The orthocenter is the point where the three altitudes meet.
sketch the graph and you will see that AB is a horizontal line, the altitude is a vertical line through the point (1,3), so the equation of this altitude is x=1
next, find another altitude. I'll use the altitude of BC.
the slope of BC is (6-3)/(4-1)=1, so the slope of the altitude, which is perpendicular to BC going through the point A (0,6), is -1, the equation of the altitude of BC is y=-x+6
the system of equation : x=1
y=-x+6
has a solution (1, 5)
the solution is where the two lines meet, the meeting point is the orthocenter.
Double check by find the equation for other altitude:
slope of AC: (3-6)/(1-0)=-3
slope of altitude of AC: 1/3
equation of altitude of AC: y=(1/3)x+b
the altitude of AC goes through point B (4,6), so we can find out b by plug x=4, y=6 in the equation: 6=(1/3)*4+b, b=14/3
y=(1/3)x+14/3
Is (1,5) also a solution to this equation? Plug x=1 in the equation, we get y=5, so yes, (1,5) is a point on the third altitude.
Answer:
w = 2
Step-by-step explanation:
6w + 4w - 9 + 6 = 9w - 7+ 6
6w + 4w - 9w = - 7 + 6 + 9 - 6
w = 2
Answer :
no. of days used to repair 1250m of road = 5
no. of days used to repair 1m of road = 5/1250
no. of days used to repair 8000m of road (8*1000m ) = 5/1250*8000= 32 days
I'm reading this as
with
.
The value of the integral will be independent of the path if we can find a function
that satisfies the gradient equation above.
You have
Integrate
with respect to
. You get
Differentiate with respect to
. You get
![\dfrac{\partial f}{\partial y}=\dfrac{\partial}{\partial y}[x^2e^{-y}+g(y)]](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D%5Cdfrac%7B%5Cpartial%7D%7B%5Cpartial%20y%7D%5Bx%5E2e%5E%7B-y%7D%2Bg%28y%29%5D)
Integrate both sides with respect to
to arrive at
So you have
The gradient is continuous for all
, so the fundamental theorem of calculus applies, and so the value of the integral, regardless of the path taken, is
4*8*(3/4)=3*8=24
the volume is 24.
