1. The numbers a, b, and c sum to 90. b is twice the value of a. Write out the t

Question

1. The numbers a, b, and c sum to 90. b is twice the value of a. Write out the two equations that correspond to these two sentences. utoring We want to maximize the product of these three P abc add slider a 2. Solve your equations above for a and then replace b and c in the product. Rename the function P(a). 3. What is the domain of the function P(a)? (Make sure that a, b, and c are not negative.) × 4. Find the maximum possible product of the three numbers. Be sure to use calculus methods and include a test Use the graph of P(a) to test your answer 21500

Explanation / Answer

Solution1)

Three numbers are a, b, and c

a + b + c = 90 ..................... (1)

b = 2 * a .......................(2)

P = abc ......................(3)

put b = 2a into equation (1)

a + 2a + c = 90

3a + c = 90

c = 90 - 3a ................... (4)

Hence we have b = 2a, c = 90 - 3a

put these values into equation (3)

P = a * 2a * (90-3a)

= 180a2 - 6a3

To maximize P

dP/da = 0

dP/da = 360a - 18a2 = 0

18a(20 - a) = 0

i.e. a = 0 and a = 20

dP2/da = 360 - 36a

Double derivative at a = 0 is positive i.e P has a minima at a=0

Double derivative at a = 20 is negatice i.e. P has a maxima at a= 20

a = 20, b = 40 and c = (90 - 3*20) = 30

Maximum P = 20*40*30

= 24000

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