3. The left side of the heart can be thought of as a spherical balloon with a thick wall. In the resting state, the volume of the heart is given by

while the volume available for blood is

a. If the radius of the heart, R_H = 5.5 cm and the thickness of the heart muscle, W, is 41% of the radius of heart, then determine the volume of the heart, V_H, the internal radius, R_B, and the volume available for blood, V_B.

b. The muscles of the heart generate energy to pump the blood through the arteries. We assume that the muscles wrap around the heart, so have length, L = 2pR_H, at rest. Suppose that if the heart muscles shortens by an amount x, then the energy generated for pumping blood satisfies the relation

Find the value of x that maximizes the energy from the heart muscle and give this energy level. Graph this function for 0 < x < 10.

c. As the muscle contracts, the volume filled with blood decreases, pumping blood through the arteries. Determine the amount of blood expelled from the heart for the value of x, which maximizes the energy. The volume of the cavity for blood is given by

where z = x/2p is the decrease in the radius due to the contracting muscle.

d. One characteristic of heart disease is an enlarged heart. Write a brief paragraph explaining how the information above could be used to show why an enlarged heart is less efficient. Describe which variables would change and how they would affect the energy function and the various volumes.