Problem 7. Let R be a commutative ring. Prove that there is no isomorphism of R-modules between R² and R³. Note: your proof must use that R is commutative since the statement is false without it (see 10.3 Problem 27).

1-18 The three economic criteria for choosing the best alternative are maximize the difference between output and input, minimize input, and maximize output. For each of the following situations, what is the correct economic criterion? (a) A manufacturer can sell up to two full shifts of production at a fixed price. As production is increased, unit costs increase as a result of overtime pay and so forth. The manufacturer's criterion should be _________. (6) An architectural and engineering firm has been awarded the contract to design a wharf with fixed performance specifications for a petroleum company. The engineering firm's criterion for its client should be ____________. (c) An off-campus bookstore is choosing its target used'new split for next year. Its criterion should be _________. (d) At an auction of antiques, a bidder for a particular porcelain statue would he trying to _________.

3. Estimate the modulus of subgrade reaction for the following subgrade solls! 3. Clayey sandy gravel with a dry density of 128 Ib/ft3. 4. Plastic clay with a CBR of 6.

2. (10 points) Compute the negative moment capacity for the beam shown below. Use f'c = 3500 psi and fy = 40 ksi. (Note: tension is in the top flange and that is why the negative moment capacity is to be determined).

4. Design a round spiral column to support an axial dead load of 300 k and an axial live load of 400 k. Begin using 1.5 percent longitudinal steel, a concrete strength of 5,000 psi and Grade 60 steel. (10 points)

7.3.For the beam in Problem 7.1: (a) Calculate the increment of deflection resulting from the first application of the short-term live load. (b) Find the creep portion of the sustained load deflection plus the immediate deflection due to live load. (c) Compare your results with the limitations imposed by the ACI Code, as summarized in Table 7.2. Assume that the beam is a part of a floor system and supports cinder block partitions susceptible to cracking if deflections are excessive.

1. A reinforced concrete beam has a width of 12 in and an effective depth (from the extreme compression fiber to the centroid of the tension steel) of 22 in. The concrete compressive strength is 4000 psi and steel has a yield strength of 60,000 psi. Determine: a) The depth of neutral axis corresponding to balanced condition, the area of steel corresponding to balanced condition, and the balanced condition reinforcement ratio b) Area of tension steel corresponding to: (i) a design tensile steel strain of 0.004, and (ii)tension controlled failure with a steel strain of 0.005

[10] 1. A 25 ft. reinforced concrete beam is being designed to support a uniformly distributed load. This load has an average value of 1.4 kip/ft with a standard deviation of 0.3 kip/ft. The average flexural resistance of the beam is 230 ft-kip with a standard deviation of 15 ft-kip, and the average shear resistance of the beam is 45 kip with a standard deviation of 7 kip. 1) [1] Find the global safety factor in both flexure and shear (note: safety factor resistance/load effect) 2) [2] Find the safety index in both flexure and shear 3) [3] If the average flexural resistance is increased to 270 ft-kip with the same standard deviation of 15 ft-kip, find the new safety index in flexure 4) [3] If the standard deviation of the shear resistance is decreased to 5 kip with the same shear resistance, find the new safety index in shear 5) [1] Comment on your results

Consider a drilled, rough concrete pile with diameter B = 0.8m and length D = 7m embedded in a site underlain by a 5m thick layer of sand with fiction angle = 38 degrees and Ko = 0.5 that lies over a 6m thick layer of clay with fiction angle 28 degrees, Ko = 0.38, and Su 80 kPa.5. Draw the profile with the pile. Remember that placing the water table at the ground surface is th emost conservative option for foundation design and analysis b. Determine the short term side friction resistance for the segment of the pile that rests in the sand laver. Determine the long term side friction resistance for the segment of the pile that rests in the claylayer.C.

3. (20 points) A concrete floor system consists of parallel T beams spaced 10 feet on center (see sketch below)and span 32 feet between center points of columns. L-beams are used at the edges of the slabs. The 6 inch thick slab was poured monolithically with the beams that have a web width of 14 inches and a total depth (measured from the top of the slab to the bottom of the web) of 28 inches. Assume the effective depth "d" to be H - 3inches. In addition to it's own weight, each beam must carry an unfactored superimposed dead load of 50pounds per square foot and a live load of 225 pounds per square foot. The concrete compressive strength is 3ksi and the yield strength of steel is 60 ksi. Calculate As for the T-beams and the edge L-beams, select bars and check spacing, minimum steel requirements, equilibrium and if tension control.

1-33 A student accepts a full-time job in November, but a better job comes before graduation in May. What are the ethical dimensions of the student's decision? Would you take the better job? Why or why not?