Problem 45 - Thinking Mathematically

Question

You are about to sue your contractor who promised to install a water tank that holds 500 gallons of water. You know that 500 gallons is the capacity of a tank that holds 67 cubic feet. The cylindrical tank has a radius of 3 feet and a height of 2 feet 4 inches. Does the evidence indicate you can win the case against the contractor if it goes to court?

Step-by-Step Solution

Answer

No, the evidence does not support the case against the contractor. The volume of the tank in question is marginally smaller than the promised volume of 67 cubic feet. Depending on standards of deviation or measurement error, this could be in compliance. It is advised to seek specific legal counsel.

Step 1: Convert inches to feet

To be able to work with a consistent unit of measurement in calculating volume, we need to convert the given height from feet and inches to feet only. There are 12 inches in a foot, so 4 inches represents \(\frac{4}{12} = 0.33\) feet. Thus, the total height of the tank is \(2 + 0.33 = 2.33\) feet.

Step 2: Calculate the volume of the tank

The formula for the volume of a cylinder is given by \( V = \pi r^2 h'\) where \( r\) is the radius and \( h'\) is the height. Substituting the given radius and height, \( V = \pi (3)^2 (2.33) = 66.2\) cubic feet.

Step 3: Convert gallons to cubic feet

We also need the capacity in cubic feet to make a fair comparison. We've been given that 500 gallons equals 67 cubic feet, meaning 1 gallon equals \(\frac{67}{500} = 0.134\) cubic feet. So, the promised tank size in cubic feet would be \(500 * 0.134 = 67\) cubic feet.

Step 4: Compare tank volume with the promised volume

The calculated volume of the tank is 66.2 cubic feet while the promised volume was 67 cubic feet, the difference is 0.8 cubic feet. However, they are very close.

Key Concepts

Cylinder Volume

Understanding the volume of a cylinder is essential when determining the capacity of cylindrical objects like water tanks. Cylinders are common shapes, so knowing how to calculate their volume can be practical. The volume of a cylinder is determined by two key measurements: its radius and its height. The formula to compute the volume is \[ V = \pi r^2 h \] where \( V \) represents the volume, \( r \) is the radius of the base, and \( h \) is the height of the cylinder.When calculating the volume, it's vital that all measurements are in the same unit. Only then does the equation provide an accurate result. In our case, the volume calculated for the tank with a radius of 3 feet and height of 2.33 feet was found to be approximately 66.2 cubic feet. Understanding this formula helps you compare how much space different cylindrical tanks can hold.

Unit Conversion

Unit conversion plays a crucial role in geometry and volume calculations, especially when units are mixed, like feet and inches. When working with objects measured in different units, we must convert all dimensions to the same unit before calculation. In this problem, the height is given as 2 feet 4 inches. To simplify, we convert 4 inches into feet by dividing by 12 (since there are 12 inches in a foot), which results in about 0.33 feet. Adding the 2 feet, the complete height becomes 2.33 feet. Having consistent units makes volume calculations straightforward and accurate. This practice of conversion is a vital skill, whether calculating a tank’s capacity at home or in an exam.

Volume Calculation

Calculating the volume of any shape involves using the right formula and ensuring all measurements align correctly with these formulas. After converting units to feet, we plug these values into the cylinder volume formula.For our cylindrical water tank:- Radius \( r = 3 \) feet- Height \( h = 2.33 \) feetThe formula \( V = \pi r^2 h \) gives us the volume as 66.2 cubic feet. Using the value of \( \pi \approx 3.14159 \) ensures precision; however, using approximations (like \( \pi \approx 3.14 \)) can also provide a sufficiently close estimate in many practical scenarios.This calculation isn't just about using the formula; it's also about ensuring every step is meticulously done to reach a logical conclusion.

Capacity Comparison

Once you have calculated the volume, comparing it with the expected capacity helps verify accuracy and promises. For the dealing with the contractor, the tank's capacity in gallons needed converting to volume in cubic feet, ensuring fair comparison. Given that 500 gallons equals 67 cubic feet: - Each gallon is approximately 0.134 cubic feet Now, you match this against the calculated 66.2 cubic feet from your volume formula. While the difference is small, understanding these tiny variations can be crucial legally and practically. Ultimately, this comparison demonstrates how geometry assists in real-world scenarios. Whether in legal settings or everyday usage, volume calculation and comparison, clarified by unit conversion, are foundational skills.