We are given D (7, -3) and D'(2, 5).
SAppy the transformation
D'(x,y) → D(x-5, y+8).
Then
x=7 → x=7-5 = 2
y=-3 → y=-3+8 = 5
Answer: D (x-5, y+8) → D'
Find an equation of the plane that contains the points p(5,−1,1),q(9,1,5),and r(8,−6,0)p(5,−1,1),q(9,1,5),and r(8,−6,0).
topjm [15]
Given plane passes through:
p(5,-1,1), q(9,1,5), r(8,-6,0)
We need to find a plane that is parallel to the plane through all three points, we form the vectors of any two sides of the triangle pqr:
pq=p-q=<5-9,-1-1,1-5>=<-4,-2,-4>
pr=p-r=<5-8,-1-6,1-0>=<-3,5,1>
The vector product pq x pr gives a vector perpendicular to both pq and pr. This vector is the normal vector of a plane passing through all three points
pq x pr
=
i j k
-4 -2 -4
-3 5 1
=<-2+20,12+4,-20-6>
=<18,16,-26>
Since the length of the normal vector does not change the direction, we simplify the normal vector as
N = <9,8,-13>
The required plane must pass through all three points.
We know that the normal vector is perpendicular to the plane through the three points, so we just need to make sure the plane passes through one of the three points, say q(9,1,5).
The equation of the required plane is therefore
Π : 9(x-9)+8(y-1)-13(z-5)=0
expand and simplify, we get the equation
Π : 9x+8y-13z=24
Check to see that the plane passes through all three points:
at p: 9(5)+8(-1)-13(1)=45-8-13=24
at q: 9(9)+8(1)-13(5)=81+9-65=24
at r: 9(8)+8(-6)-13(0)=72-48-0=24
So plane passes through all three points, as required.
Are you asking for a name?
M is a subset of u.
Is there something further?
9514 1404 393
Answer:
- 320 m after 8 seconds
- 5.6 seconds, 10.4 seconds to height of 290 m
Step-by-step explanation:
To find the height at 8 seconds, evaluate the formula for t=8.
S(t) = -5t^2 +80t
S(8) = -5(8^2) +80(8) = -320 +640 = 320
The height of the rocket is 320 meters 8 seconds after takeoff.
__
To find the time to 290 meters height, solve ...
S(t) = 290
290 = -5t^2 +80t
-58 = t^2 -16t . . . . . . . divide by -5
6 = t^2 -16t +64 . . . . . complete the square by adding 64
±√6 = t -8 . . . . . . . . . take the square root
t = 8 ±√6 ≈ {5.551, 10.449}
The rocket is at 290 meters height after 5.6 seconds and again after 10.4 seconds.
