Mathematics • Year 10 • Unit 1 • Lesson 14
Surds in the Real World
Apply Lesson 14's surd operations to the geometry and engineering contexts where exact values matter: square diagonals, special right triangles (1 : √3 : 2), Pythagoras with surd answers, and AC peak-vs-RMS voltage. Then explain in your own words why engineers keep surd form until the very last step.
1. Word problems
Each problem uses surd simplification, like-surd combining, or rationalising, and the answer should stay in exact form unless told otherwise.
1.1, Side of a square paver. A square concrete paver has area 50 cm². The lesson's SAQ 3 explores this exact scenario.
(a) Find the exact side length in simplest surd form.
(b) Find the exact diagonal length in simplest surd form. (Hint: diagonal = side × √2.)
(c) Explain in one sentence why a paving contractor cutting many of these pavers would use the surd form rather than the decimal approximation. 3 marks
1.2, A 30°-60°-90° set-square triangle. The lesson's Real-Life Link cites the side-ratio 1 : √3 : 2 for these triangles. An architect's set-square has its shortest side 6 cm long.
(a) State the lengths of the other two sides in exact form (using a surd where needed).
(b) Calculate the perimeter in simplest surd form.
(c) Calculate the area in exact form. 3 marks
1.3, Pythagoras with a surd hypotenuse. A roof rafter forms a right-angled triangle with legs 4 m and 6 m. Find the exact length of the rafter in simplest surd form.
(a) Apply Pythagoras: c² = a² + b².
(b) Simplify the surd answer.
(c) Compute the decimal value to 2 decimal places, but state which form a carpenter on site would actually cut to. 3 marks
1.4, AC voltage peak vs RMS. Lesson 14's Real-Life Link cites that the relationship between peak and RMS voltage in an AC circuit involves √2: V_peak = V_RMS × √2. A standard Australian power point delivers V_RMS = 230 V.
(a) Write V_peak in exact surd form.
(b) Calculate the decimal value to the nearest volt.
(c) Conversely, a peak voltage of 100√2 V corresponds to what V_RMS in exact form? 3 marks
1.5, Combining surds in a perimeter. A rectangular garden has length 5√2 m and width 3√2 m.
(a) Calculate the perimeter in simplest surd form.
(b) Calculate the area in exact form (you should get a whole number).
(c) Compare the area calculation method to what would happen if you used decimal approximations of √2 first. 3 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A friend says: "Why bother with surds? My calculator gives √2 = 1.41421356, so just use that everywhere." Using everything from Lesson 14, explain (i) what is lost when you replace √2 with a decimal approximation, (ii) why squaring 1.414 does NOT give exactly 2 (test this on your calculator), and what this tells you about compounding error, and (iii) why architects working with 30°-60°-90° triangles or engineers working with AC voltages keep surd form until the final step. Refer to "exact value" somewhere in your answer.
1.1, Square paver
(a) Side = √50 = √(25 × 2) = 5√2 cm.
(b) Diagonal = 5√2 × √2 = 5 × 2 = 10 cm exactly.
(c) The exact form means every paver's diagonal is exactly 10 cm, no compounding rounding errors across dozens of cuts. A decimal approximation of 5√2 ≈ 7.07 cm would propagate small errors into the diagonal calculation.
1.2-30°-60°-90° set-square
(a) Sides (in the 1 : √3 : 2 ratio scaled by 6): 6, 6√3, 12 cm.
(b) Perimeter = 6 + 6√3 + 12 = (18 + 6√3) cm.
(c) Area = ½ × 6 × 6√3 = 18√3 cm².
The lesson's Real-Life Link names the 1 : √3 : 2 ratio specifically.
1.3, Roof rafter
(a) c² = 4² + 6² = 16 + 36 = 52. So c = √52.
(b) 52 = 4 × 13. √52 = √4 × √13 = 2√13 m.
(c) Decimal: 2√13 ≈ 2 × 3.606 ≈ 7.21 m. A carpenter on site would cut to the decimal (with a tape measure marked in cm), but a structural engineer would calculate using 2√13 to keep precision in subsequent load calculations.
1.4, Australian power point
(a) V_peak = 230 × √2 = 230√2 V exactly.
(b) 230√2 ≈ 230 × 1.4142 ≈ 325 V.
(c) V_RMS = (100√2) / √2 = 100 V. (The √2's cancel cleanly, no need to rationalise here because they appear in both numerator and denominator.)
1.5, Rectangular garden in surds
(a) Perimeter = 2(5√2 + 3√2) = 2(8√2) = 16√2 m.
(b) Area = (5√2)(3√2) = 5 × 3 × (√2)² = 15 × 2 = 30 m² (whole number, exact).
(c) Using decimals: (5 × 1.414)(3 × 1.414) ≈ 7.07 × 4.243 ≈ 29.99, close to 30 but no longer exact. Surd form preserves the answer perfectly because (√2)² collapses exactly to 2.
2.1, Explain your thinking (sample response)
(i) When you replace √2 with 1.41421356 you lose exactness: the decimal terminates somewhere, but √2 has an infinite, non-repeating expansion, so any decimal cuts off real information. (ii) Squaring 1.414 on a calculator gives 1.999396, not 2, the small rounding error in the decimal got amplified by squaring, and in a longer chain of multiplications the error grows further. (iii) Architects designing 30°-60°-90° trusses, and electrical engineers converting between V_peak and V_RMS in AC circuits, keep the surd until the last step because the exact value protects against this compounding error; only at the very end (when ordering a length of timber or specifying a wire gauge) do they convert to a decimal. That way the geometry stays mathematically perfect and the only rounding happens once.
Marking: 1 for naming what is lost (exactness / infinite non-repeating decimal), 1 for showing that (1.414)² ≠ 2 demonstrates compounding error, 1 for a real-world example (architect or engineer), 1 for using "exact value" correctly.