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Lostsunrise [7]
3 years ago
9

P is the point on the line 2x+y-10=0 such that the length of OP, the line segment from the origin O to P, is a minimum. Find the

coordinates of P and this minimum length
Mathematics
1 answer:
nirvana33 [79]3 years ago
8 0
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).
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What is the next term of the following geometric sequence:
LenKa [72]

Answer:

D

Step-by-step explanation:

Each term has been multiplied by 2/5.

-1/2 * 2/5 = -2/10=-1/5

-1/5 * 2/5 = -2/25

-2/25 * 2/5 = -4/125

-4/125 * 2/5 = -8/625

4 0
3 years ago
The area of a rectangle is
ANEK [815]
Area = length x width
length = 3 x width - 5

i.e. 50 = w(3w - 5)
3w^2 - 5w - 50 = 0
(3w + 10)(w - 5) = 0
w = -10/3 or w = 5
but width cannot be negative, hence
width = 5 yd
Length = 3 x 5 - 5 = 15 - 5 = 10 yd
8 0
3 years ago
Given that the measure of angle GHJ = 154 degrees find the angle of HIJ
vichka [17]

Answer:

the correct answer is 77°

Step-by-step explanation:

Angle GHJ and IHJ form supplementary angles which add up to 180°

therefore angle IHJ=180-154

angle IHJ=26°

Triangle IHJ is an isosceles triangle whose base angles HIJand IJH are equal. Lets assign them the same value, x° each

since the sum of all interior angles of a triangle add up to 180°, then x°+x°+26°=180°

2x+26=180

2x=154°

x=77°

Therefore both HIJ and IJH measure 77°

6 0
3 years ago
There are 15 coins, which are nickels and dimes only, in Ediths purse. The total value of the coins in her purse is $1.15. If n=
Goshia [24]

Answer:

n=7 d=8

Step-by-step explanation:

n+d=15 d=15-n

0.05n+0.1(15-n)=1.15

0.05n-0.1n+1.5=1.15

-0.05n=-0.35

n=7 d=8

5 0
2 years ago
Solve. -X - 11- 2x+4 help pls​
cestrela7 [59]

Answer:

i got -2x - x - 7

Step-by-step explanation:

if im wrong sorry

6 0
2 years ago
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