Problem 48 - Algebra A Combined Function

Question

Find the maximum number of goldfish you can put in a cylindrical tank whose diameter is 8 meters and whose height is 3 meters, if each goldfish needs 2 cubic meters of water. \(\left(V=\pi r^{2} h\right)\)

Step-by-Step Solution

Answer

The maximum number of goldfish is 75.

Step 1: Understanding the problem

We need to find the maximum number of goldfish that can fit in a cylindrical tank. Each goldfish requires 2 cubic meters of water, so first, we need to calculate the total volume of the tank using the formula for the volume of a cylinder.

Step 2: Determine the radius of the tank

The tank has a diameter of 8 meters. Since the radius is half of the diameter, we calculate the radius as follows:\[\text{Radius} = \frac{8}{2} = 4 \text{ meters}\]

Step 3: Calculate the volume of the tank

Using the formula for the volume of a cylinder \(V = \pi r^2 h\), we substitute the radius and height into the formula:\[V = \pi \times 4^2 \times 3 = \pi \times 16 \times 3 = 48\pi \text{ cubic meters}\]

Step 4: Calculate the maximum number of goldfish

Since each goldfish requires 2 cubic meters of water, we divide the total volume of the tank by 2 to find the maximum number of goldfish the tank can hold:\[\text{Maximum number of goldfish} = \frac{48\pi}{2} = 24\pi\]Since \(\pi\) is approximately 3.14, we calculate it as:\[24 \times 3.14 = 75.36\]

Step 5: Round down to the nearest whole number

Since we can't have a fraction of a goldfish, we round down 75.36 to the nearest whole number, which is 75.

Key Concepts

Radius and Diameter

To understand cylinder volume, it's essential to grasp the difference between radius and diameter. The diameter refers to the distance across the circle, passing through the center point. In our case, the tank’s diameter is given as 8 meters.

The radius, however, is the distance from the center of the circle to any point on its edge. It's exactly half of the diameter. By dividing the diameter by two, you can easily find the radius:

• Diameter of the tank: 8 meters
• Radius = Diameter ÷ 2 = 8 ÷ 2 = 4 meters

Understanding the radius is crucial since it's used in the formula to calculate the volume of a cylinder, which is our next focus.

Volume Calculation

The volume of a cylinder, like our tank, is how much space it can hold. For a cylinder, this volume is calculated using the formula: \[ V = \pi r^2 h \]where:

• \( V \) represents the volume
• \( r \) is the radius
• \( h \) is the height

For our tank:

• Radius \( (r) = 4 \; \text{meters} \)
• Height \( (h) = 3 \; \text{meters} \)

Substitute these values into the formula:\[ V = \pi \times 4^2 \times 3 = \pi \times 16 \times 3 = 48\pi \; \text{cubic meters} \]The volume tells us the total space inside the tank, which is essential for figuring out how many goldfish it can hold, as each needs a specific amount of water.

Mathematical Problem-Solving

Solving mathematical problems often involves breaking them down into manageable steps. For our task, we follow a structured approach to find out how many goldfish can fit in the tank. Once we know the total volume of the tank, our next step is using this volume efficiently.Each goldfish needs 2 cubic meters of water, meaning we must divide the tank's total volume by this amount. This division will give us the total number of goldfish the tank can accommodate:

• Volume of the tank = \( 48\pi \; \text{cubic meters} \)
• Each goldfish needs 2 cubic meters, so:
• Maximum number of goldfish = \( \frac{48\pi}{2} = 24\pi \)

Since \( \pi \approx 3.14 \), calculating 24 times \( \pi \) gives us 75.36. However, since you can't have a fraction of a goldfish, we round down to 75 goldfish in total! This systematic approach ensures that we've accounted for all details in calculation, producing a precise and practical solution.