Question

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.
$v = \frac{4}{3}\pi r^3$
$v = \frac{4}{3}\pi(9)^3$
$v = \frac{4}{3}\pi(729)$
$v = 972\pi$ cubic units
is noah’s work correct? explain.
\bigcirc yes, noah’s calculations are correct.
\bigcirc no, noah did not simplify correctly.
\bigcirc no, noah used the diameter instead of the radius in the calculations.
\bigcirc no, noah did not use the correct formula for volume of a sphere.

Explanation:

Step1: Recall sphere volume formula

The formula for the volume of a sphere is \( V=\frac{4}{3}\pi r^{3} \), where \( r \) is the radius.

Step2: Determine radius from diameter

Given the diameter \( d = 9 \) units, the radius \( r=\frac{d}{2}=\frac{9}{2} = 4.5 \) units.

Step3: Analyze Noah's work

Noah used \( r = 9 \) (the diameter) instead of \( r=\frac{9}{2} \) in the formula \( V=\frac{4}{3}\pi r^{3} \). His formula and simplification steps (other than using diameter as radius) are correct. So the error is using diameter instead of radius.

Answer:

No, Noah used the diameter instead of the radius in the calculations.