Use Lagrange multipliers to find three positive numbers whose sum is 210 and whose product is maximum. (Enter your answers as a
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Consider the line y=5x-3. Two solutions (x, y) of this equation, are points on the line.
For example, let x=0, then y=-3. So (0, -3) is a point on the line.
If x=1, then y=5*1-3=5-3=2, so (1, 2) is a point on the line.
a)
Consider the transformations of these two points under <span>(x,y) --> (x, y-6):
</span>(0,-3) --> (0, -3-6)=(0, -9),
(1, 2) --> (1, 2-6)=(1, -4).
So (0, -3) and (1, 2) are translated to respectively (0, -9) and (1, -4).
The whole line is translated to the line containing (0, -9) and (1, -4).
The slope of the new line is
. The equation of the line is:
,
that is, y=5x-9.
b)
Consider the transformations of these two points under (x,y) --> (x+2, y):
(0,-3) --> (0+2, -3)=(2, -3),
(1, 2) --> (1+2, 2)=(3, 2).
The slope of the new line is
.
The equation of the line is
,
which can be written as
y=5x-13.
Answer:
a) y=5x-9
b) y=5x-13
For example, let x=0, then y=-3. So (0, -3) is a point on the line.
If x=1, then y=5*1-3=5-3=2, so (1, 2) is a point on the line.
a)
Consider the transformations of these two points under <span>(x,y) --> (x, y-6):
</span>(0,-3) --> (0, -3-6)=(0, -9),
(1, 2) --> (1, 2-6)=(1, -4).
So (0, -3) and (1, 2) are translated to respectively (0, -9) and (1, -4).
The whole line is translated to the line containing (0, -9) and (1, -4).
The slope of the new line is
that is, y=5x-9.
b)
Consider the transformations of these two points under (x,y) --> (x+2, y):
(0,-3) --> (0+2, -3)=(2, -3),
(1, 2) --> (1+2, 2)=(3, 2).
The slope of the new line is
The equation of the line is
which can be written as
y=5x-13.
Answer:
a) y=5x-9
b) y=5x-13