Question

A bottle manufacturing company pays $4,312 a month in utilities, and the average production cost per bottle is $0.65. The revenue that the company earns after selling x bottles is given by the function B(x) = 4.5x.
The company owners are trying to cut costs in order to make a profit selling fewer bottles. Select the cost description and revenue function that would produce a profit after selling the fewest number of bottles.

The comp;any can cut the cost of production per bottle to $0.60.
The revenue function would then be defines as B(x) = 5.5x

The comp;any can cut the cost of production per bottle to $0.55.
The revenue function would then be defines as B(x) = 4.75x

The comp;any can cut the cost of production per bottle to $0.50.
The revenue function would then be defines as B(x) = 5x

Answer 1

Let
x-----------> number of bottles produced in a month

we know that

revenue
B(x) = 4.5x.

expenses
E(x)=0.65x+$4312

equals revenue and expenses
4.5x=0.65x+4312-------> 4.5x-0.65x=4312----------> x=4312/3.85

x=1120 bottles  (minimum amount of bottles to start getting monthly benefits )
case a)The company can cut the cost of production per bottle to $0.60.
The revenue function would then be defines as B(x) = 5.5x

Revenue
B(x) = 5.5x.

expenses
E(x)=0.60x+$4312

equals revenue and expenses
5.5x=0.60x+4312-------> 5.5x-0.60x=4312----------> x=4312/4.9

x=880 bottles  
case b)The company can cut the cost of production per bottle to $0.55.
The revenue function would then be defines as B(x) = 4.75x

Revenue
B(x) = 4.75x.

expenses
E(x)=0.55x+$4312

equals revenue and expenses
4.75x=0.55x+4312-------> 4.75x-0.55x=4312----------> x=4312/4.2

x=1026.67 bottles  
case c) The company can cut the cost of production per bottle to $0.50.The revenue function would then be defines as B(x) = 5x

Revenue
B(x) = 5x.

expenses
E(x)=0.50x+$4312

equals revenue and expenses
5x=0.50x+4312-------> 5x-0.50x=4312----------> x=4312/4.5

x=958.22 bottles  

the fewest number of bottles is the case a) 880 bottles

the answer is
the fewest number of bottles is 880 bottles when
The company can cut the cost of production per bottle to $0.60.
The revenue function would then be defines as B(x) = 5.5x