91Ó°ÊÓ

In this exercise, we talk about a proportionality constant. This constant is a fixed number that links two variable quantities together.
When we say that the weight of an object on Earth is directly proportional to its weight on the moon, we mean that we can multiply the weight on Earth by a constant number to get the weight on the moon.
We express this relationship mathematically as: \( w_{\text{moon}} = k \times w_{\text{earth}} \) Here, \( k \) is our proportionality constant. We find \( k \) using known weights; for example, a 200-lb astronaut weighs 32 lb on the moon. By substituting these values into our equation, we can solve for \( k \): \( 32 = k \times 200 \)
So, \( k = \frac{32}{200} = 0.16 \)
Now, whenever we want to find the weight of any object on the moon, we simply multiply its Earth weight by \( 0.16 \).

Weight conversion using a proportionality constant is straightforward once you have the constant. For example, let's convert a dog's weight from Earth to the moon using the constant \( k = 0.16 \).
First, we note down the weight of the dog on Earth. In this case, the dog weighs 50 lb.
We use the proportionality equation: \( w_{\text{moon}} = k \times w_{\text{earth}} \)
Plugging in our values, we get: \( w_{\text{moon}} = 0.16 \times 50 \)
By solving this, we find: \( w_{\text{moon}} = 8 \text{ lb} \)
This simplicity is what makes proportionality constants so useful for weight conversions.

Mathematical equations are powerful tools that help us represent and solve real-world problems. In this example, our equation \( w_{\text{moon}} = k \times w_{\text{earth}} \) is a perfect illustration of direct proportionality.
The equation can be broken down into:

• \( w_{\text{moon}} \): Weight of the object on the moon.
• \( k \): Proportionality constant (fixed number).
• \( w_{\text{earth}} \): Weight of the object on Earth.

This equation tells us that if we know any two of these values, we can always find the third.
In the astronaut example, knowing both the Earth weight (200 lb) and the moon weight (32 lb), we found \( k = 0.16 \).
Using mathematical equations simplifies complex problems and lets us find answers with ease and precision.