Question

1. find the area of the circle. find the solution in terms of \\(\pi\\), and then convert the result into decimal form rounded to the nearest hundredth using 3.14 for \\(\pi\\).

the circumference is \\(40\pi\\) decimeters.
\\(\square\\) a = 2,512 \\(\text{dm}^2\\)
\\(\square\\) a = 1,256 \\(\text{dm}^2\\)
\\(\square\\) a = 5,024 \\(\text{dm}^2\\)
\\(\square\\) a = 400\\(\pi\\) \\(\text{dm}^2\\)
\\(\square\\) a = 1,600\\(\pi\\) \\(\text{dm}^2\\)
\\(\square\\) a = 800\\(\pi\\)

Explanation:

Step1: Find radius from circumference

Circumference formula: $C = 2\pi r$. Substitute $C=40\pi$:
$$40\pi = 2\pi r$$
Solve for $r$: $r = \frac{40\pi}{2\pi} = 20$ dm

Step2: Calculate area in terms of $\pi$

Area formula: $A = \pi r^2$. Substitute $r=20$:
$$A = \pi (20)^2 = 400\pi \ \text{dm}^2$$

Step3: Convert to decimal form

Substitute $\pi=3.14$:
$$A = 400 \times 3.14 = 1256.00 \ \text{dm}^2$$

Answer:

In terms of $\pi$: $A = 400\pi \ \text{dm}^2$
Decimal form: $A = 1,256 \ \text{dm}^2$
(The correct options are $\boldsymbol{A = 1,256 \ \text{dm}^2}$ and $\boldsymbol{A = 400\pi \ \text{dm}^2}$)