Equipartition Theorem A very sensitive spring balance consists of a quartz spring suspended vertically from a fixed support. The spring constant is k. The balance is at temperature T in a location where the gravitational acceleration is g. (a) When a small object of mass m is suspended from the spring, the spring will be stretched out by an amount x and will reach an equilibrium position. However, thermal energy will cause fluctuations in x. Ignoring the mass of the spring, what is the average value of x,

The second largest public utility in the nation is the sole provider of electricity in 32 counties of southern florida. to meet the monthly demand for electricity in these counties, which is given by the inverse demand function p = 1,200 – 4q, the utility company has set up two electric generating facilities: q1 kilowatts are produced at facility 1, and q2 kilowatts are produced at facility 2 (so q = q1 + q2). the costs of producing electricity at each facility are given by c1(q1) = 8,000 + 6q12 and c2(q2) = 6,000 + 3q22, respectively. determine the profit-maximizing amounts of electricity to produce at the two facilities, the optimal price, and the utility company’s profit