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Sagot :
Certainly! Let's analyze the given sphere with a diameter of 5 meters and deduce the correct statements about it step by step.
1. Determining the Radius
- The diameter of the sphere is given as 5 meters.
- The radius is half of the diameter.
- Therefore, the radius [tex]\( r \)[/tex] is calculated as:
[tex]\[ r = \frac{\text{diameter}}{2} = \frac{5 \text{ meters}}{2} = 2.5 \text{ meters} \][/tex]
2. Checking the Statements
a. The sphere has a radius of 10 cm.
- Since 10 cm is equivalent to 0.1 meters, we compare this with the radius we calculated.
- The radius is 2.5 meters, not 0.1 meters.
- This statement is False.
b. The diameter measure is substituted into the formula to find the volume.
- The correct formula to find the volume of a sphere uses the radius, not the diameter directly:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
- Substituting the diameter directly into the volume formula is incorrect.
- This statement is False.
c. The radius is half the diameter.
- As we calculated, the radius is exactly half of the diameter.
- This statement is True.
d. The formula to apply is [tex]\( V = \frac{4}{3} B h \)[/tex]
- The correct formula to calculate the volume of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
- The formula [tex]\( V = \frac{4}{3} B h \)[/tex] is not applicable to spheres; it seems to be a reference to volumes of other geometric shapes.
- This statement is False.
e. The volume of the sphere is two-thirds the volume of a cylinder with the same radius and height.
- The volume of a sphere is given by:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \][/tex]
- The volume of a cylinder with the same radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] is:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
- There is no such direct relationship that the volume of a sphere is two-thirds the volume of such a cylinder.
- This statement is False.
Conclusion:
After analyzing all the given statements, the conclusions are:
1. The sphere does not have a radius of 10 cm. (False)
2. The diameter measure is not substituted into the formula to find the volume. (False)
3. The radius is half the diameter. (True)
4. The formula to apply is not [tex]\( V = \frac{4}{3} B h \)[/tex]. (False)
5. The volume of the sphere is not two-thirds the volume of a cylinder with the same radius and height. (False)
1. Determining the Radius
- The diameter of the sphere is given as 5 meters.
- The radius is half of the diameter.
- Therefore, the radius [tex]\( r \)[/tex] is calculated as:
[tex]\[ r = \frac{\text{diameter}}{2} = \frac{5 \text{ meters}}{2} = 2.5 \text{ meters} \][/tex]
2. Checking the Statements
a. The sphere has a radius of 10 cm.
- Since 10 cm is equivalent to 0.1 meters, we compare this with the radius we calculated.
- The radius is 2.5 meters, not 0.1 meters.
- This statement is False.
b. The diameter measure is substituted into the formula to find the volume.
- The correct formula to find the volume of a sphere uses the radius, not the diameter directly:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
- Substituting the diameter directly into the volume formula is incorrect.
- This statement is False.
c. The radius is half the diameter.
- As we calculated, the radius is exactly half of the diameter.
- This statement is True.
d. The formula to apply is [tex]\( V = \frac{4}{3} B h \)[/tex]
- The correct formula to calculate the volume of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
- The formula [tex]\( V = \frac{4}{3} B h \)[/tex] is not applicable to spheres; it seems to be a reference to volumes of other geometric shapes.
- This statement is False.
e. The volume of the sphere is two-thirds the volume of a cylinder with the same radius and height.
- The volume of a sphere is given by:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \][/tex]
- The volume of a cylinder with the same radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] is:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
- There is no such direct relationship that the volume of a sphere is two-thirds the volume of such a cylinder.
- This statement is False.
Conclusion:
After analyzing all the given statements, the conclusions are:
1. The sphere does not have a radius of 10 cm. (False)
2. The diameter measure is not substituted into the formula to find the volume. (False)
3. The radius is half the diameter. (True)
4. The formula to apply is not [tex]\( V = \frac{4}{3} B h \)[/tex]. (False)
5. The volume of the sphere is not two-thirds the volume of a cylinder with the same radius and height. (False)
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