Math Problem How Much Cardboard Will Santiago Have Left Over After Making 10 Piñata Hats

Hey guys! Let's dive into a fun math problem with a crafty twist. Santiago wants to make some awesome piñata hats for a party, and we need to figure out how much cardboard he'll have left over. This is a cool mix of geometry and practical problem-solving, so let's get started!

Understanding the Project: Piñata Hats and Cardboard

Our main goal is to calculate how much cardboard Santiago will have leftover after making 10 piñata hats. Each hat needs to have a diameter of 14 cm and a height of 24 cm. Santiago has one big sheet of cardboard that measures 70 cm by 100 cm. This means we're dealing with a classic surface area problem, but with a real-world application. So, grab your thinking caps, and let’s break this down step by step.

Figuring Out the Hat Shape

First things first, we need to recognize what shape we're working with. A piñata hat, in this case, is essentially a cone (or, more accurately, the lateral surface of a cone, since we don’t need the base). To figure out how much cardboard each hat needs, we need to calculate the lateral surface area of a cone. This involves a bit of geometry, but don’t worry, we’ll walk through it together.

The Lateral Surface Area Formula

The formula for the lateral surface area (*LSA*) of a cone is:

**LSA = πrl**

Where:

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

We know the diameter of the base (14 cm), so we can easily find the radius (*r*). The radius is half the diameter, so:

**r = 14 cm / 2 = 7 cm**

But what about the slant height (*l*)? We have the height of the cone (24 cm), and we can use the Pythagorean theorem to find the slant height. Imagine a right triangle inside the cone, where the height of the cone and the radius form the two shorter sides, and the slant height is the hypotenuse.

Calculating the Slant Height

The Pythagorean theorem states:

**a² + b² = c²**

In our case:

• a = height of the cone = 24 cm
• b = radius = 7 cm
• c = slant height (*l*), which we want to find

So, let's plug in the numbers:

**24² + 7² = l²**

**576 + 49 = l²**

**625 = l²**

To find *l*, we take the square root of 625:

**l = √625 = 25 cm**

Now we have all the pieces we need to calculate the lateral surface area of one hat!

Calculating the Surface Area for One Hat

Let's plug the values we found into the lateral surface area formula:

**LSA = πrl**

**LSA = 3.14159 × 7 cm × 25 cm**

**LSA ≈ 549.78 cm²**

So, each hat needs approximately 549.78 square centimeters of cardboard. That's a crucial number! Now, let’s see how this translates when making ten hats.

Scaling Up: Ten Hats and Total Cardboard Needed

Now that we know how much cardboard one hat requires, let's figure out the total amount of cardboard Santiago needs for all ten hats. This is a simple multiplication, but it’s a vital step in solving the problem.

Total Cardboard for Ten Hats

To find the total area, we multiply the area for one hat by the number of hats:

**Total Area = 549.78 cm² / hat × 10 hats**

**Total Area ≈ 5497.8 cm²**

So, Santiago needs approximately 5497.8 square centimeters of cardboard to make all ten hats. That’s quite a bit! But don’t worry, we’re almost there. Next, we need to figure out how much cardboard Santiago has to begin with.

Cardboard Sheet Area

Santiago has a sheet of cardboard that measures 70 cm by 100 cm. To find the total area of the cardboard sheet, we simply multiply the length by the width. This will tell us the total amount of cardboard Santiago has available.

Calculating the Total Cardboard Area

The area of the cardboard sheet is:

**Area = length × width**

**Area = 70 cm × 100 cm**

**Area = 7000 cm²**

So, Santiago has 7000 square centimeters of cardboard to work with. Now we're talking! We know the total cardboard available and the total cardboard needed. The last step is to find out how much is left over.

The Grand Finale: Cardboard Leftover Calculation

We're in the home stretch! We know how much cardboard Santiago started with and how much he needs for the hats. The final step is to subtract the total area needed for the hats from the total area of the cardboard sheet. This will tell us how much cardboard Santiago has left over. This is where we get the answer to our original question.

Calculating the Leftover Cardboard

To find the leftover cardboard, we subtract the total area needed from the total area available:

**Leftover Area = Total Cardboard Area - Total Area Needed**

**Leftover Area = 7000 cm² - 5497.8 cm²**

**Leftover Area ≈ 1502.2 cm²**

So, Santiago will have approximately 1502.2 square centimeters of cardboard leftover. That’s a pretty good amount! He might even be able to make a few extra decorations.

Conclusion: Santiago's Cardboard Success!

We did it! We figured out that Santiago will have approximately 1502.2 square centimeters of cardboard left over after making his 10 piñata hats. This problem was a great way to see how math, especially geometry and surface area calculations, can be applied to everyday situations. Remember, guys, math is all around us, even when we’re making piñata hats!

This problem showed us the importance of breaking down a big task into smaller, manageable steps. We started by understanding the problem, then calculated the surface area of one hat, scaled it up for ten hats, figured out the total cardboard area, and finally, calculated the leftover cardboard. Each step was crucial to reaching the final answer. So next time you're tackling a tricky problem, remember to take it one step at a time, and you'll get there!

And that’s a wrap! Hope you enjoyed this math adventure. Keep exploring, keep learning, and most importantly, keep having fun with math!