Semicircle Perimeter Cff476

1. **State the problem:** We have a figure composed of a rectangle and a semicircle. The area of the rectangle is given as $128\ \text{cm}^2$. We need to find the perimeter of the semicircle. 2. **Identify known values and formulas:** - Area of rectangle $= \text{length} \times \text{width} = 128\ \text{cm}^2$ - The semicircle is attached to one side of the rectangle, so the diameter of the semicircle equals one side of the rectangle. - Perimeter of a semicircle $= \pi r + 2r$ where $r$ is the radius. 3. **Find the dimensions of the rectangle:** Since the semicircle is attached to the rectangle, the diameter of the semicircle equals the length of one side of the rectangle. Assuming the diameter of the semicircle is the length of the rectangle, let: - Diameter $d = l$ - Width $w$ Area of rectangle: $$128 = l \times w$$ 4. **Find the radius of the semicircle:** $$r = \frac{d}{2} = \frac{l}{2}$$ 5. **Calculate the perimeter of the semicircle:** $$\text{Perimeter} = \pi r + 2r = r(\pi + 2)$$ 6. **Use the given value of $\pi = 3.14$:** $$\text{Perimeter} = r(3.14 + 2) = r(5.14)$$ 7. **Find $l$ and $r$ using the rectangle area:** Since the semicircle is attached to the rectangle, the diameter $d = l$. We need more information to find $l$ and $w$ individually, but since the problem only gives the area and asks for the perimeter of the semicircle, we assume the diameter equals the width of the rectangle. If the diameter $d = w$, then: $$128 = l \times d$$ But without $l$ or $d$, we cannot find $r$ directly. **Assuming the diameter equals the width $w$, and the length $l$ is such that $l = \frac{128}{d}$** 8. **Since the semicircle is attached to the rectangle, the diameter equals the width $w$.** We can express the perimeter of the semicircle in terms of $d$: $$r = \frac{d}{2}$$ $$\text{Perimeter} = \frac{d}{2} \times 5.14 = 2.57d$$ 9. **Without the exact diameter, the problem cannot be solved numerically.** **However, if the diameter is given or can be found from the figure, substitute it here to find the perimeter.** **Final formula for perimeter of semicircle:** $$\boxed{\text{Perimeter} = 2.57 \times d}$$ **Note:** To find the exact numerical value, the diameter $d$ must be known or given.