1. **Problem:** Calculate the correct distance between points A and B given a measured distance and tape error. 2. **Given:** Measured distance AB = 165.20 m, tape length = 50 m, tape is 0.01 m too short. 3. **Step 1:** Determine the number of full tapes used: $$\text{number of tapes} = \frac{165.20}{50} = 3.304$$ 4. **Step 2:** Each tape is 0.01 m too short, so actual tape length = 50 + 0.01 = 50.01 m. 5. **Step 3:** Correct distance = number of tapes \( \times \) actual tape length: $$3.304 \times 50.01 = 165.36504\, m$$ --- 6. **Problem:** Find the error in the area of a rectangular lot with a true area of 2 hectares, length twice the width, measured using a tape 0.02 m too long. 7. **Given:** Area = 2 hectares = 20,000 m², length = 2 \(\times\) width, tape length = 50 m, tape error = +0.02 m. 8. **Step 1:** Let width = $w$, then length = $2w$. 9. **Step 2:** Calculate $w$: $$w \times 2w = 20,000 \,\Rightarrow 2w^2 = 20,000\Rightarrow w^2=10,000\Rightarrow w=100 \, m$$ 10. **Step 3:** True length = $200$ m. 11. **Step 4:** Number of tapes for width = $100/50 = 2$, number of tapes for length = $200/50 = 4$. 12. **Step 5:** Actual tape length = $50 + 0.02 = 50.02$ m. 13. **Step 6:** Measured width: $$2 \times 50.02 = 100.04 \, m$$ 14. **Step 7:** Measured length: $$4 \times 50.02 = 200.08 \, m$$ 15. **Step 8:** Measured area: $$100.04 \times 200.08 = 20,016.0032 \, m^2$$ 16. **Step 9:** Error in area: $$20,016.0032 - 20,000 = 16.0032 \, m^2$$ --- 17. **Problem:** Calculate correct distance BC measured with a tape 0.015 m too short. 18. **Given:** Measured BC = 146.5 m, tape length = 100 m, tape error = -0.015 m. 19. **Step 1:** Number of tapes: $$\frac{146.5}{100} = 1.465$$ 20. **Step 2:** Actual tape length: $$100 + 0.015 = 100.015 \, m$$ 21. **Step 3:** Correct distance: $$1.465 \times 100.015 = 146.521975 \, m$$ --- 22. **Problem:** Calculate measured distance AB using a tape 0.025 m too long. 23. **Given:** Actual distance AB = 213.50 m, tape length = 100 m, tape error = +0.025 m. 24. **Step 1:** Number of tapes using actual length: $$\frac{213.50}{100} = 2.135$$ 25. **Step 2:** Tape length used for measurement = 100 + 0.025 = 100.025 m. 26. **Step 3:** Measured distance: $$2.135 \times 100 = 213.5 \, \text{(true tapes number)}$$ 27. **Step 4:** Alternatively, to find measured distance: $$\text{measured distance} = \frac{\text{actual distance}}{\text{tape corrected length}} \times \text{tape nominal length}$$ 28. **Step 5:** Number of tapes measured: $$\frac{213.5}{100.025} = 2.13466$$ 29. **Step 6:** Multiply by nominal tape length: $$2.13466 \times 100 = 213.466 \, m$$ **Final answers:** 1. Correct distance AB = $165.365$ m 2. Error in area = $16.0032$ sq. m 3. Correct distance BC = $146.522$ m 4. Measured distance AB = $213.466$ m