To determine which shape has the greater area, we need to calculate the area of each shape.
Shape A:
Shape A is a circle with a radius [tex]\(\frac{\sqrt{17}}{2}\)[/tex] cm. The formula for the area of a circle is:
[tex]\[
\text{Area} = \pi r^2
\][/tex]
where [tex]\(r\)[/tex] is the radius.
For Shape A:
[tex]\[
r = \frac{\sqrt{17}}{2}
\][/tex]
Substituting this into the formula for the area:
[tex]\[
\text{Area}_A = \pi \left(\frac{\sqrt{17}}{2}\right)^2
\][/tex]
Calculating the radius squared:
[tex]\[
\left(\frac{\sqrt{17}}{2}\right)^2 = \frac{17}{4}
\][/tex]
Then, substituting back into the area formula:
[tex]\[
\text{Area}_A = \pi \cdot \frac{17}{4} = \frac{17\pi}{4}
\][/tex]
Numerically, this area is approximately:
[tex]\[
\text{Area}_A \approx 13.35 \, \text{cm}^2
\][/tex]
Shape B:
Shape B is a sector of a circle with radius 5 cm. Since no additional details are provided about the angle of the sector, we'll assume Shape B is a full circle.
The formula for the area of a circle is again:
[tex]\[
\text{Area} = \pi r^2
\][/tex]
where [tex]\(r\)[/tex] is the radius.
For Shape B:
[tex]\[
r = 5 \, \text{cm}
\][/tex]
Substituting this into the formula for the area:
[tex]\[
\text{Area}_B = \pi (5)^2 = 25\pi
\][/tex]
Numerically, this area is approximately:
[tex]\[
\text{Area}_B \approx 78.54 \, \text{cm}^2
\][/tex]
Comparison:
[tex]\[
\text{Area}_A \approx 13.35 \, \text{cm}^2
\][/tex]
[tex]\[
\text{Area}_B \approx 78.54 \, \text{cm}^2
\][/tex]
Since [tex]\(78.54 \, \text{cm}^2 > 13.35 \, \text{cm}^2\)[/tex], we conclude that Shape B has the greater area.
Therefore, Shape B has the greater area.
