Question 1192199
Here's how to solve this problem:

**(i) Find the values of e^y when x = 0:**

1. **Recognize the linear relationship:** Since plotting *eʸ* against *x²* gives a straight line, we can express this relationship as:

   *eʸ = mx² + c*

   where *m* is the slope and *c* is the y-intercept.

2. **Calculate the slope (m):**  We are given two points on the line: (x², eʸ) = (0.2², 1) and (0.5², 1.6).

   *m = (change in eʸ) / (change in x²)*
   *m = (1.6 - 1) / (0.5² - 0.2²)*
   *m = 0.6 / (0.25 - 0.04)*
   *m = 0.6 / 0.21*
   *m = 20/7*

3. **Calculate the y-intercept (c):** We can use either of the given points and the slope to find *c*. Let's use (0.2², 1):

   *1 = (20/7)(0.2²) + c*
   *1 = (20/7)(0.04) + c*
   *1 = 0.8/7 + c*
   *c = 1 - 0.8/7*
   *c = 6.2/7*

4. **Find *eʸ* when *x = 0*:** Substitute *x = 0* into the equation *eʸ = mx² + c*:

   *eʸ = (20/7)(0²) + 6.2/7*
   *eʸ = 6.2/7*

   Therefore, when *x = 0*, *eʸ = 6.2/7* (approximately 0.886).

**(ii) Express y in terms of x:**

1. **We have the equation:** *eʸ = mx² + c*

2. **Substitute the values of *m* and *c*:**

   *eʸ = (20/7)x² + 6.2/7*

3. **Take the natural logarithm of both sides:**

   *ln(eʸ) = ln((20/7)x² + 6.2/7)*

4. **Simplify:**

   *y = ln((20/7)x² + 6.2/7)*

Therefore, *y = ln((20/7)x² + 6.2/7)*.