Need help with a complicated word problem...

[ProtoflareX]

One of the world's most adorable marine mammals is at risk of extinction. Your marine mammal research team has received a $1,000 grant to carry out an intervention to increase its population by this time next year. Your goal then is to maximize the population one year from now, hoping to publish a breakthrough paper and get more funding to do a bigger project to save the species.

Based on your research data, your team estimates that if M gallons of medicine and N gallons of nutrients are added to their marine habitat, then by next year this will save M^2 + MN + 12N lives. (The habitat is huge, so there is no need to worry about "overdose"). A gallon of medicine costs $25 and a gallon of nutrient costs $5.

a.) What is the combined cost of M gallons of medicine and N gallons of nutrients? (Give a single formula)

b.) To maximize, within your budget, the number of lives saved by next year, which combinations of medicine and nutrients would you consider putting into the habitat?

c.) Which combination would you use? How many marine mammals will you have saved by this time next year?

I would like somebody to tell me if my answers are correct.

A: 25M + 5N = 1,000
B: Not sure what the difference between this and C is, so I didn't answer it.
C: I would use 8.75 gallons of medicine and 156.25 gallons of nutrients. By this time next year, I will have saved 3,318.75 mammals.

 

[ksdhart]

Well, since you've only given your answer and shown none of your work, all I can do is tell you that your answer is not correct. Please reply back showing all of your work, so we can determine where you went wrong. Thanks.

 

[ProtoflareX]

Well, since you've only given your answer and shown none of your work, all I can do is tell you that your answer is not correct. Please reply back showing all of your work, so we can determine where you went wrong. Thanks.

After creating the equation:

. . .25M + 5N = 1,000

I attempted to solve for N by creating the equation:

. . .5N = 1,000 - 25M

I then divided both sides by 5 in order to simplify it and got:

. . .N = 200 - 5M

I then put the new N into the equation:

. . .M^2 + NM + 12N

. . .M^2 + (200 - 5M)M + 12(200 - 5M)

I then distributed M and 12 into their respective parenthesis and got:

. . .M^2 + 200M - 5M^2 + 2,400 - 60M

After combining like terms, I got:

. . .-4M^2 + 140M + 2,400

I then took the derivative of that equation and got:

. . .16m + 140

I then divided both sides by 16 to get M = 8.75. Afterwards, I put the new M into the equation for N and got 156.25.

That is how I got the answers I arrived at, but as you mentioned, I must have done something incorrectly.

 

[stapel]

Your goal then is to maximize the population one year from now....

...Your team estimates that if M gallons of medicine and N gallons of nutrients are added to their marine habitat, then by next year this will save M^2 + MN + 12N lives.... A gallon of medicine costs $25 and a gallon of nutrient costs $5.

a.) What is the combined cost of M gallons of medicine and N gallons of nutrients? (Give a single formula)

After creating the equation:

. . .25M + 5N = 1,000

This would be the "cost" equation, relating the total amount of funding ($1,000.00) and the per-gallon costs of the inputs ($25/gal M and $5/gal N).

b.) To maximize, within your budget, the number of lives saved by next year, which combinations of medicine and nutrients would you consider putting into the habitat?

I attempted to solve for N by creating the equation:

. . .5N = 1,000 - 25M

I then divided both sides by 5 in order to simplify it and got:

. . .N = 200 - 5M

I then put the new N into the equation:

. . .M^2 + NM + 12N

This is not actually an "equation", as there is no "equals" sign or "other side". Perhaps you meant to have done something along the lines of the following, specifying that you are plugging the results from the "cost" equation into the "saved lives" equation:

. . . . .f(M, N) = M² + NM + 12N

. . . . .f(M) = M² + (200 - 5M)M + 12(200 - 5M)

. . . . .f(M) = -4M² + 140M + 2,400

So your functional result looks good.

I then took the derivative of that equation and got:

. . .16m + 140

I will guess that you're using "m", contrary to standard mathematical practice, to mean "M". I would suggest that you review the Power Rule, as the derivative of x² is not 4x, and the derivative of -x is not +1.

 

[Ishuda]

One of the world's most adorable marine mammals is at risk of extinction. Your marine mammal research team has received a $1,000 grant to carry out an intervention to increase its population by this time next year. Your goal then is to maximize the population one year from now, hoping to publish a breakthrough paper and get more funding to do a bigger project to save the species.

Based on your research data, your team estimates that if M gallons of medicine and N gallons of nutrients are added to their marine habitat, then by next year this will save M^2 + MN + 12N lives. (The habitat is huge, so there is no need to worry about "overdose"). A gallon of medicine costs $25 and a gallon of nutrient costs $5.

a.) What is the combined cost of M gallons of medicine and N gallons of nutrients? (Give a single formula)

b.) To maximize, within your budget, the number of lives saved by next year, which combinations of medicine and nutrients would you consider putting into the habitat?

c.) Which combination would you use? How many marine mammals will you have saved by this time next year?

I would like somebody to tell me if my answers are correct.

A: 25M + 5N = 1,000
B: Not sure what the difference between this and C is, so I didn't answer it.
C: I would use 8.75 gallons of medicine and 156.25 gallons of nutrients. By this time next year, I will have saved 3,318.75 mammals.

A: Your answer is the proper answer for B as you indicated. However it is not the general equation for cost. The general equation is not limited to the $1000.
B: See A
C: It appears that you almost got there. The proper 'benefit function' is
B(M) = .-4M^2 + 140M + 2,400; 0 \(\displaystyle \le\) M \(\displaystyle \le\) 40
But, as stapel points out, you also did the derivative incorrectly.