Problem 39 - Algebra and Trigonometry

Question

Growing Cabbages In the short growing season of the Canadian arctic territory of Nunavut, some gardeners find it possible to grow gigantic cabbages in the midnight sun. Assume that the final size of a cabbage is proportional to the amount of nutrients it receives, and inversely proportional to the number of other cabbages surrounding it. A cabbage that received 20 oz of nutrients and had 12 other cabbages around it grew to 30 lb. What size would it grow to if it received 10 oz of nutrients and had only 5 cabbage “neighbors”?

Step-by-Step Solution

Answer

The cabbage would grow to 36 lb.

Step 1: Identify the Relationship

We are told the final size of a cabbage is proportional to the amount of nutrients it receives and inversely proportional to the number of other cabbages around it. This can be expressed with the formula \( S = k \frac{N}{C} \), where \( S \) is the size of the cabbage, \( N \) is the nutrients, \( C \) is the number of cabbage neighbors, and \( k \) is the constant of proportionality.

Step 2: Calculate the Constant of Proportionality

Given that a cabbage with 20 oz of nutrients and 12 neighbors grows to 30 lb, substitute these values into the equation: \( 30 = k \frac{20}{12} \). Simplify and solve for \( k \): \( 30 = k \frac{5}{3} \), leading to \( k = 18 \).

Step 3: Find the New Size

Using the derived constant \( k = 18 \), find the new cabbage size with 10 oz of nutrients and 5 neighbors. Substitute into the formula: \( S = 18 \frac{10}{5} = 18 \times 2 = 36 \).

Step 4: Conclusion

The new size of the cabbage is 36 lb when it receives 10 oz of nutrients and there are 5 neighborhoods.

Key Concepts

Inverse Proportionality

Imagine you have a single cake to share among friends. The more friends you have, the smaller each piece of cake will be. This is similar to the concept of inverse proportionality in math. Inverse proportionality means that when one value increases, the other decreases. In algebraic terms, if you double the number of friends, the cake pieces will be half the size.
For example, in our cabbage problem, the final size of the cabbage is inversely proportional to the number of other cabbages around it. This is expressed by the formula: \[ S = k \frac{N}{C} \]Here, \(S\) is the size of the cabbage, \(N\) represents the amount of nutrients, and \(C\) is the number of neighboring cabbages. When the number of surrounding cabbages increases, the size of our cabbage decreases if other factors remain the same.
An understanding of inverse proportionality is helpful because it shows how changes in one factor can negatively affect another. If resources are fixed, sharing them among more entities often results in smaller individual portions.

Nutrient Distribution

When we talk about nutrient distribution in gardening, we’re essentially discussing how resources like water, sunlight, and soil nutrients are allocated to each plant.
In the case of the giant cabbages in Nunavut, sufficient nutrient distribution is key. It's analogous to dividing a budget among several projects; the more evenly you’re able to distribute nutrients, the better the overall growth.
In the provided exercise, each cabbage's final size depends directly on the nutrients it receives. More nutrients lead to a bigger size, holding other conditions constant. This is a classic example of direct proportionality, where increasing one factor leads to an increase in another.

• A cabbage getting more nutrients will generally grow larger.
• The fewer cabbages sharing those nutrients, the bigger each one can get.

Understanding nutrient distribution can help when planning your garden layout – ensuring each plant gets what it needs to thrive.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario to predict future outcomes or understand a situation.
In our cabbage problem, modeling helps predict the effect of nutrients and neighboring competition on cabbage growth. By establishing a formula \( S = k \frac{N}{C} \), we quantify how each factor influences size.
Mathematical models are useful because they can:

• Help visualize complex relationships through equations.
• Assist in making predictions about future scenarios.
• Provide a basis for scientific and economic decisions.

Models are built on data, assumptions, and ideally, lots of testing. In cases like the cabbage exercise, once the model is set up using known variables, you can predict sizes by changing inputs and seeing corresponding results. This reduces guesswork in gardening, allowing for a more systematic approach to plant growth management.