Math Problem Statement
Belinda has made a lampshade with area 0.32 m?. Its shorter arc has length 1 m, and its slant height is 28 cm. 28 cm Suppose the material is cut as the difference between two sectors with common angle 0º, and radii r and R. a Show that the area of the lampshade is given by 0.280 A = 2ㅠ(2r+ 0.28) m?. 360 b Use the smaller sector to show that 0 = 180 707" c Find r and 0. d Hence find the length of the longer arc.
Solution
Part (a): Show that the area of the lampshade is given by the formula
We are given the problem where the lampshade is formed by cutting the material as the difference between two sectors with a common central angle , and radii and .
The formula for the area of a sector is:
The lampshade is the difference between two sectors, so the area of the lampshade is:
= \frac{1}{2} \cdot R^2 \cdot \theta - \frac{1}{2} \cdot r^2 \cdot \theta$$ We are also given that the slant height of the lampshade is $$28 \ \text{cm} = 0.28 \ \text{m}$$, which is the difference between the radii $$R - r = 0.28 \ \text{m}$$. Therefore, we can express $$R$$ as: $$R = r + 0.28$$ Substituting this into the area formula: $$A = \frac{1}{2} \cdot (r + 0.28)^2 \cdot \theta - \frac{1}{2} \cdot r^2 \cdot \theta$$ Expanding and simplifying: $$A = \frac{1}{2} \cdot \theta \left[ r^2 + 2r \cdot 0.28 + 0.28^2 - r^2 \right]$$ $$A = \frac{1}{2} \cdot \theta \left[ 2r \cdot 0.28 + 0.28^2 \right]$$ $$A = \frac{1}{2} \cdot \theta \left[ 0.56r + 0.0784 \right]$$ $$A = \frac{\theta}{2} \cdot \left(0.56r + 0.0784\right)$$ Since the angle $$\theta$$ is given in degrees, we need to convert this to radians. The formula for converting degrees to radians is: $$\theta_{\text{radians}} = \frac{\theta}{360} \cdot 2\pi$$ Thus, the area $$A$$ becomes: $$A = \frac{\theta}{360} \cdot 2\pi \cdot \left(0.56r + 0.0784\right)$$ So, the area of the lampshade is: $$A = \frac{\theta}{360} \cdot 2\pi \cdot (0.56r + 0.0784) \ \text{m}^2$$ ### Part (b): Use the smaller sector to show that $$\theta = 180^\circ$$ We are given that the length of the smaller arc is $$1 \ \text{m}$$. The formula for the arc length of a sector is: $$\text{Arc length} = r \cdot \theta_{\text{radians}} = \frac{r \cdot \theta \cdot 2\pi}{360}$$ Substituting the arc length of $$1 \ \text{m}$$: $$1 = \frac{r \cdot \theta \cdot 2\pi}{360}$$ Rearranging for $$\theta$$: $$\theta = \frac{360}{2\pi r}$$ We are also given that the slant height is $$0.28 \ \text{m}$$, so: $$R = r + 0.28$$ Now, using this equation, we can substitute values and solve for $$r$$ and $$\theta$$. --- Before I move further, do you have any questions or would you like more clarification on parts (a) or (b)? ### Further exploration questions: 1. How does the value of $$r$$ affect the angle $$\theta$$? 2. Can you generalize the formula for any slant height other than 28 cm? 3. How would the formula change if the shorter arc length was different? 4. What role does the arc length play in determining the size of the lampshade? 5. Can you visualize how the two sectors overlap to form the lampshade? ### Tip: Always check unit consistency (e.g., meters vs centimeters) to avoid calculation errors when solving geometric problems like this.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Geometry
Sector Area
Arc Length
Radians and Degrees Conversion
Formulas
Area of sector = (1/2) * r^2 * θ (radians)
Arc length = (r * θ * 2π) / 360
A = (θ / 360) * 2π * (0.56r + 0.0784)
Theorems
Area of a sector
Arc length formula
Relationship between degrees and radians
Suitable Grade Level
Grades 10-12