An insulated pipe carries a hot fluid. The setup is shown in Figure 6.18. The copper pipe has radius R₀ = 1 cm and carries a liquid at T₀ = 100 ^◦C. The pipe is encased in a cylindrical layer of insulation of outer radius R₁; the insulation has been chosen to be closed-cell polyurethane spray foam with an R-value of 1.00m² K/W per inch of thickness. How large must R₁ be so that the heat loss to the surrounding room (at 20 ^◦C) is less than 10W/m? First explain why the heat flux must be q = cr̂/r. You may find it useful to make the analogy between heat propagation through the insulator and current flowing outward through a cylindrical conductor. ∇f (r) = r̂ f ′(r).