Plane R and plane P intersect at AB.
Planes are when 3 points that are not on the same line can be used to describe a plane. Plane P is horizontal and Plane R is vertical and these 2 planes intersect at line AB. Intersecting is where these 2 planes meet, therefore line AB is common to both planes.
a) Let f be the cost of each ticket. We know that 2 tickets cost a total of $24.95, so therefore we have this equation:
2f = 24.95
b) Solve this equation by dividing both sides by 2:
f = 12.475
This value isn't possible with money so to make it realistic, we'll round to the nearest cent:
f = $12.48
c) We're ordering one more ticket, so we'll just add f to our total cost:
64.50 + f
= 64.50 + 12.48
= $76.98
The dimension of the container of this size that has the minimum cost is
,
and ![z =\frac{4}{3} \sqrt[3]{360}](https://tex.z-dn.net/?f=z%20%3D%5Cfrac%7B4%7D%7B3%7D%20%5Csqrt%5B3%5D%7B360%7D)
The given parameters are:
- Volume = 480m^3
- Cost: $5 per square meter for the bottom, and $3 per square meter for the sides.
Assume the dimensions of the container are: x, y and z.
The volume (V) would be:
![V = xyz](https://tex.z-dn.net/?f=V%20%3D%20xyz)
Substitute 480 for V
![xyz =480](https://tex.z-dn.net/?f=xyz%20%3D480)
And the objective cost function would be:
![C =8xy + 6xz + 6yz](https://tex.z-dn.net/?f=C%20%3D8xy%20%2B%206xz%20%2B%206yz)
Differentiate the cost function using Lagrange multipliers
![8x + 6z = \lambda xz](https://tex.z-dn.net/?f=8x%20%2B%206z%20%3D%20%5Clambda%20xz)
![8y + 6z = \lambda yz](https://tex.z-dn.net/?f=8y%20%2B%206z%20%3D%20%5Clambda%20yz)
![6x + 6y = \lambda xy](https://tex.z-dn.net/?f=6x%20%2B%206y%20%3D%20%5Clambda%20xy)
Divide equation (1) by (2)
![\frac{8x + 6z}{8y + 6z} = \frac{\lambda xz}{\lambda yz}](https://tex.z-dn.net/?f=%5Cfrac%7B8x%20%2B%206z%7D%7B8y%20%2B%206z%7D%20%3D%20%5Cfrac%7B%5Clambda%20xz%7D%7B%5Clambda%20yz%7D)
![\frac{8x + 6z}{8y + 6z} = \frac{x}{y}](https://tex.z-dn.net/?f=%5Cfrac%7B8x%20%2B%206z%7D%7B8y%20%2B%206z%7D%20%3D%20%5Cfrac%7Bx%7D%7By%7D)
Factor out 2
![\frac{4x + 3z}{4y + 3z} = \frac{x}{y}](https://tex.z-dn.net/?f=%5Cfrac%7B4x%20%2B%203z%7D%7B4y%20%2B%203z%7D%20%3D%20%5Cfrac%7Bx%7D%7By%7D)
Cross multiply
![4xy + 3yz = 4xy + 3xz](https://tex.z-dn.net/?f=4xy%20%2B%203yz%20%3D%204xy%20%2B%203xz)
Evaluate the like terms
![3yz = 3xz](https://tex.z-dn.net/?f=3yz%20%3D%203xz)
Divide both sides by 3z
![y = x](https://tex.z-dn.net/?f=y%20%3D%20x)
Divide the first equation by the third
![\frac{8y + 6z}{6x + 6y} = \frac{\lambda yz}{\lambda xy}](https://tex.z-dn.net/?f=%5Cfrac%7B8y%20%2B%206z%7D%7B6x%20%2B%206y%7D%20%3D%20%5Cfrac%7B%5Clambda%20yz%7D%7B%5Clambda%20xy%7D)
![\frac{8y + 6z}{6x + 6y} = \frac{z}{x}](https://tex.z-dn.net/?f=%5Cfrac%7B8y%20%2B%206z%7D%7B6x%20%2B%206y%7D%20%3D%20%5Cfrac%7Bz%7D%7Bx%7D)
Factor out 2
![\frac{4y + 3z}{3x + 3y} = \frac{z}{x}](https://tex.z-dn.net/?f=%5Cfrac%7B4y%20%2B%203z%7D%7B3x%20%2B%203y%7D%20%3D%20%5Cfrac%7Bz%7D%7Bx%7D)
Cross multiply
![4xy+3xz = 3xz + 3yz](https://tex.z-dn.net/?f=4xy%2B3xz%20%3D%203xz%20%2B%203yz)
Cancel out the common terms
![4xy = 3yz](https://tex.z-dn.net/?f=4xy%20%3D%203yz)
Divide both sides by y
![4x = 3z](https://tex.z-dn.net/?f=4x%20%3D%203z)
Make z the subject
![z =\frac{4}{3}x\\](https://tex.z-dn.net/?f=z%20%3D%5Cfrac%7B4%7D%7B3%7Dx%5C%5C)
So, we have:
and ![z =\frac{4}{3}x\\](https://tex.z-dn.net/?f=z%20%3D%5Cfrac%7B4%7D%7B3%7Dx%5C%5C)
Recall that:
![xyz =480](https://tex.z-dn.net/?f=xyz%20%3D480)
Substitute
and ![z =\frac{4}{3}x\\](https://tex.z-dn.net/?f=z%20%3D%5Cfrac%7B4%7D%7B3%7Dx%5C%5C)
![x \times x \times \frac 43x = 480](https://tex.z-dn.net/?f=x%20%5Ctimes%20x%20%5Ctimes%20%5Cfrac%2043x%20%3D%20480)
So, we have:
![\frac 43x^3 = 480](https://tex.z-dn.net/?f=%5Cfrac%2043x%5E3%20%3D%20480)
Multiply both sides by 3/4
![x^3 = 360](https://tex.z-dn.net/?f=x%5E3%20%3D%20360)
Take the cube roots of both sides
![x = \sqrt[3]{360}](https://tex.z-dn.net/?f=x%20%3D%20%5Csqrt%5B3%5D%7B360%7D)
Recall that:
![y = x](https://tex.z-dn.net/?f=y%20%3D%20x)
So, we have:
![y = \sqrt[3]{360}](https://tex.z-dn.net/?f=y%20%3D%20%5Csqrt%5B3%5D%7B360%7D)
Also, we have:
![z =\frac{4}{3}x\\](https://tex.z-dn.net/?f=z%20%3D%5Cfrac%7B4%7D%7B3%7Dx%5C%5C)
So, we have:
![z =\frac{4}{3} \sqrt[3]{360}](https://tex.z-dn.net/?f=z%20%3D%5Cfrac%7B4%7D%7B3%7D%20%5Csqrt%5B3%5D%7B360%7D)
Hence, the dimension of the container of this size that has the minimum cost is
,
and ![z =\frac{4}{3} \sqrt[3]{360}](https://tex.z-dn.net/?f=z%20%3D%5Cfrac%7B4%7D%7B3%7D%20%5Csqrt%5B3%5D%7B360%7D)
Read more about Lagrange multipliers at:
brainly.com/question/4609414
Answer: she waters 4 plants
Step-by-step explanation:
5cups /quart=20/x(x=#of quarts)
5x=20
x=20/5
x=4
<span>To find the standard form for 200,000+80,000+700+6, simply add these numbers together:
</span>200,000+80,000+700+6= <span>280,706
Hope this helps!
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