Question

a chocolatier makes chocolate covered cherries in the form of spheres. hannah measures the outer diameter of these chocolates to be 4 cm. if she measures the thickness of the chocolate to be 0.5 cm, how much chocolate is used in one of the chocolate covered cherries? round your answer to the nearest hundredth if necessary. (note: diagram is not drawn to scale.)

Explanation:

Step1: Find outer sphere radius

Outer diameter = 4 cm, so outer radius $R = \frac{4}{2} = 2$ cm

Step2: Find inner sphere radius

Chocolate thickness = 0.5 cm, so inner radius $r = 2 - 0.5 = 1.5$ cm

Step3: Calculate outer sphere volume

Volume formula for sphere: $V = \frac{4}{3}\pi r^3$
Outer volume $V_o = \frac{4}{3}\pi (2)^3 = \frac{32}{3}\pi$

Step4: Calculate inner sphere volume

Inner volume $V_i = \frac{4}{3}\pi (1.5)^3 = \frac{4}{3}\pi (3.375) = 4.5\pi$

Step5: Find chocolate volume (difference)

Chocolate volume $V = V_o - V_i = \frac{32}{3}\pi - 4.5\pi = \frac{32}{3}\pi - \frac{9}{2}\pi = \frac{64\pi - 27\pi}{6} = \frac{37}{6}\pi$

Step6: Compute numerical value

$\frac{37}{6}\pi \approx \frac{37}{6} \times 3.1416 \approx 19.37$

Answer:

$19.37$ cm³