The correct answer is D. 337.5
This part of the plane is a triangle. Call it
![\mathcal S](https://tex.z-dn.net/?f=%5Cmathcal%20S)
. We can find the intercepts by setting two variables to 0 simultaneously; we'd find, for instance, that
![y=z=0](https://tex.z-dn.net/?f=y%3Dz%3D0)
means
![5x=20\implies x=4](https://tex.z-dn.net/?f=5x%3D20%5Cimplies%20x%3D4)
, so that (4, 0, 0) is one vertex of the triangle. Similarly, we'd find that (0, 5, 0) and (0, 0, 20) are the other two vertices.
Next, we can parameterize the surface by
![\mathbf s(u,v)=\langle4(1-u)(1-v),5u(1-v),20v\rangle](https://tex.z-dn.net/?f=%5Cmathbf%20s%28u%2Cv%29%3D%5Clangle4%281-u%29%281-v%29%2C5u%281-v%29%2C20v%5Crangle)
so that the surface element is
![\mathrm dS=\|\mathbf s_u\times\mathbf s_v\|=20\sqrt{42}(1-v)\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%5Cmathrm%20dS%3D%5C%7C%5Cmathbf%20s_u%5Ctimes%5Cmathbf%20s_v%5C%7C%3D20%5Csqrt%7B42%7D%281-v%29%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
Then the area of
![\mathcal S](https://tex.z-dn.net/?f=%5Cmathcal%20S)
is given by the surface integral
![\displaystyle\iint_{\mathcal S}\mathrm dS=20\sqrt{42}\int_{u=0}^{u=1}\int_{v=0}^{v=1}(1-v)\,\mathrm dv\,\mathrm du](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_%7B%5Cmathcal%20S%7D%5Cmathrm%20dS%3D20%5Csqrt%7B42%7D%5Cint_%7Bu%3D0%7D%5E%7Bu%3D1%7D%5Cint_%7Bv%3D0%7D%5E%7Bv%3D1%7D%281-v%29%5C%2C%5Cmathrm%20dv%5C%2C%5Cmathrm%20du)
Answer: Longest tool life is at A-, B+ and C+, for an average predicted life of 552.5. From examination of the cube plot, we see that the low level of cutting speed and the high level of cutting angle gives good results regardless of metal hardness.
B. 7/-8
because the whole entire fraction is still a negative.