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nadezda [96]
3 years ago
8

If the height and base length of a square pyramid are each doubled, how does the volume of the pyramid change?

Mathematics
1 answer:
Vikentia [17]3 years ago
5 0
It quadruples. (Multiplies by 4)
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Use Lagrange multipliers to find the dimensions of the box with volume 1728 cm3 that has minimal surface area. (Enter the dimens
Dima020 [189]

Answer:

(x,y,z) = (12,12,12) cm

Step-by-step explanation:

The box is assumed to be a closed box.

The surface area of a box of dimension x, y and z is given by

S = 2xy + 2xz + 2yz

We're to minimize this function subject to the constraint that

xyz = 1728

The constraint can be rewritten as

xyz - 1728 = 0

Using Lagrange multiplier, we then write the equation in Lagrange form

Lagrange function = Function - λ(constraint)

where λ = Lagrange factor, which can be a function of x, y and z

L(x,y,z) = 2xy + 2xz + 2yz - λ(xyz - 1728)

We then take the partial derivatives of the Lagrange function with respect to x, y, z and λ. Because these are turning points, each of the partial derivatives is equal to 0.

(∂L/∂x) = 2y + 2z - λyz = 0

λ = (2y + 2z)/yz = (2/z) + (2/y)

(∂L/∂y) = 2x + 2z - λxz = 0

λ = (2x + 2z)/xz = (2/z) + (2/x)

(∂L/∂z) = 2x + 2y - λxy = 0

λ = (2x + 2y)/xy = (2/y) + (2/x)

(∂L/∂λ) = xyz - 1728 = 0

We can then equate the values of λ from the first 3 partial derivatives and solve for the values of x, y and z

(2/z) + (2/y) = (2/z) + (2/x)

(2/y) = (2/x)

y = x

Also,

(2/z) + (2/x) = (2/y) + (2/x)

(2/z) = (2/y)

z = y

Hence, at the point where the box has minimal area,

x = y = z

Putting these into the constraint equation or the solution of the fourth partial derivative,

xyz - 1728 = 0

x³ = 1728

x = 12 cm

x = y = z = 12 cm.

7 0
2 years ago
A music store sells used cds for $5 each and buys used CDs for$1.50 each. you go to the store with $20 and some CDs to sell. you
olya-2409 [2.1K]
Let x be the number of CDs we buy and y the number of CDs we sell. 
Each CD sell for $1.5, then the total of money we earn is $1.5y.
Each CD bought for $5, then the total money spent is $-5x
Add the above values like this:
1.5y-5x
We have $20, add them like this:
1.5y-5x+20
Since we want to have<span> at least $10 left when you leave the store, then 
we deduce the equation:
</span>1.5y-5x+20\leq 10
5 0
3 years ago
Find the equation of the line with the given properties: slope of 2, contains the point ( 4, -3).
Minchanka [31]
The answer is y=2(x-4)-3
6 0
3 years ago
Find x, y, z, so that 41, 46, x, y, z has a mean of 50 and a mode of 45
nevsk [136]

Answer:

Step-by-step explanation:

as mode is 45 ,two of three is 45

i assume x =y=45

now (41+46+45+45+z)/5=50

177+z=50×5=250

z=250-177=73

so x=45

y=45

z=73

5 0
2 years ago
Read 2 more answers
By hiring 45 new employees, a department store increased its help by 18% during the Christmas season. How many employees did the
bagirrra123 [75]
Hi!

We can set up a proportion then cross multiply to solve this.

\frac{45}{x}  \frac{18}{100}

45 x 100 = 4500
4500/18 = 250

The answer is 250

Hope this helps! :)
7 0
3 years ago
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