L=R*angle, but the angle *must* be in radians. So, just translate 43 degrees into radians: 43 * pi / 180, and you'll get:
L = 42 * ( 43 * pi / 180 ) ~ 31.504666, which rounded to the nearest hundredth is 31.50 cm.
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.
ANSWER:
24 minutes
40% of 1 hour is equivalent to 24 minutes.
STEP-BY-STEP EXPLANATION:
1 hour = 60 minutes
If:
100% = 60 minutes
Then:
100% ÷ 2.5 = 60 minutes ÷ 2.5
40% = 24 minutes
THEREFORE:
40% of 1 hour is equivalent to 24 minutes.
To solve this find the common number that can be divided by both and use that number.
In this case, 4 is divisible by both 4 and 12, and thus the final expression would be
4(d + 3e).
