Solve the following problems involving direct and inverse proportions. Round answer to two decimal places as necessary.
A cylindrical tank holds 3,915 liters of water when filled to a height of 2.1 feet. How many liters of water would the tank hold if filled to a height of 12 feet?
Answer (1)
Establish the direct proportion: Volume V is directly proportional to height h , expressed as V = k × h .
Calculate the constant of proportionality: Using the initial values V = 3915 liters and h = 2.1 feet, find k = 2.1 3915 = 1864.285714 .
Determine the new volume: Substitute h = 12 feet into the equation V = k × h to find V = 1864.285714 × 12 = 22371.42857 .
State the final answer: Round the volume to two decimal places, resulting in 22371.43 liters.
Explanation
Understanding the Problem We are given that a cylindrical tank holds 3,915 liters of water when filled to a height of 2.1 feet. We need to find out how many liters of water the tank would hold if filled to a height of 12 feet, assuming the tank has a uniform cross-section.
Setting up the Proportion Since the volume of water in the tank is directly proportional to the height of the water, we can express this relationship as V = k × h , where V is the volume of water, h is the height, and k is the constant of proportionality.
Finding the Constant of Proportionality We can find the constant of proportionality k using the given information. When the height is 2.1 feet, the volume is 3,915 liters. So, we have: 3915 = k × 2.1
Calculating k To solve for k , we divide both sides of the equation by 2.1: k = 2.1 3915 k = 1864.285714
Finding the Volume Now that we have the value of k , we can find the volume of water when the height is 12 feet. We substitute h = 12 into the equation V = k × h :
V = 1864.285714 × 12 V = 22371.42857 Rounding to two decimal places, we get V = 22371.43 liters.
Final Answer Therefore, the tank would hold approximately 22371.43 liters of water when filled to a height of 12 feet.
Examples
Direct proportion is a fundamental concept that applies to various real-world scenarios. For instance, consider baking a cake. If you need to double the recipe, you'll need to double all the ingredients. If the original recipe calls for 2 cups of flour, doubling the recipe means you'll need 4 cups of flour. This direct relationship helps ensure the cake turns out perfectly, demonstrating how understanding direct proportion can be useful in everyday tasks like cooking and baking.
