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Locate the centroid z¯ of the hemisphere. Take a = 7 m .
(Figure 1)

Express your answer to three significant figures and include the appropriate units.

To find the centroid \( \bar{z} \) of a hemisphere, we can use a known result from geometry.

For a solid hemisphere of radius \( a \), the centroid \( \bar{z} \) (distance from the base along the \( z \)-axis) is given by the formula:

\[
\bar{z} = \frac{3a}{8}
\]

Given that \( a = 7 \, \text{m} \), we substitute the value of \( a \) into the formula:

\[
\bar{z} = \frac{3(7)}{8} = \frac{21}{8} = 2.625 \, \text{m}
\]

Thus, the centroid \( \bar{z} \) of the hemisphere is \( 2.63 \, \text{m} \) (rounded to three significant figures).

Would you like more details on this result, or do you have any further questions?

Here are 5 related questions to explore:
1. How is the centroid of a full sphere calculated?
2. What is the formula for the surface area of a hemisphere?
3. How does the centroid differ between a hollow and solid hemisphere?
4. How can we compute the volume of a hemisphere?
5. How would the centroid formula change if the hemisphere were filled with a material of varying density?

**Tip:** For many symmetrical objects, centroid formulas are derived using integration over the object's volume or surface. Understanding symmetry can simplify centroid calculations.

Locate the centroid z¯ of the hemisphere. Take a = 7 m.

This solution helps in locating the centroid of a solid hemisphere with a radius of 7 meters using the formula z¯ = (3a)/8. The answer is expressed to three significant figures, giving a centroid of 2.63 meters. This problem covers key geometric concepts suitable for advanced students in Grades 11-12 or college-level courses.

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