1. **State the problem:** We have an orange grove with 40 trees, each producing 500 oranges. For every extra tree planted, the yield per tree decreases by 10 oranges. We want to find the total number of trees that maximizes the total yield. 2. **Define variables:** Let $x$ be the number of extra trees planted beyond 40. Then the total number of trees is $40 + x$. 3. **Express yield per tree:** The yield per tree decreases by 10 oranges for each extra tree, so yield per tree is $500 - 10x$. 4. **Write total yield function:** Total yield $Y$ is number of trees times yield per tree: $$Y = (40 + x)(500 - 10x)$$ 5. **Expand the expression:** $$Y = 40 \times 500 - 40 \times 10x + x \times 500 - 10x^2 = 20000 - 400x + 500x - 10x^2$$ 6. **Simplify:** $$Y = 20000 + 100x - 10x^2$$ 7. **Rewrite as quadratic:** $$Y = -10x^2 + 100x + 20000$$ 8. **Find vertex to maximize yield:** For quadratic $ax^2 + bx + c$, vertex at $x = -\frac{b}{2a}$. Here, $a = -10$, $b = 100$. $$x = -\frac{100}{2 \times (-10)} = -\frac{100}{-20} = 5$$ 9. **Calculate total trees:** $$40 + x = 40 + 5 = 45$$ 10. **Conclusion:** Maximum yield occurs when there are 45 trees. **Answer: d) 45 trees**