kbac98
Answered

Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.

Find two positive numbers whose product is 192 and their sum is a minimum.

Sagot :

ha101958

Answer:

Since we want a positive number, x = 24.  y = 192/x = 8

Step-by-step explanation:

Let x and y be the two positive numbers.

-  Their product is 192: x*y = 192

-  the sum of the first plus 3 times the second is a minimum:  x + 3y

From the first equation, y = 192/x.  Substitute that into the second equation:

x + 3y = x + 3(192/x) = x + 576/x

Now take the first derivative, set it to zero, and solve for x:

d(x + 576/x)/dx = 1 - 576/x2

0 = 1 - 576/x2

0 = x2 - 576                         [Multiplied both sides by x2]

576 = x2

√576 = √x2

±24 = x

Since we want a positive number, x = 24.  y = 192/x = 8

As a check, the factors of 192 are 192*1, 96*2, 48*4, 24*8, 16*12

192 + 3*1 = 195

96 + 3*2 = 102

48 + 3*4 = 60

24 + 3*8 = 48

16 + 3*12 = 52

24 + 3*8 is indeed the minimum

Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.