## Answer :

### Part (A): Exact Profit from the Sale of the 26th Machine

The given profit function is:

[tex]\[ P(x) = 20x - 0.5x^2 - 270 \][/tex]

To find the exact profit from the sale of the 26th machine, we evaluate this function at [tex]\( x = 26 \)[/tex].

[tex]\[ P(26) = 20(26) - 0.5(26)^2 - 270 \][/tex]

Breaking it down:

1. Calculate [tex]\( 20 \times 26 \)[/tex]:

[tex]\[ 20 \times 26 = 520 \][/tex]

2. Calculate [tex]\( 0.5 \times 26^2 \)[/tex]:

[tex]\[ 26^2 = 676 \][/tex]

[tex]\[ 0.5 \times 676 = 338 \][/tex]

3. Substitute these into the function:

[tex]\[ P(26) = 520 - 338 - 270 \][/tex]

4. Simplify the expression:

[tex]\[ P(26) = 520 - 338 = 182 \][/tex]

[tex]\[ P(26) = 182 - 270 = -88 \][/tex]

So, the exact profit from the sale of the 26th machine is:

[tex]\[ \boxed{-88} \][/tex]

### Part (B): Approximate Profit Using Marginal Profit

The marginal profit is the derivative of the total profit function [tex]\( P(x) \)[/tex]. First, we need to compute the derivative of [tex]\( P(x) \)[/tex]:

Given:

[tex]\[ P(x) = 20x - 0.5x^2 - 270 \][/tex]

The derivative [tex]\( P'(x) \)[/tex] (marginal profit function) is:

[tex]\[ P'(x) = \frac{d}{dx}(20x - 0.5x^2 - 270) \][/tex]

To get the marginal profit:

1. The derivative of [tex]\( 20x \)[/tex] is [tex]\( 20 \)[/tex]

2. The derivative of [tex]\( -0.5x^2 \)[/tex] is [tex]\( -x \)[/tex]

3. The derivative of the constant [tex]\( -270 \)[/tex] is [tex]\( 0 \)[/tex]

So,

[tex]\[ P'(x) = 20 - x \][/tex]

Now, we evaluate [tex]\( P'(x) \)[/tex] at [tex]\( x = 26 \)[/tex]:

[tex]\[ P'(26) = 20 - 26 = -6 \][/tex]

Therefore, the approximate profit from the sale of the 26th machine using the marginal profit is:

[tex]\[ \boxed{-6} \][/tex]

In summary:

- Exact profit on the 26th machine: [tex]\( \boxed{-88} \)[/tex]

- Approximate profit on the 26th machine: [tex]\( \boxed{-6} \)[/tex]