Question

some information is given about each sphere. order them from least volume to greatest volume. you may sketch a sphere to help you visualize if you prefer. sphere a has a radius of 4. sphere b has as a diameter of 6. sphere c has a volume of 64π. sphere d has a radius double that of sphere b.

Explanation:

Step1: Find Sphere B radius

Radius $r_B = \frac{\text{Diameter}}{2} = \frac{6}{2} = 3$

Step2: Calculate Sphere A volume

Volume formula: $V = \frac{4}{3}\pi r^3$
$V_A = \frac{4}{3}\pi (4)^3 = \frac{4}{3}\pi(64) = \frac{256}{3}\pi \approx 85.33\pi$

Step3: Calculate Sphere B volume

$V_B = \frac{4}{3}\pi (3)^3 = \frac{4}{3}\pi(27) = 36\pi$

Step4: Find Sphere C radius (verify)

Given $V_C = 64\pi = \frac{4}{3}\pi r_C^3$
Solve for $r_C$: $r_C^3 = 64\pi \times \frac{3}{4\pi} = 48$, $r_C = \sqrt[3]{48} \approx 3.63$ (volume is given as $64\pi$)

Step5: Find Sphere D radius & volume

$r_D = 2 \times r_B = 2\times3=6$
$V_D = \frac{4}{3}\pi (6)^3 = \frac{4}{3}\pi(216) = 288\pi$

Step6: Compare all volumes

$36\pi < 64\pi < \frac{256}{3}\pi < 288\pi$

Answer:

Sphere B < Sphere C < Sphere A < Sphere D