SOLUTION: Mr. Jones plans to invest up to $22,000 in Wachovia and CCB banks. He will invest at least $2000, but no more that $14,000, in Wachovia. He will invest no more than $15,000 in CCB.

Question 246839: Mr. Jones plans to invest up to $22,000 in Wachovia and CCB banks. He will invest at least $2000, but no more that $14,000, in Wachovia. He will invest no more than $15,000 in CCB. Wachovia pays 6% simple interest and CCB pays 6 � % simple interest. How much should he invest in each to maximize income (interest)? What is the maximum interest?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
since ccb give the best interest, it makes sense to invest as much in ccb as possible.

I would start with 15,000 invested in ccb.

That leaves 22,000 minus 15,000 equals 7,000 to invest in wachovia.

interest per year from ccb would be 15,000 * .065 = 975

interest per year from wachovia would be 7,000 * .06 = 420

total interest per year would be 975 + 420 = 1395

any other combination would give him less interest so this has to be the max.

this looks something like a linear engineering problem but I'm not quite sure how to solve it as such.

You should be able to do it graphically.

your maximization equation is:

.06*x + .065*y = p where x is the amount invested in wachovia and y is the amount invested in ccb and p is the profit that you want to maximize.

your constraint equations are:

x + y <= 22000 where x is the amount invested in wachovia and y is the amount invested in ccb.
x >= 2000
x <= 14000
y <= 15000

solve for y in x + y <= 22000 and you ge3t:

y <= 22000 - x

you can graph the following equations:

y <= 22000 -x
y <= 15000
x >= 2000
x <= 14000

your graph will look like the following:

The scale is in thousands, i.e. 1 = 1000, 10 = 10,000, etc.

You would need to draw a vertical line at x = 2 and a vertical line at x = 14.

the x-axis represents the amount invested in wachovia.

the y-axis represents the amount invested in ccb.

the domain is x = 2000 to 14000 which is the minimum and maximum value to invest in wachovia.

your range is y <= 15000 which is the maximum amount to invest in ccb.

your slanted line is the amount invested in ccb which is equal to the equation of y = 22000 - x.

you find the maximum profit at one of the intersections of the lines that form the permissible range.

those intersections would be at:

(x,y) = (7,15)
(x,y) = (14,8)

you could theoretically invest less than 22000 but you would never be able to maximize your profit that way because a portion of your money would be earning zero percent, so an implicit assumption is that all of the money needs to be invested.

this means that investing 15000 in ccb and nothing in wachovia is not an option so the portion of the graph where x is less than 7000 is not considered.

once you get past 7000 invested in wachovia, then the balance has to be invested in ccb to make the total of 22000 invested.

your intersection point are used to solve the equation of .06*x + .065*y = p

at (x,y) = (7,15) your profit is .06*7000 + .065*15000 = 1395

at (x,y) = (14,8) your profit is .06*14000 + .065*8000 = 1360

the winner looks to be (x,y) = .06*7000 + .065*15000 = 1395

the overall interest rate is 1395 / 22000 = .063409091 * 1005 = 6.34% rounded to the nearest hundredth of a percent.

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