TSA of a hemisphere is [tex]27\pi[/tex] [tex]m^3[/tex]. What is its radius?
A. 2m
B. 5m
C. 3m
D. 9m
Answer (2)
The problem provides the total surface area of a hemisphere.
Recall the formula for the total surface area of a hemisphere: TS A = 3 π r 2 .
Set up the equation 3 π r 2 = 27 π and solve for r .
The radius of the hemisphere is 3 m .
Explanation
Problem Analysis We are given the total surface area (TSA) of a hemisphere as 27 π m 3 . Our goal is to find the radius of this hemisphere.
Formula Recall The formula for the total surface area of a hemisphere is given by: TS A = 3 π r 2 where r is the radius of the hemisphere.
Equation Setup We are given that TS A = 27 π m 3 . So, we can set up the equation: 3 π r 2 = 27 π
Isolate r^2 To solve for r , we first divide both sides of the equation by 3 π :
3 π 3 π r 2 = 3 π 27 π r 2 = 9
Solve for r Now, we take the square root of both sides of the equation to find r :
r = 9 Since the radius must be a positive value, we have: r = 3
Final Answer Therefore, the radius of the hemisphere is 3 meters.
Examples
Understanding the surface area of hemispheres is useful in various real-world applications. For example, when designing domes for buildings, engineers need to calculate the surface area to determine the amount of material required. Similarly, in manufacturing spherical tanks or containers, knowing the radius helps in estimating the material needed for production. This concept is also applicable in fields like geography, where calculating the surface area of a hemispherical region on Earth can be important for climate studies or resource management.
The radius of the hemisphere, when the total surface area is 27 π m², is 3 meters. Therefore, the choice is option C: 3m.
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