20. SR = 17
22. TR = 5 and PR= 10
23. ST=9.5
24. X=1.5
for 23, the length of QS is 19, and ST is half of that length. so you divide 19/2 and get 9.5
for 24, i found it difficult to explain but i included a picture of my work. i know it may not help but it's all i've got.
32 x 1 x 1
16 x 1 x 2
8 x 2 x 2
4 x 2 x 4
8 x 4 x 1
These are the ones that come to my mind
The point
![(x,y,z)](https://tex.z-dn.net/?f=%28x%2Cy%2Cz%29)
on the plane
![x+2y+3z=9](https://tex.z-dn.net/?f=x%2B2y%2B3z%3D9)
determines the volume of the box, since
![V(x,y,z)=xyz](https://tex.z-dn.net/?f=V%28x%2Cy%2Cz%29%3Dxyz)
. Restricting the box to lie within the first octant is to say that
![x,y,z>0](https://tex.z-dn.net/?f=x%2Cy%2Cz%3E0)
.
Let's do it via Lagrange multipliers. The Lagrangian is
![L(x,y,z,\lambda)=xyz+\lambda(x+2y+3z-9)](https://tex.z-dn.net/?f=L%28x%2Cy%2Cz%2C%5Clambda%29%3Dxyz%2B%5Clambda%28x%2B2y%2B3z-9%29)
with partial derivatives (set equal to 0)
![L_x=yz+\lambda=0](https://tex.z-dn.net/?f=L_x%3Dyz%2B%5Clambda%3D0)
![L_y=xz+2\lambda=0](https://tex.z-dn.net/?f=L_y%3Dxz%2B2%5Clambda%3D0)
![L_z=xy+3\lambda=0](https://tex.z-dn.net/?f=L_z%3Dxy%2B3%5Clambda%3D0)
![L_\lambda=x+2y+3z-9=0](https://tex.z-dn.net/?f=L_%5Clambda%3Dx%2B2y%2B3z-9%3D0)
We have
![L_y-2L_x=xz-2yz=z(x-2y)=0\implies z=0\text{ or }x=2y](https://tex.z-dn.net/?f=L_y-2L_x%3Dxz-2yz%3Dz%28x-2y%29%3D0%5Cimplies%20z%3D0%5Ctext%7B%20or%20%7Dx%3D2y)
![L_z-3L_x=xy-3yz=y(x-3z)=0\implies y=0\text{ or }x=3z](https://tex.z-dn.net/?f=L_z-3L_x%3Dxy-3yz%3Dy%28x-3z%29%3D0%5Cimplies%20y%3D0%5Ctext%7B%20or%20%7Dx%3D3z)
![2L_z-3L_y=2xy-3xz=x(2y-3z)=0\implies x=0\text{ or }2y=3z](https://tex.z-dn.net/?f=2L_z-3L_y%3D2xy-3xz%3Dx%282y-3z%29%3D0%5Cimplies%20x%3D0%5Ctext%7B%20or%20%7D2y%3D3z)
We already assume
![x,y,z>0](https://tex.z-dn.net/?f=x%2Cy%2Cz%3E0)
, so we can ignore those options, leaving us with
![x=x](https://tex.z-dn.net/?f=x%3Dx)
,
![y=\dfrac x2](https://tex.z-dn.net/?f=y%3D%5Cdfrac%20x2)
, and
![z=\dfrac x3](https://tex.z-dn.net/?f=z%3D%5Cdfrac%20x3)
. Substituting into the plane equation, we get
![x+2\dfrac x2+3\dfrac x3=3x=9\implies x=3\implies y=\dfrac32\text{ and }z=1](https://tex.z-dn.net/?f=x%2B2%5Cdfrac%20x2%2B3%5Cdfrac%20x3%3D3x%3D9%5Cimplies%20x%3D3%5Cimplies%20y%3D%5Cdfrac32%5Ctext%7B%20and%20%7Dz%3D1)
So the box with largest volume has its vertex (the one opposite the vertex at the origin) in the plane at
![\left(3,\dfrac32,1\right)](https://tex.z-dn.net/?f=%5Cleft%283%2C%5Cdfrac32%2C1%5Cright%29)
, giving a volume of
![\dfrac92](https://tex.z-dn.net/?f=%5Cdfrac92)
.
Answer:A
Step-by-step explanation: