When an electronics store prices a certain brand of stereos at p hundred dollars per set, it is found that q sets will be sold each month, where q2 + 2 p2 = 41.

a. Find the elasticity of demand for the stereos using implicit differentiation.
b. For a unit price of p = 4 ($400), is the demand elastic, inelastic, or of unit elasticity?
c. Find the number of stereo sets that will maximize revenue (prove your answer).

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a. To find the elasticity of demand for the stereos, we need to first differentiate the equation q^2 + 2p^2 = 41 with respect to p. This will give us:

2q * dq/dp + 4p * dp/dp = 0

Simplifying this equation gives us:

2q * dq/dp + 4p = 0

Next, we can solve for dq/dp:

dq/dp = -4p / 2q

Finally, we can define the elasticity of demand (E) as the absolute value of dq/dp times the ratio of p to q:

E = |dq/dp| * (p/q)

b. To determine whether the demand is elastic, inelastic, or of unit elasticity at a unit price of p = 4 ($400), we need to calculate the elasticity of demand (E). Since we have the value of p, we can plug it into the equation from part a:

E = |dq/dp| * (p/q)
= |-4p / 2q| * (p/q)
= 4p^2 / 2q^2

Substituting p = 4, we get:

E = 4(4)^2 / 2q^2
= 64 / 2q^2
= 32 / q^2

To determine the elasticity at a unit price of p = 4, we need to find the value of q. We can substitute p = 4 into the original equation q^2 + 2p^2 = 41 and solve for q:

q^2 + 2(4)^2 = 41
q^2 + 32 = 41
q^2 = 9
q = 3

Now we can substitute q = 3 into the equation for E:

E = 32 / (3^2)
= 32 / 9

Since E is greater than 1 (specifically, E is approximately 3.56), the demand is elastic.

c. To find the number of stereo sets that will maximize revenue, we need to consider the revenue function. Revenue is calculated by multiplying the price (p) by the number of sets sold (q). In this case, the price is given as p hundred dollars and the number of sets sold is q.

Therefore, the revenue function, R, can be defined as:

R = p * q

To maximize the revenue, we need to find the critical points of the revenue function. Critical points occur when the derivative of the revenue function with respect to q equals zero.

dR/dq = p = 0

Since the critical point occurs at p = 4, we substitute p = 4 into the original equation q^2 + 2p^2 = 41 to find q:

q^2 + 2(4)^2 = 41
q^2 + 32 = 41
q^2 = 9
q = 3

Therefore, the number of stereo sets that will maximize revenue is q = 3, which we found in part b.