The minimum distance is the perpendicular distance. So establish the distance from the origin to the line using the distance formula.
The distance here is: <span><span>d2</span>=(x−0<span>)^2</span>+(y−0<span>)^2
</span> =<span>x^2</span>+<span>y^2
</span></span>
To minimize this function d^2 subject to the constraint, <span>2x+y−10=0
</span>If we substitute, the y-values the distance function can take will be related to the x-values by the line:<span>y=10−2x
</span>You can substitute this in for y in the distance function and take the derivative:
<span>d=sqrt [<span><span><span>x2</span>+(10−2x<span>)^2]
</span></span></span></span>
d′=1/2 (5x2−40x+100)^(−1/2) (10x−40)<span>
</span>Setting the derivative to zero to find optimal x,
<span><span>d′</span>=0→10x−40=0→x=4
</span>
This will be the x-value on the line such that the distance between the origin and line will be EITHER a maximum or minimum (technically, it should be checked afterward).
For x = 4, the corresponding y-value is found from the equation of the line (since we need the corresponding y-value on the line for this x-value).
Then y = 10 - 2(4) = 2.
So the point, P, is (4,2).
Answer:
The volume of the box is 3/4 cubic meters.
Step-by-step explanation:
We know that volume = length * width * height. We have length, width, and height already, so all we have to do is multiply.
5/6 * 1 1/8 * 4/5. First we convert 1 1/8 from a mixed fraction, to give us 9/8, as 8/8 = 1.
5/6 * 9/8 * 4/5. 5 * 9 = 45, and 6 * 8 = 48. 45/48 reduces to 15/16.
15/16 * 4/5. 15 * 4 = 60, and 16 * 5 = 80. 60/80 reduces to 6/8, which reduces to 3/4.
Answer:

Step-by-step explanation:
<u>Given</u>
<u />
<u>Swap x and y and solve for y</u>
<u />
<u>Final answer</u>
<u />
Keep in mind that we need to swap the variables because a function and its inverse are symmetrical about the line y=x. See the graph below.
Answer:
44 or 45
Step-by-step explanation:
400 divided by nine
Answer:
Step-by-step explanation:
Let “x” be the weight of the bottle and “y” is weight of honey.
We know the total weight of bottle full of honey is 1000 g.
So Equation 1 is x + y = 1000 g
And with half of the honey, we know the total weight is 600g.
Equation 2: x + (y/2) = 600 g
y/2=600 - x
y= 2 (600 - x)
y= 1200 - 2x
We can replace “y” with 1200 - 2x in Equation 1:
x + 1200 - 2x = 1000
x = 200 g
Weight of the bottle is 200 g.