Let's approach each part of the question step-by-step.
### Problem 4: Regular polygon with an exterior angle of 60 degrees
The formula to calculate the exterior angle of a regular polygon is:
[tex]\[ \text{Exterior angle} = \frac{360^\circ}{n} \][/tex]
where [tex]\( n \)[/tex] is the number of sides of the polygon. We need to find the polygon with an exterior angle measuring 60 degrees.
Given:
[tex]\[ \text{Exterior angle} = 60^\circ \][/tex]
Using the formula:
[tex]\[ 60 = \frac{360}{n} \][/tex]
To find [tex]\( n \)[/tex]:
[tex]\[ n = \frac{360}{60} \][/tex]
[tex]\[ n = 6 \][/tex]
A regular polygon with 6 sides is a regular hexagon.
Therefore, the answer is:
B. Regular hexagon
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### Problem 5: Volume of a sphere with a radius of 30 cm
The formula to calculate the volume of a sphere is:
[tex]\[ \text{Volume} = \frac{4}{3} \pi r^3 \][/tex]
where [tex]\( r \)[/tex] is the radius of the sphere. Given that the radius [tex]\( r = 30 \)[/tex] cm, we can plug this value into the formula.
[tex]\[ \text{Volume} = \frac{4}{3} \pi (30)^3 \][/tex]
Now, let's calculate it step-by-step:
1. Calculate [tex]\( 30^3 \)[/tex]:
[tex]\[ 30^3 = 30 \times 30 \times 30 = 27000 \][/tex]
2. Multiply by [tex]\( \frac{4}{3} \)[/tex]:
[tex]\[ \frac{4}{3} \times 27000 = \frac{108000}{3} = 36000 \][/tex]
3. Multiply by [tex]\( \pi \)[/tex] (approximately [tex]\( 3.14159 \)[/tex]):
[tex]\[ 36000 \times 3.14159 \approx 113097.24 \][/tex]
4. Round to the nearest whole number:
[tex]\[ 113097.24 \approx 113097 \][/tex]
Therefore, the volume of the sphere to the nearest whole number is:
113097 cm³
