Calculating Volume Percentage Increase Cylinder Radius Decrease Height Increase
In this article, we will explore the concept of how changes in the dimensions of a right circular cylinder affect its volume. Specifically, we will delve into a scenario where the radius of the cylinder's base is decreased by 12%, and its height is significantly increased by 175%. Our main goal is to determine the resulting percentage increase in the cylinder's volume, rounding our answer to the closest integer. This involves understanding the formula for the volume of a cylinder and how it relates to changes in radius and height. Let's dive into the mathematical principles and calculations required to solve this problem effectively.
Understanding Cylinder Volume and Percentage Change
To accurately calculate the percentage increase in the volume of a right circular cylinder, it's crucial to first understand the fundamental formula for the volume of a cylinder. The volume (V) of a cylinder is given by the formula V = πr²h, where r represents the radius of the base and h represents the height of the cylinder. This formula highlights the direct relationship between the volume and both the radius (squared) and the height. Any changes in these dimensions will directly impact the volume. When dealing with percentage changes, it’s essential to express the new dimensions in terms of the original dimensions to effectively compare the initial and final volumes. For instance, a 12% decrease in the radius means the new radius is 88% of the original radius, and a 175% increase in height means the new height is 275% of the original height. By applying these percentage changes to the volume formula, we can derive a new volume expression. This new volume is then compared to the original volume to calculate the percentage increase. This process involves a few key steps. First, we define the initial volume using the original radius and height. Then, we calculate the new radius and height based on the given percentage changes. Next, we substitute these new dimensions into the volume formula to find the new volume. Finally, we determine the percentage change by comparing the new volume to the original volume, which gives us the answer to our problem. This step-by-step approach ensures accuracy and clarity in understanding the impact of dimensional changes on the volume of the cylinder.
Calculating the New Dimensions
In addressing the problem of finding the percentage increase in volume, the initial step involves calculating the new dimensions of the cylinder after the specified changes. The problem states that the radius of the base is decreased by 12%, and the height is increased by 175%. Let's denote the original radius as r and the original height as h. A 12% decrease in the radius means that the new radius, r', will be 100% - 12% = 88% of the original radius. Mathematically, this is expressed as r' = 0.88r. This calculation is straightforward, converting the percentage decrease into a decimal multiplier. Similarly, a 175% increase in the height means that the new height, h', will be the original height plus 175% of the original height, which is 100% + 175% = 275% of the original height. This can be written as h' = 2.75h. Understanding these conversions from percentage changes to numerical multipliers is crucial for accurately calculating the new volume. These new dimensions, r' = 0.88r and h' = 2.75h, will be used in the next step to calculate the new volume of the cylinder. Accurate calculation of these new dimensions is essential as they directly influence the final result of the percentage increase in volume. Failing to accurately determine these values will lead to an incorrect final answer. This stage sets the foundation for the rest of the calculation, ensuring that the subsequent steps are based on correct values. By expressing the new dimensions in terms of the original dimensions, we can easily compare the original and new volumes, which is necessary for finding the percentage increase.
Determining the New Volume
Once the new dimensions of the cylinder are determined, the next crucial step is to determine the new volume (V') using the updated radius (r') and height (h'). As previously established, the formula for the volume of a cylinder is V = πr²h. We have calculated the new radius r' as 0.88r and the new height h' as 2.75h. Substituting these values into the volume formula, we get V' = π(0.88r)²(2.75h). The next step involves simplifying this expression to make it easier to compare with the original volume. First, square the term 0.88r, which gives us (0.88)² * r² = 0.7744r². Then, multiply this by 2.75h. Thus, the new volume V' becomes V' = π(0.7744r²)(2.75h). Now, multiply the numerical coefficients 0.7744 and 2.75. This multiplication results in 2.1296. Therefore, the expression for the new volume simplifies to V' = 2.1296πr²h. This expression is crucial because it directly relates the new volume to the original volume (which is V = πr²h). The coefficient 2.1296 indicates how the new volume compares to the original volume. By having the new volume expressed in terms of π, r², and h, we can easily see the proportional change and calculate the percentage increase, which is the ultimate goal of the problem. This step of determining the new volume is essential as it provides the necessary information for the final calculation of the percentage increase.
Calculating the Percentage Increase in Volume
With the original volume (V) and the new volume (V') now defined, the final step is to calculate the percentage increase in volume. We have the original volume as V = πr²h and the new volume as V' = 2.1296πr²h. To find the percentage increase, we use the formula: Percentage Increase = [(New Volume - Original Volume) / Original Volume] * 100. Substituting the expressions for V and V' into this formula, we get: Percentage Increase = [(2.1296πr²h - πr²h) / πr²h] * 100. Notice that πr²h is a common factor in both terms in the numerator, and it is also in the denominator. This allows us to simplify the expression by factoring out πr²h from the numerator: Percentage Increase = [πr²h (2.1296 - 1) / πr²h] * 100. Now, we can cancel out the πr²h terms from the numerator and denominator: Percentage Increase = (2.1296 - 1) * 100. Subtract 1 from 2.1296 to get 1.1296. So, the expression simplifies to: Percentage Increase = 1.1296 * 100. Multiplying 1.1296 by 100 gives us 112.96%. The problem asks for the closest integer, so we round 112.96% to the nearest whole number, which is 113%. Therefore, the percentage increase in the volume of the cylinder is approximately 113%. This final calculation provides the answer to the problem, showing the overall impact of the changes in radius and height on the cylinder’s volume. Understanding each step in this process, from calculating the new dimensions to determining the final percentage increase, is crucial for solving similar problems in geometry and mathematical reasoning.
Conclusion
In conclusion, determining the percentage increase in the volume of a cylinder after changes to its dimensions involves a multi-step process that combines geometric principles with algebraic manipulation. By carefully calculating the new dimensions, finding the new volume, and applying the percentage increase formula, we arrived at the solution. The key takeaways from this problem include the importance of understanding the formula for the volume of a cylinder, how percentage changes affect dimensions, and the method for comparing initial and final volumes. In this specific scenario, a 12% decrease in the radius and a 175% increase in the height resulted in an approximate 113% increase in the cylinder's volume. This demonstrates the significant impact that changes in dimensions can have on the volume of a three-dimensional object. The methodology used here can be applied to various similar problems, emphasizing the importance of a systematic approach in solving mathematical questions. By breaking down the problem into smaller, manageable steps, we can ensure accuracy and clarity in our calculations, leading to a correct and well-understood solution. This type of problem is not only a mathematical exercise but also enhances our understanding of spatial relationships and proportional changes in real-world applications. Understanding these concepts is crucial for various fields, including engineering, physics, and even everyday problem-solving.
