![\vec r(t)=\langle6t,1+3t,4t\rangle](https://tex.z-dn.net/?f=%5Cvec%20r%28t%29%3D%5Clangle6t%2C1%2B3t%2C4t%5Crangle)
![\vec R(s)=\langle2+s,-8+3s,-12+4s\rangle](https://tex.z-dn.net/?f=%5Cvec%20R%28s%29%3D%5Clangle2%2Bs%2C-8%2B3s%2C-12%2B4s%5Crangle)
Take the derivatives of each to get the tangent vectors:
![\dfrac{\mathrm d\vec r(t)}{\mathrm dt}=\langle6,3,4\rangle](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%5Cvec%20r%28t%29%7D%7B%5Cmathrm%20dt%7D%3D%5Clangle6%2C3%2C4%5Crangle)
![\dfrac{\mathrm d\vec R(s)}{\mathrm ds}=\langle1,3,4\rangle](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%5Cvec%20R%28s%29%7D%7B%5Cmathrm%20ds%7D%3D%5Clangle1%2C3%2C4%5Crangle)
Take the cross product of the tangent vectors to get a vector that is normal to both lines:
![\langle6,3,4\rangle\times\langle1,3,4\rangle=\langle0,-20,15\rangle](https://tex.z-dn.net/?f=%5Clangle6%2C3%2C4%5Crangle%5Ctimes%5Clangle1%2C3%2C4%5Crangle%3D%5Clangle0%2C-20%2C15%5Crangle)
The two given lines intersect when
:
![\langle6t,1+3t,4t\rangle=\langle2+s,-8+3s,-12+4s\rangle\implies t=1,s=4](https://tex.z-dn.net/?f=%5Clangle6t%2C1%2B3t%2C4t%5Crangle%3D%5Clangle2%2Bs%2C-8%2B3s%2C-12%2B4s%5Crangle%5Cimplies%20t%3D1%2Cs%3D4)
that is, at the point (6, 4, 4).
The line perpendicular to both of the given lines through the origin is obtained by scaling the normal vector found earlier by
; translate this line by adding the vector
to get the line we want,
![\vec\rho(\tau)=\langle6,4,4\rangle+\langle0,-20,15\rangle\tau](https://tex.z-dn.net/?f=%5Cvec%5Crho%28%5Ctau%29%3D%5Clangle6%2C4%2C4%5Crangle%2B%5Clangle0%2C-20%2C15%5Crangle%5Ctau)
![\boxed{\vec\rho(\tau)=\langle6,4-20\tau,4+15\tau\rangle}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cvec%5Crho%28%5Ctau%29%3D%5Clangle6%2C4-20%5Ctau%2C4%2B15%5Ctau%5Crangle%7D)
Answer/Step-by-step explanation:
m<MAH = 90° (linear pair. It is also supplementary to m<MAK = 90°)
m<MAL + m<HAL = m<MAH
Which is:
(9x + 1) + (x + 9) = 90°
Solve for x
9x + 1 + x + 9 = 90
Collect like terms
9x + x + 1 + 9 = 90
10x + 10 = 90
Subtract both sides by 10
10x + 10 - 10 = 90 - 10
10x = 80
Divide both sides by 10
x = 8
Find m<MAL
m<MAL = 9x + 1
Plug in the value of x
m<MAL = 9(8) + 1 = 72 + 1 = 73°
Find m<HAL
m<HAL = x + 9
m<HAL = 8 + 9 = 17°
Answer:
Step-by-step explanation:
Derivative is the slope of the tangent line at any point.
That slope is positive for x < -5 and x > 0
That slope is negative for -5 < x < 0
The Median for this graph is 21.6
The formula is 1/2 x ( a + b ) x h
A and b are the top and bottom sides and h is always the perpendicular height.
Now replace the variables...
1/2 x (6 + 14 ) x 8.4
And then...
1/2 x 20 x 8.4 which will give you 84cm^2
Hope this helps! Anymore questions let me know :)