Problem 40 - Algebra 2

Question

Many areas of Northern California depend on the snowpack of the Sierra Nevada Mountains for their water supply. If 250 cubic centimeters of snow will melt to 28 cubic centimeters of water, how much water does 900 cubic centimeters of snow produce?

Step-by-Step Solution

Answer

900 cc of snow produces 100.8 cc of water.

Step 1: Understanding the Problem

The problem involves finding out how much water can be produced from a given volume of snow. We know that 250 cubic centimeters (cc) of snow melts into 28 cc of water.

Step 2: Setting Up a Proportion

We need to figure out the relationship between snow and water for the given amounts. The proportion can be set as \( \frac{250}{28} = \frac{900}{x} \), where \(x\) is the amount of water from 900 cc of snow.

Step 3: Cross-multiplication

To solve the proportion \( \frac{250}{28} = \frac{900}{x} \), we cross-multiply: \( 250 \times x = 28 \times 900 \).

Step 4: Solving for Unknown

Calculate the product of the numbers on the right side: \( 28 \times 900 = 25200 \). Now solve for \(x\): divide by 250, \( x = \frac{25200}{250} \).

Step 5: Calculating Final Value

Performing the division, \( x = 100.8 \). Therefore, 900 cubic centimeters of snow will produce 100.8 cubic centimeters of water.

Key Concepts

Cross-Multiplication

Cross-multiplication is a valuable technique used primarily to solve proportions. A proportion is an equation that states two ratios are equivalent. To apply cross-multiplication, you multiply across the equal sign in an 'X' pattern. Consider a proportion set up as \( \frac{a}{b} = \frac{c}{d} \). Cross-multiplication helps to eliminate fractions: you multiply \( a \) by \( d \) and \( b \) by \( c \), resulting in \( a \cdot d = b \cdot c \). This method is efficient and easy to use, making it popular in finding unknown values in a proportion.

• Simplifies solving proportions
• Eliminates fraction operations
• Directly compares related quantities

Once you have cross-multiplied, you can easily solve for the unknown variable by dividing both sides by the coefficient of the variable. In our exercise, knowing cross-multiplication makes it simple to find how much water can be produced from snow by using already known amounts.

Ratios

Ratios express the relationship between two quantities, demonstrating how many times one value contains or is contained within the other. They are typically written as \( \frac{a}{b} \) or using colon notation as \( a:b \). Ratios are fundamental in solving problems involving proportions, as they allow comparisons and scaling between different items or scenarios.In our context, the problem states that 250 cubic centimeters of snow correlates to 28 cubic centimeters of water. This is a ratio of the volume of snow to water, 250:28. When solving real-world problems like snowpack conversion, such ratios help us understand the quantities involved.

• Helps compare related items
• Used to solve proportion problems
• Crucial for understanding scaling and conversion

Ratios are instrumental when setting up the initial equation to solve for an unknown value, as in our exercise with converting snow to water.

Snowpack Water Conversion

Snowpack water conversion refers to calculating the amount of water obtained from melting a specific volume of snow. This process involves understanding the relationship between snow and water, which naturally involves the concept of proportions and ratios. In areas like Northern California, knowing how much water is derived from snowpacks is vital, often influencing water resource management and planning. Why is this important? As snow is less dense than water, it takes a larger volume of snow to yield a small amount of water. Thus, converting snow volume to water volume gives practical insights into water supply.

• Essential for water resource planning
• Engages scientific and mathematical concepts
• Relevant in climate and environmental studies

By calculating how snow converts to water, one can predict water availability based on seasonal snowpacks, a crucial metric for many regions dependent on this natural resource for their annual water needs.