Noah is working on a geometry problem that involves finding the volume of a sphere with a diameter of 9 units. His work is shown below.
[tex]\begin{array}{l}
V=\frac{4}{3} \pi r^3 \\
V=\frac{4}{3} \pi(9)^3 \\
V=\frac{4}{3} \pi(729) \\
V=972 \pi \text { cubic units }\\
\end{array}[/tex]
Is Noah's work correct? Explain.
A. Yes, Noah's calculations are correct.
B. No, Noah did not simplify correctly.
C. No, Noah used the diameter instead of the radius in the calculations.
D. No, Noah did not use the correct formula for volume of a sphere.
Answer (1)
Calculate the radius from the diameter: r = 2 9 = 4.5 .
Substitute the radius into the volume formula: V = 3 4 π ( 4.5 ) 3 .
Calculate the volume: V = 121.5 π .
Noah's mistake was using the diameter instead of the radius, so his work is incorrect: No, Noah used the diameter instead of the radius in the calculations .
Explanation
Problem Analysis and Setup The problem asks us to evaluate Noah's work in calculating the volume of a sphere with a diameter of 9 units. We need to check if Noah used the correct formula and if he substituted the values correctly. The formula for the volume of a sphere is given by V = 3 4 π r 3 , where r is the radius of the sphere.
Calculate the Radius First, we need to find the radius of the sphere. The diameter is given as 9 units, so the radius is half of the diameter: r = 2 d = 2 9 = 4.5 units.
Substitute Radius into Volume Formula Now, we substitute the correct radius value into the volume formula: V = 3 4 π ( 4.5 ) 3 .
Calculate 4.5 cubed We calculate ( 4.5 ) 3 = 4.5 × 4.5 × 4.5 = 91.125 .
Calculate the Volume Then, we calculate V = 3 4 π ( 91.125 ) = 3 4 × 91.125 π = 3 364.5 π = 121.5 π .
Compare and Identify the Error Comparing the calculated volume 121.5 π with Noah's result 972 π , we see that Noah's calculation is incorrect. Noah used the diameter instead of the radius in the volume formula. He calculated V = 3 4 π ( 9 ) 3 instead of V = 3 4 π ( 4.5 ) 3 .
Conclusion Therefore, Noah's work is incorrect because he used the diameter instead of the radius in the volume calculation.
Examples
Understanding the volume of spheres is crucial in many real-world applications. For instance, when designing spherical tanks for storing liquids or gases, engineers need to accurately calculate the volume to determine the tank's capacity. Similarly, in the pharmaceutical industry, calculating the volume of spherical capsules is essential for precise drug delivery. Even in sports, knowing the volume of a ball (like a basketball or soccer ball) helps in understanding its aerodynamic properties and performance characteristics. Therefore, mastering the formula for the volume of a sphere is not just an academic exercise but a practical skill with wide-ranging implications.
