Understanding the Tank Volume Calculation

This problem asks us to find the volume of a tank with a specific shape using the mean-area method. The tank has a depth of 6 meters. Its top surface is a rectangle measuring 6 meters by 4 meters, and its bottom surface is a rectangle measuring 4 meters by 2 meters.

Applying the Mean-Area Method

The mean-area method is a way to estimate the volume of shapes that taper, like a frustum or this tank. It works by calculating the average of the areas of the two parallel surfaces (top and bottom) and then multiplying this average area by the perpendicular distance between them (the depth).

Calculating Surface Areas

First, let's calculate the area of the top and bottom rectangular surfaces:

• Area of the Top Surface (A_top): The top is a rectangle with dimensions 6 m x 4 m. $ A_{top} = \text{Length}_{top} \times \text{Width}_{top} $ $ A_{top} = 6 \, \text{m} \times 4 \, \text{m} $ $ A_{top} = 24 \, \text{m}^2 $
• Area of the Bottom Surface (A_bottom): The bottom is a rectangle with dimensions 4 m x 2 m. $ A_{bottom} = \text{Length}_{bottom} \times \text{Width}_{bottom} $ $ A_{bottom} = 4 \, \text{m} \times 2 \, \text{m} $ $ A_{bottom} = 8 \, \text{m}^2 $

Calculating the Mean Area

Next, we find the average area (mean area) between the top and bottom surfaces:

• Mean Area (A_mean): $ A_{mean} = \frac{A_{top} + A_{bottom}}{2} $ $ A_{mean} = \frac{24 \, \text{m}^2 + 8 \, \text{m}^2}{2} $ $ A_{mean} = \frac{32 \, \text{m}^2}{2} $ $ A_{mean} = 16 \, \text{m}^2 $

Calculating the Tank Volume

Finally, we calculate the volume of the tank by multiplying the mean area by the depth of the tank:

• Volume (V): The depth (h) is given as 6 m. $ V = A_{mean} \times h $ $ V = 16 \, \text{m}^2 \times 6 \, \text{m} $ $ V = 96 \, \text{m}^3 $

Therefore, the volume of the tank, calculated using the mean-area method, is 96 cubic meters.