Among all pairs of numbers whose sum is 16, find a pair whose pro...

Question

Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?

Accepted Answer

Two numbers add up to 118 and will multiply to give our product that is a maximum solve for this product. So it's gonna give us a maximum which means this is a part of a parabola. Two numbers are adding up to 1 18. Now let's say we have one number X. So some number now our other number, you know our numbers add up to 1 18. The other number will be 1 18 minus X. So some other number now we can set up our multiplication. So we have X times 1 18 months X. For our two numbers we simplify this even further. We have 1 18 X minus X squared. And rewriting we get negative X squared in the front plus 1 X. Now we have a problem. And the form A X squared plus Bx plus C. A. is a number and never x squared which is negative one B. Is number four over X which is 1 18 and C is just zero since there's no C in this equation. So let's go ahead and find the maximum. Find the maximum. We're gonna find our vertex. So go ahead and make this. The equation for vertex is X equals negative B. Number two A. In sulfur X. RB is 1 18. So we have negative 1 18 over to Times are K which is negative one negative 1, 18 over negative two. Which is equal to 59. So are x equals 59. Now to find our maximum we're gonna take this 59 and plug it into our equation. So I go back over here, We have negative, we have X. now 39 square plus 1 18 times 59. We're gonna play this in our calculator and simplify. We would get 34 81. Maybe the answer to our problem is the answer. A Okay. I hope to help you solve the problem. Thank you for watching. Goodbye.

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Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?