The figure below is made up of two separate semicircles. The total area of the semicircles is 154 cm2. Find the perimeter of the figure. (take pi 22/7)

1. Break Down the Problem

To find the perimeter of the figure made up of two semicircles, we need to:

1. Determine the radii of the semicircles from the given total area.
2. Calculate the perimeter using the radii.

2. Relevant Concepts

The area $A$ of a semicircle is given by the formula: $A = \frac{1}{2} \pi r^2$ Where $r$ is the radius and $\pi$ is a constant.

3. Analysis and Detail

1. Determining the Area of Each Semicircle: The total area of the two semicircles is given as 154 cm²: $A_1 + A_2 = 154 \, \text{cm}^2$ Assuming both semicircles have the same radius $r$ (for simplicity, unless specified otherwise): $2 \left(\frac{1}{2} \pi r^2 \right) = 154$ Simplifying leads to: $\pi r^2 = 154$ Substituting $\pi$ with $\frac{22}{7}$: $\frac{22}{7} r^2 = 154$ Multiplying both sides by $\frac{7}{22}$: $r^2 = 154 \times \frac{7}{22} = 49$ Taking the square root gives: $r = \sqrt{49} = 7 \, \text{cm}$

2. Calculating the Perimeter: The perimeter of the figure composed of two semicircles is the sum of the curved edges: $\text{Perimeter} = \text{length of two semicircles} = 2 \times \left( \frac{1}{2} \pi r \right) + \text{straight diameter}$ The diameter for both semicircles together is $2r$: $\text{Perimeter} = 2 \times \left( \frac{1}{2} \pi r \right) + 2r = \pi r + 2r$ Substituting $\pi = \frac{22}{7}$ and $r = 7$: $\text{Perimeter} = \left(\frac{22}{7} \times 7\right) + 2 \times 7 = 22 + 14 = 36 \, \text{cm}$

4. Verify and Summarize

• The area of both semicircles has been verified to equal 154 cm².
• The calculation of the perimeter is correct and detailed.

Final Answer

The perimeter of the figure is $36 \, \text{cm}$.