Calculating Remaining Space In A Box: Ball & Can Problem

by SLV Team 57 views

Hey guys! Let's dive into a cool math problem today. We're going to figure out how much space is left in a box after we put a ball and a can inside. It sounds like a puzzle, right? So, grab your thinking caps, and let's get started!

Understanding the Problem

So, here's the deal: We have a square box with a volume of 1 cubic meter (1 m³). Inside this box, we need to fit a ball and a can. The ball has a radius of 40 centimeters (cm), and the can has a radius of 50 cm and a height of 65 cm. The big question is: How much space will be left in the box after we put these two items inside? To solve this, we need to calculate the volumes of the ball and the can and then subtract those volumes from the total volume of the box. But remember, the question gives us the key: we must convert all measurements to meters before we start calculating. This is super important because we need to work with consistent units to get the correct answer. Failing to convert units would be a classic mistake, like mixing apples and oranges! So, let's make sure we're all on the same page and convert those centimeters to meters before moving forward. Think of it as setting the stage for a flawless calculation!

Step 1: Converting to Meters

Before we can calculate any volumes, we need to convert all measurements from centimeters (cm) to meters (m). This is crucial because our box volume is given in cubic meters (m³), and we need to work with consistent units. Remember, there are 100 centimeters in 1 meter. So, to convert from centimeters to meters, we divide by 100. Let's start with the ball. The ball has a radius of 40 cm. To convert this to meters, we divide 40 by 100, which gives us 0.4 meters. So, the radius of the ball is 0.4 m. Next, let's convert the dimensions of the can. The can has a radius of 50 cm. Dividing 50 by 100, we get 0.5 meters. So, the radius of the can is 0.5 m. The can also has a height of 65 cm. Dividing 65 by 100, we get 0.65 meters. So, the height of the can is 0.65 m. Now that we've converted all the measurements to meters, we're ready to move on to the next step: calculating the volumes of the ball and the can. This is where the fun really begins, as we get to apply our geometry knowledge to solve the problem!

Step 2: Calculating the Volume of the Ball

Now that we have the radius of the ball in meters (0.4 m), we can calculate its volume. The formula for the volume of a sphere (which is the shape of our ball) is V = (4/3)πr³, where V is the volume, π (pi) is approximately 3.14159, and r is the radius. So, let's plug in the values: V = (4/3) * 3.14159 * (0.4)³. First, we need to calculate 0.4 cubed (0.4³), which means 0.4 * 0.4 * 0.4. This equals 0.064. Now we have: V = (4/3) * 3.14159 * 0.064. Next, let's multiply 3.14159 by 0.064, which gives us approximately 0.20106. So, the equation becomes: V = (4/3) * 0.20106. Now, multiply 0.20106 by 4, which equals 0.80424. Finally, divide 0.80424 by 3 to get the volume: V ≈ 0.268 m³. So, the volume of the ball is approximately 0.268 cubic meters. We've successfully calculated the volume of the ball, which is a significant step towards solving the overall problem. Next, we'll tackle the volume of the can, and then we'll be able to determine how much space is left in the box. Keep up the great work, guys!

Step 3: Calculating the Volume of the Can

Alright, let's move on to calculating the volume of the can. Since the can is shaped like a cylinder, we'll use the formula for the volume of a cylinder: V = πr²h, where V is the volume, π (pi) is approximately 3.14159, r is the radius, and h is the height. We already know the radius of the can is 0.5 meters and the height is 0.65 meters. Let's plug those values into the formula: V = 3.14159 * (0.5)² * 0.65. First, we need to calculate 0.5 squared (0.5²), which means 0.5 * 0.5. This equals 0.25. So, the equation becomes: V = 3.14159 * 0.25 * 0.65. Next, let's multiply 3.14159 by 0.25, which gives us approximately 0.7854. Now we have: V = 0.7854 * 0.65. Finally, multiply 0.7854 by 0.65 to get the volume: V ≈ 0.5105 m³. So, the volume of the can is approximately 0.5105 cubic meters. We've now calculated the volumes of both the ball and the can. This is fantastic progress! We're just one step away from finding the remaining space in the box. In the next step, we'll combine these volumes and subtract them from the total volume of the box to get our final answer. You guys are doing awesome!

Step 4: Calculating the Remaining Space

We're in the home stretch now! We've calculated the volume of the ball (approximately 0.268 m³) and the volume of the can (approximately 0.5105 m³). Now, we need to find the total volume occupied by both the ball and the can. To do this, we simply add the two volumes together: Total volume occupied = Volume of ball + Volume of can Total volume occupied = 0.268 m³ + 0.5105 m³ Total volume occupied ≈ 0.7785 m³ So, the ball and the can together take up approximately 0.7785 cubic meters of space. Now, let's find out how much space is left in the box. We know the box has a total volume of 1 m³. To find the remaining space, we subtract the total volume occupied by the ball and the can from the total volume of the box: Remaining space = Total volume of box - Total volume occupied Remaining space = 1 m³ - 0.7785 m³ Remaining space ≈ 0.2215 m³ Therefore, there is approximately 0.2215 cubic meters of space left in the box. We've solved it! You guys did an amazing job following along and working through the steps. High five!

Final Answer

So, to wrap it all up, after placing the ball and the can inside the 1 m³ box, there is approximately 0.2215 m³ of space remaining. Isn't it cool how we used math to solve a real-world problem? We converted units, calculated volumes, and subtracted to find our answer. Math is like a superpower, guys! It helps us understand and solve all sorts of interesting puzzles. I hope you enjoyed this exercise and learned something new today. Keep exploring the world of math, and who knows what other exciting challenges you'll conquer! Remember, every problem is just a chance to learn and grow. Keep up the awesome work!