iii Compute an apporximate depth using Newtons Method with s

iii. Compute an apporximate depth using Newton\'s Method with starting value x0=0.01

(1) (30 pts) In laying water mains, utilities must be concerned with the possibility of freezing Although soil and weather conditions are complicated, reasonable approximations can be made on the basis of the assumption that soil is uniform in all directions. In that case the temperature in degrees Celsius T(x, t) at a distance r (in meters) below the surface t seconds after the beginning of a cold snap, approximately satisfies T(x, t) - T Ti -Ts er where T, is the constant temperature during a cold period, Ti is the initial soil tem- perature before the cold snap, is the thermal conductivity (in meters per second), an Jo exp(-82)ds Assume that Ti = 20 [degrees C], T, =-15[degrees C], = 0.138-10-6 meters2 per second For parts ii. and iii., run your experiments with a tolerance ofe = 10-13 i. (10 pts) We want to determine how deep a water main should be buried so that it will only freeze after 60 days expos ure at this constant surface temperature Formulate the problem as a root finding problem f(x) = 0·What is f and what is Plot the function f on [0, T], where is chosen so that f() >0 values ao = 0[meters] and bo meters] xo = 0.01 [meters] ii. (10 pts) Compute an approximate depth using the Bisection Method with starting iii. (10 pts) Compute an approximate depth using Newton\'s Method with starting value What happens if you start with zo = z?

Solution

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