Substitute x = 1000 and y = 800 into the profit formula: π = 20(800) - 5(1000) - 1500 = 16000 - 5000 - 1500 = 9500. Thus, the profit is 9500.

For part (c), the firm cannot sell more pots than it produces, so y ≤ x. Both x and y must be non-negative integers (x, y ≥ 0 and x, y Z). For part (d), if the firm sells all that it manufactures, then y = x, simplifying the formula to π = 20x - 5x - 1500 = 15x - 1500.

For part (c), the firm cannot sell more pots than it produces, so y ≤ x. Both x and y must be non-negative integers (x, y ≥ 0 and x, y Z). For part (d), if the firm sells all that it manufactures, then y = x, simplifying the formula to π = 20x - 5x - 1500 = 15x - 1500.

10. Evaluate the following without using a calculator: (a) $50\,563^2 - 49\,437^2$ (b) $90^2 - 89.99^2$ (c) $759^2 - 541^2$ (d) $123\,456\,789^2 - 123\,456\,788^2$ 11. A specialist paint manufacturer receives $\$20 $for each pot sold. The initial set-up cost for the production run is$ \$1500 $and the cost of making each pot of paint is$ \$5$. (a) Write down a formula for the total profit, $\pi$ , if the firm manufactures $x$ pots of paint and sells $y$ pots. (b) Use your formula to calculate the profit when $x = 1000$ and $y = 800$ . (c) State any restrictions on the variables in the mathematical formula in part (a). (d) Simplify the formula in the case when the firm sells all that it manufactures.

Answer: 10. (a) $112\,600\,000$ , (b) $1.7999$ , (c) $283\,400$ , (d) $246\,913\,577 11. (a)\pi = 20y - 5x - 1500, (b)9500 $, (c)$ 0\le y \le x $with x, y non-negative integers, (d)$ \pi = 15x - 1500$

Step-by-step solution

Step 1: Evaluate Question 10 (a) and (b) using Difference of Squares

We use the algebraic identity for the difference of two squares, $a^2 - b^2 = (a - b)(a + b)$ . For part (a), $a = 50563$ and $b = 49437$ , so $(50563 - 49437)(50563 + 49437) = 1126 \times 100000 = 112600000$ . For part (b), $a = 90$ and $b = 89.99$ , giving $(90 - 89.99)(90 + 89.99) = 0.01 \times 179.99 = 1.7999$ .

Step 2: Evaluate Question 10 (c) and (d)

Applying $a^2 - b^2 = (a - b)(a + b)$ to part (c), $(759 - 541)(759 + 541) = 218 \times 1300 = 283400$ . For part (d), since the numbers differ by $1$ , $(123456789 - 123456788)(123456789 + 123456788) = 1 \times 246913577 = 246913577$ .

Step 3: Formulate Profit Function (Question 11a)

The revenue is obtained from selling $y$ pots at 20 each, giving 20y. The total cost includes a fixed set-up cost of 1500 plus a variable cost of 5 for each of the x pots manufactured, which is 1500 + 5x. Therefore, the total profit formula is π = $20y - 5x - 1500$ .

Step 4: Calculate Profit for Given Values (Question 11b)

Substitute $x = 1000$ and $y = 800$ into the profit formula: $\pi = 20(800) - 5(1000) - 1500 = 16000 - 5000 - 1500 = 9500$ . Thus, the profit is 9500.

Step 5: Determine Restrictions and Simplify (Question 11c & 11d)

For part (c), the firm cannot sell more pots than it produces, so $y \le x$ . Both $x$ and $y$ must be non-negative integers ( $x, y \ge 0$ and $x, y \in \mathbb{Z}$ ). For part (d), if the firm sells all that it manufactures, then $y = x$ , simplifying the formula to $\pi = 20x - 5x - 1500 = 15x - 1500$ .

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