Sure, let's break down the problem step-by-step to find the uncovered area of the table.
1. Calculate the Area of the Table Top:
- The radius of the table top is given as 36 cm.
- The formula for the area of a circle is [tex]\( A = \pi r^2 \)[/tex].
- Using [tex]\(\pi = 3.142\)[/tex] and [tex]\(r = 36\)[/tex] cm, the area of the table top is:
[tex]\[
A_{\text{table}} = 3.142 \times (36)^2 = 4072.032 \text{ cm}^2
\][/tex]
2. Determine the Radius of Each Coaster:
- Six coasters fit in a straight line across the diameter of the table.
- Hence, the diameter of the circle formed by the six coasters is equal to the diameter of the table.
- The diameter of the table is [tex]\(2 \times 36 \text{ cm} = 72 \text{ cm}\)[/tex].
- Therefore, the diameter of each coaster is [tex]\( \frac{72 \text{ cm}}{6} = 12 \text{ cm}\)[/tex].
- The radius of each coaster is [tex]\( \frac{12 \text{ cm}}{2} = 6 \text{ cm}\)[/tex].
3. Calculate the Area of One Coaster:
- Using the radius of one coaster which is 6 cm, and the same formula for the area of a circle:
[tex]\[
A_{\text{coaster}} = 3.142 \times (6)^2 = 113.112 \text{ cm}^2
\][/tex]
4. Calculate the Total Area Covered by Six Coasters:
- Since there are six coasters, the total area covered by the coasters is:
[tex]\[
A_{\text{total coasters}} = 6 \times 113.112 = 678.672 \text{ cm}^2
\][/tex]
5. Calculate the Uncovered Area:
- The area of the table top that is not covered by the coasters is:
[tex]\[
A_{\text{uncovered}} = A_{\text{table}} - A_{\text{total coasters}}
\][/tex]
[tex]\[
A_{\text{uncovered}} = 4072.032 - 678.672 = 3393.360 \text{ cm}^2
\][/tex]
Thus, the area of the table that is not covered by the coasters is approximately [tex]\( 3393.36 \text{ cm}^2 \)[/tex].
