Question

a sphere has a radius of 16 in. which statements about the sphere are true? choose three correct answers.
the sphere has a diameter of 32 in.
the radiuss length is one - half the length of the diameter.
the diameters length is one - half the length of the radius.
the volume of the sphere is \\( \frac { 2,048 } { 3 } \pi \mathrm { in } ^ { 3 } \\).
the volume of the sphere is \\( \frac { 16,384 } { 3 } \pi \mathrm { in } ^ { 3 } \\).
the sphere has a diameter of 8 in.

Explanation:

Step1: Calculate the diameter

The formula for the diameter \(d\) of a sphere is \(d = 2r\), where \(r\) is the radius. Given \(r=16\) in, then \(d = 2\times16=32\) in. So the statement "The sphere has a diameter of 32 in" is true.

Step2: Analyze the radius - diameter relationship

By the formula \(r=\frac{d}{2}\), the radius's length is one - half the length of the diameter. So the statement "The radius's length is one - half the length of the diameter" is true.

Step3: Calculate the volume

The formula for the volume \(V\) of a sphere is \(V=\frac{4}{3}\pi r^{3}\). Substitute \(r = 16\) in: \(V=\frac{4}{3}\pi(16)^{3}=\frac{4}{3}\pi\times4096=\frac{16384}{3}\pi\) \(in^{3}\). So the statement "The volume of the sphere is \(\frac{16384}{3}\pi\) \(in^{3}\)" is true.

Answer:

The sphere has a diameter of 32 in, The radius's length is one - half the length of the diameter, The volume of the sphere is \(\frac{16384}{3}\pi\) \(in^{3}\)