1. The numbers a, b, and c sum to 90. b is twice the value of a. Write out the t
ID: 2885947 • Letter: 1
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 21500Explanation / 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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