1. **Problem statement:** A plot of land has an area of 876 m² and is divided into three rectangular fields of equal width. The total fencing used is 177 m, which includes the perimeter and two internal fences dividing the fields. 2. **Known values:** - Total area $= 876$ m² - Height of the plot $= 36$ m - Width of the plot $= 27$ m (given at the bottom) - The plot is divided into three fields of equal width, so each field has width $w$ and length $AB$ (unknown). 3. **Find:** (a) Length $AB$ (b) Perimeter of the plot --- ### Step 1: Express the area in terms of $AB$ The total area is given by: $$\text{Area} = \text{length} \times \text{width}$$ Here, the length is $AB$ and the width is 27 m. So, $$AB \times 27 = 876$$ ### Step 2: Solve for $AB$ Divide both sides by 27: $$AB = \frac{876}{27}$$ Calculate: $$AB = 32.444... \approx 32.44 \text{ m}$$ ### Step 3: Find the perimeter of the plot The perimeter $P$ of a rectangle is: $$P = 2(\text{length} + \text{width})$$ Using $AB = 32.44$ m and width $= 27$ m: $$P = 2(32.44 + 27) = 2(59.44) = 118.88 \text{ m}$$ ### Step 4: Calculate total fencing used The fencing includes the perimeter plus two internal fences dividing the plot into three fields. Each internal fence runs parallel to the length $AB$ and has length $36$ m. Total fencing: $$\text{Fencing} = P + 2 \times 36 = 118.88 + 72 = 190.88 \text{ m}$$ But the problem states fencing used is 177 m, so we need to check the width of each field. ### Step 5: Reconsider the width of each field The plot is divided into three fields of equal width, so each field width is $w$. Total width is $3w = 27$ m, so $$w = \frac{27}{3} = 9 \text{ m}$$ ### Step 6: Calculate fencing length using $w$ and $AB$ The fencing includes: - Two lengths of $AB$ (top and bottom) - Four widths of $36$ m (two sides and two internal fences) So total fencing: $$F = 2 \times AB + 4 \times 36$$ Given $F = 177$ m, solve for $AB$: $$2AB + 144 = 177$$ $$2AB = 33$$ $$AB = 16.5 \text{ m}$$ ### Step 7: Verify area with $AB = 16.5$ m and width $27$ m Area: $$16.5 \times 27 = 445.5 \text{ m}^2$$ This does not match the given area 876 m², so the width must be different. ### Step 8: Let the width of each field be $w$, so total width is $3w$ Area: $$AB \times 3w = 876$$ Fencing: $$2AB + 4 \times 36 = 177$$ From fencing: $$2AB = 177 - 144 = 33$$ $$AB = 16.5$$ From area: $$16.5 \times 3w = 876$$ $$3w = \frac{876}{16.5} = 53.09$$ $$w = \frac{53.09}{3} = 17.7$$ ### Step 9: Final answers (a) Length $AB = 16.5$ m (b) Perimeter $P = 2(AB + 3w) = 2(16.5 + 53.09) = 2(69.59) = 139.18$ m