91Ó°ÊÓ

Understanding proportional relationships is essential in algebra. When we say that two quantities are directly proportional, it means that as one quantity increases, the other quantity increases at a constant rate.
For this exercise, the pressure exerted by a liquid is directly proportional to the depth beneath the surface.

If we denote the pressure by \(P\) and the depth by \(d\), we can write this relationship as:
\[ P = k \times d \]
Here, \(k\) represents the constant of proportionality, which stays the same regardless of the values of \(P\) and \(d\). This equation tells us that if we know the constant of proportionality, we can calculate the pressure for any given depth.

Realizing that the pressure changes consistently as the depth changes helps us understand why the problem can be solved using simple algebraic methods.

The constant of proportionality, denoted as \(k\), is crucial in solving direct proportionality problems. It shows how much one variable changes as the other variable changes.
In our exercise, we are given that at a depth of 30 meters, the pressure is 80 newtons.

To find \(k\), we set up the equation:
\[ 80 = k \times 30 \]
Solving for \(k\) involves isolating it on one side of the equation:
\[ k = \frac{80}{30} = \frac{8}{3} \]
This means that for every meter of depth, the pressure increases by \(\frac{8}{3}\) newtons. Understanding \(k\) helps us determine how other depths will affect the pressure exerted by the liquid.

Let's break down each step to solve the problem to find the pressure at 50 meters depth.
Step 1: Identify the given information and the relationship between variables. We know that pressure \(P\) is directly proportional to depth \(d\), expressed as \[ P = k \times d \]
Step 2: Find the constant of proportionality \(k\). Using the given depth of 30 meters and pressure of 80 newtons:
\[ 80 = k \times 30 \]
Solving for \(k\), we find
\[ k = \frac{80}{30} = \frac{8}{3} \]
Step 3: Use \(k\) to find the pressure at the new depth.

We need to determine the pressure at 50 meters. Using the relationship \[ P = \frac{8}{3} \times 50 \]
Finally, calculate \(P\):
\[ P = \frac{400}{3} \approx 133.33 \text{ newtons} \]
By following these steps, solving problems involving direct proportionality becomes straightforward and systematic. Providing a clear approach ensures that you can apply this method to other similar problems in algebra.