The surface area of a table tennis ball is $\frac{1}{\pi} cm^2$. What is its diameter?
a) $\frac{7}{22} cm$
b) $\frac{22}{7} cm$
c) 7 cm
d) 3 cm
Answer (2)
State the formula for the surface area of a sphere: S A = 4 π r 2 .
Substitute the given surface area π 1 into the formula and solve for the radius r .
Calculate the diameter d using the formula d = 2 r .
Approximate π as 7 22 to find the diameter: 22 7 c m .
Explanation
Problem Analysis The problem states that the surface area of a table tennis ball is π 1 c m 2 . We need to find the diameter of the ball.
Define variables and state formulas Let S A be the surface area of the sphere, which is given as π 1 c m 2 .
Let r be the radius of the sphere and d be the diameter. Then d = 2 r .
The formula for the surface area of a sphere is S A = 4 π r 2 .
Substitute the surface area Substitute the given surface area into the formula: π 1 = 4 π r 2
Solve for the radius Solve for r 2 :
r 2 = 4 π 2 1 Take the square root to solve for r :
r = 4 π 2 1 = 2 π 1
Calculate the diameter Calculate the diameter d = 2 r :
d = 2 × 2 π 1 = π 1
Approximate the diameter We can approximate π as 7 22 . Then the diameter is: d = π 1 ≈ 7 22 1 = 22 7 Therefore, the diameter of the table tennis ball is approximately 22 7 c m .
Final Answer The diameter of the table tennis ball is 22 7 c m . So the answer is (a).
Examples
Understanding the surface area and diameter relationship of spheres is crucial in various fields. For instance, in sports equipment design, knowing the surface area helps determine the amount of material needed to coat a ball, while the diameter affects its aerodynamics and handling. Similarly, in pharmaceutical science, the surface area to volume ratio of spherical drug particles impacts their dissolution rate and bioavailability. Even in astronomy, calculating the surface area and diameter of planets and stars helps scientists understand their physical properties and behavior.
The diameter of the table tennis ball is calculated to be 22 7 c m based on its surface area of π 1 c m 2 . This conclusion follows from using the surface area formula for a sphere and solving for diameter. The answer is option (a) 22 7 c m .
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