SOLUTION: 1.) Suppose that x and y are real numbers with x + 3y =20. a.)Find the smallest value of x^2 + y^2. b)Find the greatest value of xy. c.) Find the smallest value of (x-y)^2. d.)

Question 570989: 1.) Suppose that x and y are real numbers with x + 3y =20.
a.)Find the smallest value of x^2 + y^2.
b)Find the greatest value of xy.
c.) Find the smallest value of (x-y)^2.
d.)The greatest value of y^2 - x^2.

2,) Sally worked 50 hrs last week and made $495 for the week. For every hour worked over 40 hours, her job pays time and a half. What is sally's hourly pay rate.
Answer by htmentor(1343) (Show Source): You can put this solution on YOUR website!
1.) Suppose that x and y are real numbers with x + 3y =20.
a.)Find the smallest value of x^2 + y^2.
b)Find the greatest value of xy.
c.) Find the smallest value of (x-y)^2.
d.)The greatest value of y^2 - x^2.

2,) Sally worked 50 hrs last week and made $495 for the week. For every hour worked over 40 hours, her job pays time and a half. What is sally's hourly pay rate.
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a) Express y in terms of x:
y = (1/3)(20-x)
So x^2 + y^2 = x^2 + [(1/3)(20-x)]^2 = x^2 + (1/9)(400-40x+x^2)
Collecting terms gives (10/9)x^2 -(40/9)x + 400/9
This expression is minimized when the derivative equals 0:
(20/9)x - 40/9 = 0
This gives x = 2
So y = (1/3)(20-2) = 6
So the smallest value is 2^2 + 6^2 = 40
b) xy = (x/3)*(20-x) = (20/3)x - (1/3)x^2
The expression will be a maximum when the derivative=0:
0 = 20/3 - (2/3)x
This gives x = 10
Therefore y = 10/3
The maximum value is 10*10/3 = 100/3
...I'll leave the other two for you to do...
2) Let x = Sally's hourly pay
Her total pay will be the hourly pay times 40 hrs plus 10 hrs working at time and a half:
495 = 40x + 10*3x/2 = 40x + 15x = 55x
This gives x = 9
Her hourly salary is $9
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