Conical Tent Canvas Cost: A Step-by-Step Calculation

Hey guys! Ever wondered how much canvas you'd need to make a conical tent? And more importantly, how much it would cost? Well, you've come to the right place! Let's break down a problem where we calculate the cost of canvas for a conical tent. We'll go through each step, making it super easy to follow along. This will help you understand not just this specific problem, but also the general principles of calculating surface areas and costs – super useful for everyday life and maybe even your next camping trip! Let's dive in and get those math muscles working!

Understanding the Problem

First things first, let's get a grasp of what we're dealing with. The main challenge presented to us is to determine the financial cost associated with the canvas material needed to construct a conical tent. This tent has specific dimensions: a height of 16 meters and a base diameter of 24 meters. Additionally, we have the cost of the canvas itself, which is priced at Rs 210 per square meter. To solve this, we need to figure out the surface area of the tent (the part that needs canvas) and then multiply that by the cost per square meter. Easy peasy, right? Well, maybe after we break it down a bit more! We need to use the given height and diameter to calculate the slant height, and then use that to find the curved surface area. So, keep reading, we'll guide you through the steps.

To accurately calculate the cost, it is essential to recognize that the conical tent's canvas requirement corresponds to its curved surface area. Think of it this way: the canvas will wrap around the cone, but it won't cover the circular base on the ground. Therefore, we need to find the formula for the curved surface area of a cone. This formula involves the radius of the base and the slant height of the cone. We have the diameter, so getting the radius is a snap (just divide by two!). But what about the slant height? That's where our good friend the Pythagorean theorem comes in! The slant height, the height of the cone, and the radius form a right-angled triangle. Remember that? We'll use that to calculate the slant height. Once we have all the pieces, we'll plug them into the curved surface area formula and get our answer in square meters. This value represents the exact amount of canvas required to construct the tent.

Finally, to arrive at the ultimate solution – the cost – we must apply a straightforward calculation. This involves multiplying the previously determined curved surface area, measured in square meters, by the cost of the canvas per square meter. Given that the canvas costs Rs 210 for every square meter, this multiplication will yield the total expenditure required for the canvas. This final figure represents the financial investment needed to procure the material necessary for building the conical tent. So, you see, it's all connected! We started with the dimensions of the tent, worked our way through surface area, and now we're at the final cost. This methodical approach is key to solving any math problem. Keep this in mind, and you'll be a problem-solving pro in no time!

Calculating the Radius and Slant Height

Okay, let's get down to the nitty-gritty calculations! Remember, we need to find the curved surface area of the cone, and for that, we need the radius and the slant height. We already have the diameter, so finding the radius is the first easy step. The radius is simply half of the diameter. The problem states that the base diameter of the conical tent is 24 meters. To find the radius (r), we divide the diameter by 2:

r = Diameter / 2 r = 24 m / 2 r = 12 m

So, there we have it! The radius of the base of our conical tent is 12 meters. One piece of the puzzle down!

Now for the slant height – this one requires a little more Pythagorean magic. The slant height (l), the height of the cone (h), and the radius (r) form a right-angled triangle. The slant height is the hypotenuse (the longest side) of this triangle. We can use the Pythagorean theorem, which states: a² + b² = c², where a and b are the lengths of the two shorter sides, and c is the length of the hypotenuse. In our case, a = height (h), b = radius (r), and c = slant height (l). The height of the tent is given as 16 meters. So, we can plug in the values we have:

l² = h² + r² l² = (16 m)² + (12 m)² l² = 256 m² + 144 m² l² = 400 m²

To find the slant height (l), we need to take the square root of both sides:

l = √400 m² l = 20 m

Fantastic! We've calculated the slant height to be 20 meters. We now have both the radius (12 meters) and the slant height (20 meters). We're one step closer to finding the cost of the canvas!

Determining the Curved Surface Area

Alright, with the radius and slant height in hand, we're ready to calculate the curved surface area (CSA) of the conical tent. This is the area that the canvas will cover. The formula for the curved surface area of a cone is pretty straightforward:

CSA = πrl

Where:

• π (pi) is approximately 3.14159
• r is the radius of the base
• l is the slant height

We already know that r = 12 meters and l = 20 meters. So, let's plug those values into the formula:

CSA = π * 12 m * 20 m CSA = 3.14159 * 12 m * 20 m CSA ≈ 753.98 m²

Therefore, the curved surface area of the conical tent is approximately 753.98 square meters. This means we need about 753.98 square meters of canvas to make the tent. We're on the home stretch now!

Calculating the Total Cost of Canvas

We've made it to the final step – calculating the total cost of the canvas! We know the curved surface area of the tent (the amount of canvas needed) and the cost per square meter of canvas. To find the total cost, we simply multiply these two values together. The problem states that the cost of the canvas is Rs 210 per square meter. We've calculated the curved surface area to be approximately 753.98 square meters. So, the total cost is:

Total cost = CSA * Cost per square meter Total cost = 753.98 m² * Rs 210/m² Total cost ≈ Rs 158335.80

Therefore, the cost of the canvas required to make the conical tent is approximately Rs 158335.80. Wow, that's a lot of rupees! But hey, we figured it out! We took a real-world problem, broke it down into smaller steps, and used our math skills to solve it. You guys are awesome!

Conclusion

So, there you have it! We've successfully calculated the cost of the canvas needed to make a conical tent. We started by understanding the problem, then we calculated the radius and slant height, followed by the curved surface area, and finally, the total cost. This problem demonstrates how math is used in everyday situations. Knowing how to calculate surface areas and costs can be super handy, whether you're planning a camping trip or just trying to figure out how much material you need for a DIY project. Remember, the key is to break down complex problems into smaller, manageable steps. And don't be afraid to use your math skills! Keep practicing, and you'll become a math master in no time! Great job everyone! You tackled this problem like champs!