The area of the box is
![A(x,y,z)=xy+2xz+2yz](https://tex.z-dn.net/?f=A%28x%2Cy%2Cz%29%3Dxy%2B2xz%2B2yz)
which we want to minimize subject to the constraint
.
The Lagrangian is
![L(x,y,z)=xy+2xz+2yz+\lambda(xyz-5324)](https://tex.z-dn.net/?f=L%28x%2Cy%2Cz%29%3Dxy%2B2xz%2B2yz%2B%5Clambda%28xyz-5324%29)
with critical points where the partial derivatives are 0:
![L_x=y+2z+\lambda yz=0](https://tex.z-dn.net/?f=L_x%3Dy%2B2z%2B%5Clambda%20yz%3D0)
![L_y=x+2z+\lambda xz=0](https://tex.z-dn.net/?f=L_y%3Dx%2B2z%2B%5Clambda%20xz%3D0)
![L_z=2x+2y+\lambda xy=0](https://tex.z-dn.net/?f=L_z%3D2x%2B2y%2B%5Clambda%20xy%3D0)
![L_\lambda=xyz-5324=0](https://tex.z-dn.net/?f=L_%5Clambda%3Dxyz-5324%3D0)
Notice that
![L_y-L_x=(x-y)+\lambda(xz-yz)=0\implies(x-y)(1+\lambda z)=0\implies x=y\text{ or }z=-\dfrac1\lambda](https://tex.z-dn.net/?f=L_y-L_x%3D%28x-y%29%2B%5Clambda%28xz-yz%29%3D0%5Cimplies%28x-y%29%281%2B%5Clambda%20z%29%3D0%5Cimplies%20x%3Dy%5Ctext%7B%20or%20%7Dz%3D-%5Cdfrac1%5Clambda)
Substituting the latter into either
or
will end up suggesting that
is infinite, so we throw out this case.
If
, then
![L_z=0\implies4x+\lambda x^2=0\implies x=0\text{ or }x=-\dfrac4\lambda](https://tex.z-dn.net/?f=L_z%3D0%5Cimplies4x%2B%5Clambda%20x%5E2%3D0%5Cimplies%20x%3D0%5Ctext%7B%20or%20%7Dx%3D-%5Cdfrac4%5Clambda)
We ignore the case where
because that would make the volume 0. Then
![x=y=-\dfrac4\lambda\text{ and }L_x=0\implies-\dfrac4\lambda+2(1331\lambda^2)+\lambda\left(-\dfrac4\lambda\right)(1331\lambda^2)=0](https://tex.z-dn.net/?f=x%3Dy%3D-%5Cdfrac4%5Clambda%5Ctext%7B%20and%20%7DL_x%3D0%5Cimplies-%5Cdfrac4%5Clambda%2B2%281331%5Clambda%5E2%29%2B%5Clambda%5Cleft%28-%5Cdfrac4%5Clambda%5Cright%29%281331%5Clambda%5E2%29%3D0)
![\implies2662\lambda^3+4=0](https://tex.z-dn.net/?f=%5Cimplies2662%5Clambda%5E3%2B4%3D0)
![\implies\lambda=-\dfrac{\sqrt[3]{2}}{11}](https://tex.z-dn.net/?f=%5Cimplies%5Clambda%3D-%5Cdfrac%7B%5Csqrt%5B3%5D%7B2%7D%7D%7B11%7D)
so we have one critical point at
![(x,y,z)=\left(22\sqrt[3]{4},22\sqrt[3]{4},\dfrac{11}{2\sqrt[3]{2}}\right)\approx(34.9228,34.9228,4.3654)](https://tex.z-dn.net/?f=%28x%2Cy%2Cz%29%3D%5Cleft%2822%5Csqrt%5B3%5D%7B4%7D%2C22%5Csqrt%5B3%5D%7B4%7D%2C%5Cdfrac%7B11%7D%7B2%5Csqrt%5B3%5D%7B2%7D%7D%5Cright%29%5Capprox%2834.9228%2C34.9228%2C4.3654%29)
which give a minimum area of about 1829.41 sq. cm.
Answer:
![\Huge \boxed{x=5}](https://tex.z-dn.net/?f=%5CHuge%20%5Cboxed%7Bx%3D5%7D)
![\Huge \boxed{121\°}](https://tex.z-dn.net/?f=%5CHuge%20%5Cboxed%7B121%5C%C2%B0%7D)
Step-by-step explanation:
∠1 and ∠2 are vertical angles.
∠1 = ∠2
<u>Part A:</u>
<u></u>
Create an equation and solve for x.
20x + 21 = 30x - 29
Add 29 and -20x on both sides.
20x + 21 + 29 - 20x = 30x - 29 + 29 - 20x
Simplify the equation.
50 = 10x
Divide both sides by 10.
= ![\frac{10x}{10}](https://tex.z-dn.net/?f=%5Cfrac%7B10x%7D%7B10%7D)
Simplify the equation.
5 = x
Switch sides.
x = 5
<u>Part B:</u>
<u></u>
Let x = 5
20(5) + 21
Distribute.
100 + 21
Add.
= 121
Since both angles are vertical angles, they are equal.
∠1 = 121°
∠2 = 121°
You: spends $50 on supplies
charges $10 per hour ⇔
[The number that 10 & 8 both have in
Brother: spends $30 on supplies common is 40]
charges $8 per hour ⇔
↓
You: 10 x 4= 40 [You'll work 4 hours to get the same amount of money.]
↓
Brother: 8 x 5= 40 [Your brother will work 5 hours to get the same amount of money.]
The answer if $160, you times the 32 students with the $5 to donate.
Answer:
17 and where’s the c lol
Step-by-step explanation: