Question

Use the scenario below to answer all of the questions. Show all of your work on handwritten pages. Take pictures of your page. Upload the pictures to this assessment on Engage. A farmer in upstate New York has a 100-acre farm to plant red beans and romaine lettuce. Every acre planted with beans requires 50 gallons of water per acre and consumes 20 pounds of fertilizer. Every acre planted with lettuce requires 75 gallons of water and 15 pounds of fertilizer. The farmer estimates that is will take 2 hours of labor to harvest each acre planted with beans and 2.5 hours to harvest each acre planted with lettuce. She believes that beans will sell for about $3 a bushel and lettuce at $1 per 3 head bundle. Every acre planted with beans is expected to yield 90 bushels. Every acre planted with lettuce is expected to yield 300 bundles. The farmer can pump 6,000 gallons per day for irrigation. She can buy fertilizer at $10 per 50-pound bag. She can hire laborers at $5 per hour. The farmer can sell al of the beans and lettuce she produces. Let R be the number of acres of red beans planted and L the number of acres of lettuce planted. Express the farmer's constraints in terms of these decision variables. What are the units of the right-hand side constraint assumptions? 1. What is the profit per acre of beans and lettuce? Express the objective function in terms of R and L. Is this a minimization or maximization problem? 2. If the farmer produces 50 bushels of beans and 80 bundles of lettuce is the solution feasible, or not, and why? 3. Which constraints are binding if the optimal solution is 60 acres of beans and 40 acres of lettuce and why? What does an allowable increase of 1500 for the water constraint mean? 4. The farmer wants to plant black beans (B) as well on the same plot. Black beans require 45 gallons of water, 25 pounds of fertilizer per acre, and the same labor as red beans. The black beans are expected to sell for $3.25 per bushel and yield 100 bushels per acre. If the shadow price for water is 1.14 and for acreage is 199, should the farmer plant this crop? 5.

Answer 1

nswer According -to the iven injo he Srenaio ormation Set belou to anerthe uestions A former n pstade Naw york as a o0 acrefarmto plant red betnard remain ttuce etbh number acres f red beans planted ard L the number acreS 익 lettuce planted Cupress farmer's conlains in termso these ision riabes. ket noof acres of red bean plant bet no 익 acres let tuce planned e L In hors 509 allows ot Nata pe r acre . e per acre Tn beanS pounds ertier C Ccnstant s 6oo aos per day Gcoog alow The Pro:t per ocre a beansadlelue E-press +he objective 1s this e mimize Pro lem -function in terms 약 Ran tL

어익 cash toa rat bann, festilizer Cast abous c as t Lobous euired peaue ahour ard cast lobous houn 5sec cost OL we plan | っ Total cost -HR + 15.5터ㅡㅡㅡㅡㅡ Here | Since Poof -spR +8H 5L mau,mization Problem go bundles s the Soluian HeasiHe luttuce oduce of 50 burdle Bann So burdes lutture Here P required i s =(50) (50t (so)H5)-8500 allens. .*.zt is ot poss.le on acenh ands value PS 6000、

4 The constraints ng4 the Optirral Solution 60 acre s af . bens a 40 acres 퍄 lettuce il 60 oc'eç 04 benns Ao acies ece Water reau r ement (60) (50) + 140 〕(+5) 6000 gallons tHere the water is h constraint cindi ing and pretuction caneaied at Hence proved