Daily Maximum Production

To produce $x$ units of pennies, we require $3x$ kg of zinc and $x$ kg of copper.

To produce $y$ units of Doorknobs, we require $2y$ kg of zinc and $2y$ kg of copper.

The total amount of zinc equals $160$ kg and $120$ kg of copper. Thus, the total amount of stuff equals

$3x + 2y + x + 2y = 160 + 120$

or

$4x + 4y = 280$

$4(x + y) = 280$ or $x + y = 70$ .

Thus, answer is $\boxed{70}$

Alternatively, we could use the simultaneous equations $3x + 2y = 160$ and $x + 2y = 120$ to get $x$ and $y$ and then get $x + y$ .

Represent the condtions / constraints as mathematical equations

Draw a linear graph for 3x+2y-160<=0 and x+2y-120<=0, x>=0, y>=0; where x is the production of Pennies and y is the production of Doorknobs.

Shade areas for which 3x+2y-160<=0 and x+2y-120<=0, x>=0, y>=0.

We get a quadrilateral bounded by corners with coordinates O(0,0); A(53.33,0); B(20,50); C(0,60)

The solution to the problem is bound to be on one of the corners (A, B, C, O). (this can be cross checked)

Find (x+y) of each corner: O(0); A(53.33); B(70); C(60)

Applying the condition max(x+y); we get B(x=20, y=50) represents the max production per day.

suppose no.of pennies=x door knobs=y zinc used 3x+2y=160 copper used x+2y=120 solve the equations x=20 pennies y=50 doorknobs copper use by pennies= 20 1= 20kg doorknobs=50 2=100 total= 120kgs zinc used pennies=20 3 =60kgs doorknobs=50 2=100 kgs total 160 kgs

to produce penies and doorknob zink amounts 160 3x +2y=160 an similarly x+2y=120 then the value of x is 20 and value y is 50 total x+y=70

total material we have each day 280 kg. each object uses 4 kg material . either we can use all the material for making Pennies or for Doorknobs. maximum no. of pennies made = 280 divided by 4 = 70. in this situation no. of doorknobs made =0 and same is with doorknobs

Let p be the units of pennies produced, and d the units of doorknobs produced.

Two equations:

$3p + 2d \leq 160$ --- (1)

$p + 2d \leq 120$ --- (2)

From (1)

$2d \leq 160-3p$

From (2)

$2d \leq 120-p$

By elimination $2p \leq 40 p \leq 20$

Sub $p = 20$ to get $d = 50$

The largest value of $x+y$ would be $20+50 = 70$

x is pennies; y is doorknobs

3x + 2y = 160 ; x + 2y = 120 (eliminated 2y) 2x = 40 x 20

20 + 2y = 120 2y = 120 - 20 2y = 100 y = 50

x + y = 20 + 50 = 70

for both the items .. 4 kgs only required total supply is 160+120+280 so 280/4= 70... no chance of getting more than 70

Let $x$ be the amount of copper needed to produce a penny and $\frac {120-x}{2}$ be the corresponding amount of copper to produce a doorknob . Subsequently, we can denote the amount of zinc needed to produce a penny as $3x$ , so the amount of zinc for doorknob would be $\frac {160-3x}{2}$ . Since the amount of copper and zinc needed in doorknob production is the same, then to solve for $x$ we must first have:

$\frac {120-x}{2} = \frac {160-3x}{2}$

Multiply both sides by 2, to get:

$120 - x = 160 - 3x$

$3x-x=160-120$

$2x=40$

$x = 20$

By plugging in the value of x to either of the two equations for doorknob production, we would get 50. Hence: $x + y = 20 + 50 = \boxed {70}$

I solved it using linear programming. I don't Know how to explain since I would need to sketch a quick graph. Bus my constraints were 3x + 2y <= 160 AND x + 2y <=120 AND x + y >0 The largest value for the objective line was the intersection of the two lines so I solved them linearly and gave me 50 and 20 50 + 20 = 70 If anyone solved it like that let me know :)

to produce x pennies we need 3x kg zinc and x kg copper, and to produce y doorknobs we need 2y kg zinc and 2y kg copper, thus we need 3x + 2y kg zinc and x + 2y kg copper. now apply the conditions, 3x + 2y at most be equal to 160 and x + 2y at most be equal to 120, by solving these inequalities we find that x can not exceed 20 and y can not exceed 50 thus we can conclude that the max value of x + y can be 70

1. Add amount of zinc and copper needed for both.

3 + 2 = 5 kg of zinc 1 + 2 = 3 kg of copper

2. Divide by daily supply of each material

160/5 = 32 120/3 = 40

3. Add results

40 + 32 = 72

Both doorknobs and pennies require 4 kg of metal. There are 280 kg of metal supplied per day. Dividing one by another, you get 70 - this is the smallest option - so you don't have to worry about not being able to make that exact amount of X+Y.

3x+2y=160 Total zinc x+2y120. Total copper Subtract so: 2x=40 =>. x=20

=> y= 50

So x+y=70