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:
8 ft; 6 m; 28pi ft; 10pi mm
Step-by-step explanation:
1. Radius = diameter/2
16/2=8
Therefore, 8 ft
2. Diameter= radius * 2
3*2=6
Therefore, 6 m
3. Circumference = 2*pi*r
2*pi*14=28pi
Therefore, 28pi ft
4. Circumference = pi*d
Therefore, 10pi mm
I hope this helped and have a good rest of your day!
Answer:
x=12
Step-by-step explanation:
Simplify both sides by adding like terms
5/12x=x-7
subtract x from both sides
-7/12x=-7
multiply both sides by 12/-7
x=12
Answer:
91 degrees
Step-by-step explanation:
Answer:
1. 
2 a) 
b) 0
3. x = 10
Step-by-step explanation:
1.
2. a)
![2 [(\frac{-1}{4}) -12 (-1)^2] - 5[24 +2 (-12)]^2\\2[(\frac{-1}{4}) -12] - 5[24 -24]^2\\2 * [(\frac{-1}{4}) -12] -0\\\frac{-49}{2}](https://tex.z-dn.net/?f=2%20%5B%28%5Cfrac%7B-1%7D%7B4%7D%29%20-12%20%28-1%29%5E2%5D%20-%205%5B24%20%2B2%20%28-12%29%5D%5E2%5C%5C2%5B%28%5Cfrac%7B-1%7D%7B4%7D%29%20-12%5D%20-%205%5B24%20-24%5D%5E2%5C%5C2%20%2A%20%5B%28%5Cfrac%7B-1%7D%7B4%7D%29%20-12%5D%20%20-0%5C%5C%5Cfrac%7B-49%7D%7B2%7D)
b)
3.
