1. **Problem:** Muthu put a 5.4 m ladder against Mandy's window 5.2 m above the ground. Find the distance from the base of the ladder to the wall. Step 1: Identify that this forms a right triangle with ladder as the hypotenuse $c = 5.4$ m and height $a = 5.2$ m. Step 2: Use the Pythagorean theorem: $$ b = \sqrt{c^2 - a^2} = \sqrt{5.4^2 - 5.2^2} $$ Step 3: Calculate: $$ 5.4^2 = 29.16, \quad 5.2^2 = 27.04 $$ Step 4: $$ b = \sqrt{29.16 - 27.04} = \sqrt{2.12} = 1.456 $$ Step 5: Round to 1 decimal place: $1.5$ m. --- 2. **Problem:** Martha's compost bin is trapezoidal with width 2 m, back wall 1.8 m, front wall 0.9 m. Find the lid length that fits exactly. Step 1: The lid length is the top base of the trapezoid. Step 2: Draw a right triangle on the side showing height difference $1.8 - 0.9 = 0.9$ m and base $x$ is the overhang. Step 3: The bin's width bottom base = 2 m. Step 4: Use Pythagorean theorem on the right triangle formed by the side wall and difference in heights: $$ x = \sqrt{l^2 - 0.9^2} $$ But here, $l$ is the lid length sought, and the whole top base = bottom base - 2 overhangs: $$ l = 2 - 2x $$ Step 5: Substitute $x = \sqrt{l^2 - 0.9^2}$: $$ l = 2 - 2\sqrt{l^2 - 0.9^2} $$ Step 6: Let’s solve for $l$: Rearranged: $$ 2\sqrt{l^2 - 0.9^2} = 2 - l $$ Square both sides: $$ 4(l^2 - 0.81) = (2 - l)^2 $$ Expand: $$ 4l^2 - 3.24 = 4 - 4l + l^2 $$ Simplify: $$ 4l^2 - 3.24 = 4 - 4l + l^2 $$ Bring all to one side: $$ 4l^2 - l^2 + 4l - 3.24 - 4 = 0 $$ $$ 3l^2 + 4l - 7.24 = 0 $$ Step 7: Solve quadratic equation: $$ l = \frac{-4 \pm \sqrt{4^2 - 4 \times 3 \times (-7.24)}}{2 \times 3} = \frac{-4 \pm \sqrt{16 + 86.88}}{6} = \frac{-4 \pm \sqrt{102.88}}{6} $$ $$ \sqrt{102.88} \approx 10.144 $$ Choose positive root: $$ l = \frac{-4 + 10.144}{6} = \frac{6.144}{6} = 1.024 $$ Step 8: Round to 1 decimal place: $1.0$ m. --- 3. **Problem:** Find the height and area of an equilateral triangle of side 10 cm. (a) Height $h$ formula: $$ h = \frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3} $$ Calculate: $$ h = 5 \times 1.73205 = 8.660 $$ (b) Area $A$ formula: $$ A = \frac{1}{2} \times base \times height = \frac{1}{2} \times 10 \times 8.660 = 43.3 $$ Round answers to 3 significant figures: $$ h = 8.66 \text{ cm}, \quad A = 43.3 \text{ cm}^2 $$ --- **Final Answers:** 1. Distance from base to wall = 1.5 m 2. Lid length = 1.0 m 3. (a) Height = 8.66 cm (b) Area = 43.3 cm\textsuperscript{2}