on a certain island, at any given time, there are R hundred rats and S hundred s

Question

on a certain island, at any given time, there are R hundred rats and S hundred snakes. their populations are related by the equation (R-14)^2 + 16(S-11)^2 = 68. what is the maximum combined number of snakes and rats that could ever be on the island? on a certain island, at any given time, there are R hundred rats and S hundred snakes. their populations are related by the equation (R-14)^2 + 16(S-11)^2 = 68. what is the maximum combined number of snakes and rats that could ever be on the island? on a certain island, at any given time, there are R hundred rats and S hundred snakes. their populations are related by the equation (R-14)^2 + 16(S-11)^2 = 68. what is the maximum combined number of snakes and rats that could ever be on the island?

Explanation / Answer

Maximize R+S = f

condition : (R-14)^2+16(S-11)^2-68 = 0 = g

by lagrange multipliers ,

df/dR = lambda * dg/dR => 1 =lambda * 2*(R-14) => R = 14+1/(2Lambda)

df/dS = lambda * dg/dS => 1 = lambda * 16 * 2* (S-11) => S = 11+ 1/(32 Lambda)

put R and S in the g

we have

(14+1/(2Lambda) -14)^2 + 16 (11+ 1/(32 Lambda) -11)^2 = 68

1/4Lambda^2 + 16 (1/32^2Lambda^2) = 68

1/4Lambda^2 + (1/64Lambda^2) = 68

lambda = +1/16 or -1/16

so for lambda = 1/16 , R = 14+1/(2*1/16) = 14+8 = 22

S = 11+1/32(1/16) = 11.5 . so R+S = 22+11.5 = 33.5

for lambda = -1/16 , R = 14+1/(2*-1/16) = 14-8 = 6

S = 11+1/32(-1/16) = 10.5 . so R+S = 6+10.5 = 16.5

so Mamimum combined = 33.5 hundered --------ANSWER

Related Questions

Navigate

• Browse (All)
• Browse O
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.