Question

某糧油超市平時每天都將一定數(shù)量的某些品種的糧食進行包裝以便出售，已知每天包裝大黃米的質量是包裝江米質量的倍，且每天包裝大黃米和江米的質量之和為45千克．

(1)求平均每天包裝大黃米和江米的質量各是多少千克?

(2)為迎接今年6月20日的“端午節(jié)”，該超市決定在節(jié)日前20天增加每天包裝大黃米和江米的質量，二者的包裝質量與天數(shù)的變化情況如圖所示，節(jié)日后又恢復到原來每天的包裝質量．分別求出在這20天內每天包裝大黃米和江米的質量隨天數(shù)變化的函數(shù)關系式，并寫出自變量的取值范圍．

(3)假設該超市每天都會將當天包裝后的大黃米和江米全部出售，已知大黃米成本價為每千克7.9元，江米成本價為每千克9.5元，二者包裝費用平均每千克均為0.5元，大黃米售價為每千克10元，江米售價為每千克12元，那么在這20天中有哪幾天銷售大黃米和江米的利潤之和大于120元?     [總利潤=售價額－成本－包裝費用]

 

Answer 1

解：(1)解法一：

設平均每天包裝大黃米和江米的質量分別為a千克和b千克， ··············································· 1分

則 ，解得．····························································································· 3分

答：平均每天包裝大黃米和江米的質量分別為25千克和20千克． ········································ 4分

解法二：

設平均每天包裝大黃米的質量為a千克，那么包裝江米的質量（45a）千克， ····················· 1分

則  ， 解得 a=25，所以4525=20． ····························································· 3分

答：平均每天包裝大黃米和江米的質量分別為25千克和20千克． ········································ 4分

（2）觀察圖象，可設平均每天包裝大黃米的質量與天數(shù)的關系式為，平均每天包裝江米的質量與天數(shù)的關系式為．

①當時，

由題意的圖象過點，．

則可列方程組為，解得，

∴．······················································································································· 5分

由題意的圖象過點，．

則可列方程組為，解得,

∴．···················································································································· 6分

②當時，

由題意的圖象過點，．

則可列方程組為，解得，

∴．··················································································································· 7分

由題意的圖象過點，．

則可列方程組為，解得，

∴．················································································································ 8分

∴  ， ． ···························· 9分  (3)設第天銷售的總利潤為元，

①當時，

．

由題意，∴，

∴x的取值范圍為，

由題意知．································································································ 10分

②當時，

．

由題意，∴，

∴x的取值范圍為．

由題意知．················································································································· 11分

答：由①、②可知在第11，12，13，14，15，16天中銷售大黃米和江米的總利潤大于120元．     12分