Value of `y` decreases exponentially from `y =10` to `y =6`. plot `a x-y` graph.

Answer

To solve the problem of plotting the graph of \( y \) decreasing exponentially from \( y = 10 \) to \( y = 6 \), we can follow these steps:

### Step 1: Understand the Exponential Decay Formula
The general formula for exponential decay can be expressed as:
\[ y = y_0 e^{-ax} \]
where:
- \( y_0 \) is the initial value of \( y \) (which is 10 in this case),
- \( a \) is a positive constant,
- \( x \) is the independent variable,
- \( y \) is the dependent variable.

### Step 2: Set Up the Initial Condition
From the problem, we know:
- At \( x = x_1 \), \( y = 10 \).
Thus, we can write:
\[ 10 = 10 e^{-ax_1} \]
This simplifies to:
\[ 1 = e^{-ax_1} \]
Taking the natural logarithm of both sides gives:
\[ 0 = -ax_1 \implies x_1 = 0 \]

### Step 3: Set Up the Final Condition
Now, we need to find the value of \( x \) when \( y = 6 \):
\[ 6 = 10 e^{-ax_2} \]
Dividing both sides by 10 gives:
\[ 0.6 = e^{-ax_2} \]
Taking the natural logarithm of both sides results in:
\[ \ln(0.6) = -ax_2 \implies x_2 = -\frac{\ln(0.6)}{a} \]

### Step 4: Choose a Value for \( a \)
To plot the graph, we need to choose a value for \( a \). Let's assume \( a = 1 \) for simplicity. Then:
\[ x_2 = -\ln(0.6) \approx 0.5108 \]

### Step 5: Create the Data Points
Now we have two points:
1. \( (x_1, y_1) = (0, 10) \)
2. \( (x_2, y_2) \approx (0.5108, 6) \)

### Step 6: Plot the Graph
On a graph:
- The x-axis will represent the values of \( x \).
- The y-axis will represent the values of \( y \).

Plot the points \( (0, 10) \) and \( (0.5108, 6) \). Since this is an exponential decay, the curve will start at \( y = 10 \) and decrease towards \( y = 6 \) as \( x \) increases.

### Step 7: Draw the Exponential Decay Curve
The curve should be drawn smoothly connecting the two points, showing the exponential decay from \( y = 10 \) to \( y = 6 \).

### Summary of the Steps:
1. Understand the exponential decay formula.
2. Set up the initial condition and solve for \( x_1 \).
3. Set up the final condition and solve for \( x_2 \).
4. Choose a value for \( a \).
5. Create data points for plotting.
6. Plot the points on the graph.
7. Draw the exponential decay curve.