word problem with system of inequalities

Question

rick is buying souvenirs on vacation. he wants to spend $70 at most, buy only has 60 cubic inches of space availalbe in his luggage. if bracelets cost $7 and take up 3 in (3) of space and t-shirts are $5 but take up 15 in(3) of space, write and graph a system of four inequalities that model rick's possible purchases.  Let x = number of bracelets and y = number of t-shirts.  use a sacle of 2 on both axes.

 

please help solve and how do I graph this problem?  thanks1

Answer 1

This is actually easily expressed in equations first so you know you covered everything and then graphed.  They call the inequalities constraints because they *constrain* the answer. It's important to note what you're expressing to keep track.

 

You would graph it with Bracelets on the X axis and T-shirts on the Y axis. Then express Y in terms of X

 

Total cost:

7x + 5y <=70

Graphing it:  y<=14-7x/5

 

Total volume:

3x + 15y <=60

Graphing it: y<= 4 - x/5

 

Since the intersection of both lines is the max cost and volume, that will be the optimum answer. So set the two equations together:

14 - 7x/5 = 4 - x/5

10            = 6x/5

x            = 50/6

and y is 14- 7*50/6/5 = 14 - 70/6 = 14 - 11 4/6 = 2 2/6

 

Since we can't buy fractions of a product, you want 8 bracelets and 2 T-shirts. (Rounding up would cost $78 which would break our money constraint.)

 

As for graphing it, Find the x and y intercepts of both lines:

Cost: (0,14) (10,0)

Volume: (0,4) (20,0)

and graph everything below the two lines marking off the graph.

Answer 2

7x+5y≤70        This is the $ inequality

 

3x+15y<60       This is the volume inequality

 

x>0, y>0, both integers    These inequalities reflect the nature of the items. i.e. they only exist in whole positive numbers.

 

The solution set is bounded by the lines x=0, y=4-x/5, y=14-7x/5 and y=0 there are 21 possible integer solutions

Sorry there is no graphing function.

 

Regards

Jim S.

Fairport NY