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2 years ago

Find the coordinates of the point (x,y,z) on the planez=4x+3y+1 which is closest to the origin.

1 answer:

8 0

Given plane Π : f(x,y,z) = 4x+3y-z = -1
Need to find point P on Π that is closest to the origin O=(0,0,0).

Solution:
First step: check if O is on the plane Π : f(0,0,0)=0 ≠ -1 => O is not on Π
Next:
We know that the required point must lie on the normal vector <4,3,-1> passing through the origin, i.e.
P=(0,0,0)+k<4,3,-1> = (4k,3k,-k)
For P to lie on plane Π , it must satisfy
4(4k)+3(3k)-(-k)=-1
Solving for k
k=-1/26
=>
Point P is (4k,3k,-k) = (-4/26, -3/26, 1/26) = (-2/13, -3/26, 1/26)
because P is on the normal vector originating from the origin, and it satisfies the equation of plane Π
Answer: P(-2/13, -3/26, 1/26) is the point on Π closest to the origin.

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What is the average rate of change from x = 0 to x = 4?

hammer [34]

1) find the corresponding y values for when x = 0 and when x = 4,
when x = 0, y = 4
when x = 4, y = 4

the coordinates are (0,4) and (4,4)

2) to calculate the average rate of change, find the slope of the two points:

(0,4) (4,4)

(change in y) 4 - 4 = 0
(change in x) 4 - 0 = 4

0/4 = 0

the average rate of change is 0!

3 0

3 years ago

What is (f - g)(x)?<br>f(x) = 5x<br>g(x) = 3x2 - 8x

erma4kov [3.2K]

Answer:

$(5x)-(3x^2-8x)=-3x^2+13x$

Step-by-step explanation:

$(f-g)(x)=f(x)-g(x)$

Using this, we can subtract and find the answer.

$(5x)-(3x^2-8x)=-3x^2+13x$

6 0

3 years ago

Area of a triangle with points at (-9,5), (6,10), and (2,-10)

Ann [662]

First we are going to draw the triangle using the given coordinates.
Next, we are going to use the distance formula to find the sides of our triangle.
Distance formula: $d= \sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}$

Distance from point A to point B:
$d_{AB}= \sqrt{[6-(-9)]^2+(10-5)^2}$
$d_{AB}= \sqrt{(6+9)^2+(10-5)^2}$
$d_{AB}= \sqrt{(15)^2+(5)^2}$
$d_{AB}= \sqrt{225+25}$
$d_{AB}= \sqrt{250}$
$d_{AB}=15.81$

Distance from point A to point C:
$d_{AC}= \sqrt{[2-(-9)]^2+(-10-5)^2}$
$d_{AC}= \sqrt{(2+9)^2+(-10-5)^2}$
$d_{AC}= \sqrt{11^2+(-15)^2}$
$d_{AC}= \sqrt{121+225}$
$d_{AC}= \sqrt{346}$
$d_{AC}= 18.60$

Distance from point B from point C
$d_{BC}= \sqrt{(2-6)^2+(-10-10)^2}$
$d_{BC}= \sqrt{(-4)^2+(-20)^2}$
$d_{BC}= \sqrt{16+400}$
$d_{BC}= \sqrt{416}$
$d_{BC}=20.40$

Now, we are going to find the semi-perimeter of our triangle using the semi-perimeter formula:
$s= \frac{AB+AC+BC}{2}$
$s= \frac{15.81+18.60+20.40}{2}$
$s= \frac{54.81}{2}$
$s=27.41$

Finally, to find the area of our triangle, we are going to use Heron's formula:
$A= \sqrt{s(s-AB)(s-AC)(s-BC)}$
$A=\sqrt{27.41(27.41-15.81)(27.41-18.60)(27.41-20.40)}$
$A= \sqrt{27.41(11.6)(8.81)(7.01)}$
$A=140.13$

We can conclude that the perimeter of our triangle is 140.13 square units.

3 0

3 years ago

Zinnia and ruby earn $50 per week for delivering pizzas. Zinnia word for X weeks and earned an additional total bonus of $13. Ru

Pani-rosa [81]

Answer:

Zinnia = 50[x] +13

Rubu= 50[y]

8 0

3 years ago

Jeanine was making a baking. She used 1/2 cup of flour to make cookies and 2/3 cup of flour to make brownies. How much flour did

Fudgin [204]

Answer:

1 and 1/6 cups of flour

Step-by-step explanation:

We need the fractions to have common denominators so multiply 1/2 by 3 and 2/3 by 2 to get the common denominator of 6.

1/2 x 3/3 = 3/6

2/3 x 2/2 = 4/6

Add the two fractions

3/6 + 4/6 = 7/6 or 1 & 1/6

5 0

3 years ago