Problem 81 - Algebra and Trigonometry

Question

Radius of a Sphere A jeweler has three small solid spheres made of gold, of radius \(2 \mathrm{mm}, 3 \mathrm{mm},\) and 4 \(\mathrm{mm}\) . He decides to melt these down and make just one sphere out of them. What will the radius of this larger sphere be?

Step-by-Step Solution

Answer

The radius of the new sphere is approximately 4.63 mm.

Step 1: Understand the Problem

We need to find the radius of a larger sphere made by melting three smaller spheres together. This involves calculating the total volume of the three smaller spheres and then finding the radius of a new sphere that has the same volume.

Step 2: Write Down the Known Values

The radii of the small spheres are given as 2 mm, 3 mm, and 4 mm. We'll denote these radii as \(r_1 = 2\) mm, \(r_2 = 3\) mm, and \(r_3 = 4\) mm.

Step 3: Calculate the Volumes of the Small Spheres

The volume \(V\) of a sphere is calculated using the formula \(V = \frac{4}{3}\pi r^3\). So, calculate the individual volumes: - For sphere 1: \(V_1 = \frac{4}{3} \pi (2)^3 = \frac{32}{3}\pi\) - For sphere 2: \(V_2 = \frac{4}{3} \pi (3)^3 = 36\pi\) - For sphere 3: \(V_3 = \frac{4}{3} \pi (4)^3 = \frac{256}{3}\pi\)

Step 4: Sum the Volumes of the Smaller Spheres

Add up all the volumes to get the total volume of the gold: \[ V_{total} = V_1 + V_2 + V_3 = \left( \frac{32}{3} \pi + 36\pi + \frac{256}{3}\pi \right) \]Combine the terms: \[ V_{total} = \frac{32\pi + 256\pi}{3} + 36\pi = \frac{288\pi}{3} + 36\pi = 96\pi + 36\pi = 132\pi \]Ensure values are added correctly.

Step 5: Express the New Volume and Find Radius

The volume of the larger sphere is the total volume calculated: \[ V_{total} = 132\pi \]Use the volume formula for a sphere and solve for the new radius \(R\):\[ \frac{4}{3}\pi R^3 = 132\pi \]By canceling \(\pi\) and solving for \(R^3\):\[ \frac{4}{3}R^3 = 132 \]\[ R^3 = \frac{132 \times 3}{4} = 99 \]

Step 6: Calculate the Radius of the Larger Sphere

Take the cube root of both sides to find \(R\):\[ R = \sqrt[3]{99} \approx 4.63 \text{ mm} \]

Key Concepts

Radius Calculation

In measuring the dimensions of a sphere, the radius is a crucial aspect to understand. The radius is the distance from the center of the sphere to any point on its surface. To calculate the radius effectively, particularly when dealing with a sphere's volume, one can use the formula for the volume of a sphere: \(V = \frac{4}{3}\pi r^3\). Here, \(r\) represents the radius.When working on problems involving combined spheres, like in our example with different-sized gold spheres, the challenge is to determine the radius of a larger sphere made from the smaller ones. After calculating the total volume of these smaller spheres, the next step is to find the radius of a new sphere that maintains this combined volume.Often, this involves a bit of algebraic manipulation where you solve for \(r\) using the given volumes and the standard volume formula. Don't forget to take the cube root at the final step when isolating \(r\). Understanding these steps helps simplify tasks involving spherical dimensions.

Volume Addition

Adding volumes together is an essential part of working with multiple shapes. In our exercise, the jeweler melds three spheres, necessitating an understanding of volume addition. Each smaller sphere has its volume, calculated using \(V = \frac{4}{3}\pi r^3\).

• Sphere 1 with radius 2 mm has a volume of \(\frac{32}{3}\pi\).
• Sphere 2 with radius 3 mm equals \(36\pi\).
• Sphere 3 with radius 4 mm gives \(\frac{256}{3}\pi\).

These individual volumes are summed to find the total volume of gold, which becomes the volume of the new, larger sphere. Simply put, you add up the volumes like regular numbers, ensuring the units and bases are consistent. Remember, when combining volumes of like materials into a single piece, you ensure a conservation of volume.

Mathematical Problem-Solving

Problems like these involving spheres require a systematic approach to avoid errors and ensure accurate results. Let's break down the mathematical problem-solving process by highlighting key steps:

• **Understand the Problem:** Grasp what is being asked. Here, finding the new sphere's radius.
• **Set up Known Values:** Clearly write down all given data, such as radii of initial spheres.
• **Compute Individual Volumes:** Use the formula \(V = \frac{4}{3}\pi r^3\) to calculate each sphere's volume.
• **Combine Volumes:** Sum the volumes to find the total material that will make up the new sphere.
• **Solve for the New Radius:** Use the total volume to find the larger sphere's radius by reversing this sphere's volume formula.

Taking these steps ensures a logical flow of thought and actions, making even complex problems manageable. Stay organized, double-check calculations, and approach each problem as an opportunity to apply math effectively.