3. The rotunda at a state capitol is a circular region of 50 feet in diameter. The legislature wishes to appropriate money to have the floor of the rotunda tiled. If it costs $$ \$ 27 $$ per square foot, how much must the legislature spend? Use $$ \pi=3.14 $$. b) For a celebration the governor wants strings of lights to surround the outside edge of the rotunda. How many feet of lights will need to be hung? Use $$ \pi=3.14 $$.

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1. **Determine the radius of the rotunda**

The diameter of the rotunda is $$50$$ feet. The radius $$r$$ is half of the diameter:
   $$
   r = \frac{50}{2} = 25 \text{ feet}
   $$

2. **Calculate the area of the floor**

The area $$A$$ of a circle is given by:
   $$
   A = \pi r^2
   $$
   Substitute $$r = 25$$ feet and $$\pi = 3.14$$:
   $$
   A = 3.14 \times 25^2 = 3.14 \times 625 = 1962.5 \text{ square feet}
   $$

3. **Determine the cost to tile the floor**

It costs $$\$27$$ per square foot. Thus, the total cost $$C$$ is:
   $$
   C = 1962.5 \times 27 = 52987.5
   $$
   So, the legislature must spend $$\$52,\!987.50$$.

4. **Determine the circumference of the rotunda**

The circumference $$C_{\text{circ}}$$ of a circle is given by:
   $$
   C_{\text{circ}} = 2 \pi r
   $$
   Substitute $$r = 25$$ feet and $$\pi = 3.14$$:
   $$
   C_{\text{circ}} = 2 \times 3.14 \times 25 = 157 \text{ feet}
   $$

Thus, the legislature must spend $$\$52,\!987.50$$ to tile the floor, and the governor needs $$157$$ feet of lights to surround the rotunda.
To tile the rotunda's floor: - Radius: 25 feet - Area: 1,962.5 square feet - Cost: \$52,987.50 For the lights: - Circumference: 157 feet

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