 1/6 ft
B. 1/3 ft
C. 2/3 ft
D. 2 ft
Breaking Down the Problem: Circumference and Area
Let's dive into this problem, guys! At its heart, this is a geometry problem testing our understanding of circles, specifically the relationship between circumference, area, and radius. We've got a circle, and we're told its circumference (s) and area (t) are related by the equation s = 3t. Our mission, should we choose to accept it, is to find the circle's radius. This involves recalling the formulas for circumference and area, setting up an equation, and solving for the unknown radius. Let's get started and untangle this circle conundrum together! Remember those formulas for circumference and area? They're our secret weapons here. We will translate the word problem into mathematical language, expressing the circumference and area in terms of the radius (r). Setting up the equation is key, using the given relationship (s = 3t) to link the two formulas together. This will give us an equation we can actually solve. Finally, the fun part: using our algebra skills to isolate the radius and find its value. Each step carefully executed will bring us closer to the solution, demystifying the relationship between a circle's dimensions and its fundamental properties. We'll see how changing the radius dramatically affects both the circumference, which grows linearly with the radius, and the area, which grows quadratically. Understanding this interplay is fundamental not only for solving this problem but also for grasping broader geometric concepts. So, let's embark on this geometrical journey, guys, and discover the hidden value of the radius! We'll see how a single equation can unlock the secrets of a circle's size and shape, connecting the circumference and area in a beautiful dance of numbers and geometry.
Recalling the Circle Formulas
Alright, first things first, let's jot down the essential formulas for a circle. The circumference (s) of a circle is given by: s = 2πr, where r is the radius. The area (t) of a circle is given by: t = πr². These two formulas are our bread and butter for solving this problem. They connect the radius, which we're trying to find, to the given information about circumference and area. Remembering these formulas is crucial, guys, as they form the foundation for our calculations. Without them, we'd be wandering in the geometric wilderness! Thinking about these formulas, we can already see how the radius plays a central role. It's like the keystone of the circle, dictating both how far around the circle goes (circumference) and the space it occupies (area). The relationship between these two formulas is also fascinating. The circumference grows linearly with the radius – double the radius, double the circumference. But the area grows quadratically – double the radius, quadruple the area! This difference in growth rates is what makes this problem interesting. We're given a specific relationship between circumference and area, and that constraint will help us pinpoint the exact radius of this particular circle. So, with our formulas at the ready, guys, let's move on to the next step and put them to work! We'll be translating the given information into a mathematical equation, and that's where the real problem-solving fun begins. Remember, math is a language, and these formulas are part of its vocabulary. By understanding the language, we can decipher the hidden messages within the problem and reveal the answer we seek.
Setting up the Equation
Now comes the clever part! We know that s = 2πr and t = πr², and we're also given that s = 3t. Let's substitute the formulas for s and t into this equation. This gives us: 2πr = 3(πr²). See what we did there, guys? We've taken a verbal relationship (s = 3t) and turned it into a concrete mathematical equation involving only the radius r. This is a crucial step because now we have an equation we can actually solve. It's like translating a secret code into plain English – once we understand the equation, we're much closer to finding the answer. This equation beautifully captures the interplay between circumference, area, and the given relationship. It's saying that the circumference of this circle is exactly three times its area. That's a pretty specific condition, and it strongly limits the possible values for the radius. In fact, there's only one radius that will satisfy this condition, and our job is to find it. Before we start solving, let's take a moment to appreciate the power of this equation. It's a concise representation of a geometric situation, and it holds all the information we need to find the solution. Now, it's time to roll up our sleeves and put our algebra skills to the test. We'll be simplifying this equation, isolating the radius, and ultimately revealing its value. So, guys, let's move on to the next stage, where we'll tackle the equation head-on and extract the answer!
Solving for the Radius
Time to put our algebra hats on! We've got the equation 2πr = 3πr². Our goal is to isolate r and find its value. First, let's divide both sides of the equation by π. This simplifies the equation to 2r = 3r². Now, to get all the terms on one side, subtract 2r from both sides: 0 = 3r² - 2r. Next, we can factor out an r from the right side: 0 = r(3r - 2). This gives us two possible solutions: r = 0 or 3r - 2 = 0. Since a circle can't have a radius of 0, we discard that solution. Let's solve 3r - 2 = 0. Add 2 to both sides: 3r = 2. Finally, divide both sides by 3: r = 2/3. And there we have it, guys! The radius of the circle is 2/3 feet. That wasn't so bad, was it? We took a somewhat complex geometric problem, translated it into an algebraic equation, and then used our skills to solve for the unknown. This is a classic example of how math connects different areas – geometry and algebra, in this case – to solve real problems. Each step in the process was crucial, from setting up the equation to carefully simplifying and factoring. A small mistake along the way could have led us to the wrong answer. But by being methodical and paying attention to detail, we successfully navigated the problem and found the solution. Now, let's take a moment to appreciate what we've accomplished. We've not only found the radius of the circle, but we've also reinforced our understanding of the relationship between a circle's circumference, area, and radius. These are fundamental concepts in geometry, and mastering them will serve us well in future mathematical adventures. So, give yourselves a pat on the back, guys! We've conquered this circle problem, and we're ready for the next challenge!
The Answer
Therefore, the radius of the circle is 2/3 feet, which corresponds to option C. We nailed it, guys! From the initial problem setup to recalling the area and circumference formulas, and then carefully solving the algebraic equation, we successfully navigated the problem and found the correct answer. This demonstrates the power of combining geometric understanding with algebraic skills. We started with a word problem, translated it into mathematical language, and then used our problem-solving toolkit to arrive at the solution. It's a beautiful illustration of how math can be used to describe and solve real-world situations. Looking back at the problem, we can appreciate the elegance of the solution. The relationship s = 3t provided a crucial constraint that allowed us to pinpoint the exact radius of the circle. Without this constraint, there would have been infinitely many possible circles with different radii. But with it, we were able to narrow down the possibilities and find the unique circle that satisfies the given condition. This is a common theme in math – constraints often lead to specific solutions. The more information we have, the more precisely we can define the problem and find its answer. So, guys, let's celebrate our success! We've not only solved this particular problem, but we've also strengthened our understanding of circles and their properties. And that's what math is all about – not just finding the right answer, but also building our knowledge and skills along the way. Now, we're better equipped to tackle similar problems in the future, and that's a rewarding feeling. On to the next mathematical adventure!
The final answer is