To find the approximate area of a circle that has been dissected into 16 congruent sectors, where each sector has a base of 1.56 units and a height of 3.92 units, we can use the area of a triangle formula since the area of each sector resembles a triangle.
Step-by-Step Solution:
1. Determine the Area of One Sector:
Each sector of the circle can be approximated as a triangle. The area [tex]\(A\)[/tex] of a triangle is given by:
[tex]\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
Given:
- Base [tex]\( = 1.56 \)[/tex] units
- Height [tex]\( = 3.92 \)[/tex] units
Substitute these values into the formula:
[tex]\[
A_{\text{sector}} = \frac{1}{2} \times 1.56 \times 3.92
\][/tex]
This gives us:
[tex]\[
A_{\text{sector}} = 0.5 \times 1.56 \times 3.92 = 3.0576 \text{ square units}
\][/tex]
2. Calculate the Total Area of the Circle:
The circle is divided into 16 congruent sectors. Therefore, the total area of the circle [tex]\(A_{\text{circle}}\)[/tex] is:
[tex]\[
A_{\text{circle}} = 16 \times A_{\text{sector}}
\][/tex]
Substitute the area of one sector:
[tex]\[
A_{\text{circle}} = 16 \times 3.0576 = 48.9216 \text{ square units}
\][/tex]
3. Choose the Closest Approximation:
The approximate area of the circle is [tex]\(48.9216\)[/tex] square units. Among the given options:
- 27.52 units [tex]\( ^2 \)[/tex]
- 48.25 units [tex]\( ^2 \)[/tex]
- 48.92 units [tex]\( ^2 \)[/tex]
- 76.44 units [tex]\( ^2 \)[/tex]
The value [tex]\(48.92\)[/tex] square units is the closest to 48.9216 square units.
Conclusion:
The approximate area of the circle is:
[tex]\[
\boxed{48.92 \text{ units}^2}
\][/tex]
