1. **Problem:** Circle A has a circumference of $8\pi$ cm. Circle B has an area of $36\pi$ cm$^2$. Find how many times greater the radius of circle B is compared to circle A. Step 1: Write formulas: - Circumference of circle: $C = 2\pi r$ - Area of circle: $A = \pi r^2$ Step 2: Find radius of circle A using circumference: $$8\pi = 2\pi r_A \implies r_A = \frac{8\pi}{2\pi} = 4 \text{ cm}$$ Step 3: Find radius of circle B using area: $$36\pi = \pi r_B^2 \implies r_B^2 = 36 \implies r_B = 6 \text{ cm}$$ Step 4: Calculate how many times radius of B is greater than radius of A: $$\frac{r_B}{r_A} = \frac{6}{4} = 1.5$$ Answer: a) 1.5 2. **Problem:** Find the area of the shaded figure (large rectangle minus triangle). Step 1: Calculate area of large rectangle: $$\text{Area} = 25 \times 14 = 350 \text{ cm}^2$$ Step 2: Calculate area of triangle inside rectangle: Base = 11 cm, Height = 25 cm (same as rectangle's height) $$\text{Area}_{\triangle} = \frac{1}{2} \times 11 \times 25 = 137.5 \text{ cm}^2$$ Step 3: Area of shaded figure = rectangle area - triangle area: $$350 - 137.5 = 212.5 \text{ cm}^2$$ Answer: c) 212.5 sq cm 3. **Problem:** Two cylinders with the same volume contain water and milk respectively. Which statement is true? Step 1: Definition: Volume is the amount of space occupied by a liquid. Step 2: Since both cylinders have the same volume, they occupy the same space. Step 3: Weight depends on density; heights depend on dimensions; surface area depends on shape. Only the space (volume) is guaranteed same. Answer: d) The liquids in both cylinders occupy the same amount of space. 4. **Problem:** Find the coordinates of Allan's home on the map based on marker location. Step 1: The marker is at (5, 4) approximately. Step 2: Given answer options closest to this point is a) (5,3) or c) (5,7) or b) (3,5) or d) (2,3). Since marker is near x=5 and y=4, (5,3) is closest. Answer: a) (5,3) 5. **Problem:** Identify which equation shows the two rectangles are similar. Step 1: Given rectangles with dimensions: - Left: Width = 2x, Height = 1 - Right: Width = 1, Height = x Step 2: Similar rectangles have proportional sides: $$\frac{\text{Width}_1}{\text{Width}_2} = \frac{\text{Height}_1}{\text{Height}_2}$$ Step 3: Substitute: $$\frac{2x}{1} = \frac{1}{x} \implies 2x = \frac{1}{x} \implies 2x^2 = 1 \implies x^2 = \frac{1}{2}$$ Step 4: This relationship shows similarity. Answer: The equation demonstrating similarity is $$\frac{2x}{1} = \frac{1}{x}$$