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Nina [5.8K]
3 years ago
13

Expression equivalent to -2.5(4x-3)

Mathematics
1 answer:
e-lub [12.9K]3 years ago
6 0

If you distribute, then you get -10x+7.5

Note how -2.5 times 4x = -10x and how -2.5 times -3 = 7.5

The idea is that you multiply the outer term (-2.5) by each term inside

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The base of an aquarium with given volume V is made of slate and the sides are made of glass. If the slate costs seven times as
Olin [163]

Answer:

x = ∛(2V/7)

y = ∛(2V/7)

z = 3.5 [∛(2V/7)]

{x,y,z} = { ∛(2V/7), ∛(2V/7), 3.5[∛(2V/7)] }

Step-by-step explanation:

The aquarium is a cuboid open at the top.

Let the dimensions of the base of the aquarium be x and y.

The height of the aquarium is then z.

The volume of the aquarium is then

V = xyz

Area of the base of the aquarium = xy

Area of the other faces = 2xz + 2yz

The problem is to now minimize the value of the cost function.

The cost of the area of the base per area is seven times the cost of any other face per area.

With the right assumption that the cost of the other faces per area is 1 currency units, then, the cost of the base of the aquarium per area would then be 7 currency units.

Cost of the base of the aquarium = 7xy

cost of the other faces = 2xz + 2yz

Total cost function = 7xy + 2xz + 2yz

C(x,y,z) = 7xy + 2xz + 2yz

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

xyz = V

The constraint can be rewritten as

xyz - V = 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) = 7xy + 2xz + 2yz - λ(xyz - V)

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

(∂L/∂x) = 7y + 2z - λyz = 0

λ = (7y + 2z)/yz = (7/z) + (2/y) (eqn 1)

(∂L/∂y) = 7x + 2z - λxz = 0

λ = (7x + 2z)/xz = (7/z) + (2/x) (eqn 2)

(∂L/∂z) = 2x + 2y - λxy = 0

λ = (2x + 2y)/xy = (2/y) + (2/x) (eqn 3)

(∂L/∂λ) = xyz - V = 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

(eqn 1) = (eqn 2)

(7/z) + (2/y) = (7/z) + (2/x)

(2/y) = (2/x)

y = x

Also,

(eqn 1) = (eqn 3)

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

(7/z) = (2/y)

z = (7y/2)

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

y = x,

z = (7y/2) = (7x/2)

We can then substitute those into the constraint equation for y and z

xyz = V

x(x)(7x/2) = V

(7x³/2) = V

x³ = (2V/7)

x = ∛(2V/7)

y = x = ∛(2V/7)

z = (7x/2) = 3.5 [∛(2V/7)]

The values of x, y and z in terms of the volume that minimizes the cost function are

{x,y,z} = {∛(2V/7), ∛(2V/7), 3.5[∛(2V/7)]}

Hope this Helps!!!

7 0
3 years ago
Can u guys help i need to make this longer lol
Marizza181 [45]
I don’t under stand the question. What do you want me to answer
5 0
3 years ago
Solve for p. <br><br> i = prt<br><br> p = i-rt<br> p = i/(rt)<br> p = (rt)/i
Debora [2.8K]

Answer:

<h2>p = (rt)/i</h2>

Step-by-step explanation:

i=prt\to prt=i\qquad\text{divide both sides by}\ rt\neq0\\\\\dfrac{prt}{rt}=\dfrac{i}{rt}\\\\p=\dfrac{i}{rt}

5 0
3 years ago
The average of your first 6 tests is 82. If your test average after your seventh test is an 80.5, what did you score on the seve
mart [117]
So average is  of the first 6 tests is 82   well that means 

(test1 + test2+ test3+ test4 + test5+test6)/6 = 82

so now let do some cross multiplliying 

test1 + test2 + test3 + test 4 + test5 +test6 = 82*6

test1 + test2 + test3 + test 4 + test5 +test6  = 492

now lets see if we can find that 7th test score

(test1 + test2 + test3 + test4 + test5 +test6 + test7)/7 = 80.5

So look we found test1 + test2 + test3 + test4 + test5 +test6 to be 492 so lets substitute. 

(492 + test7 )/7 = 80.5

test7 = (80.5*7)-492 = 71.5 



7 0
3 years ago
Vvorth 1 points
viva [34]

Hi there!  

»»————- ★ ————-««

I believe your answer is:  

"Add 4 to both sides, and then divide by 2. The solution is x = 12."

»»————- ★ ————-««  

Here’s why:  

⸻⸻⸻⸻

  • To solve for 'x', we would have to use inverse operations.
  • We would first have to add four to both sides to undo the negative four. Addition is the opposite of subtraction.
  • We would then divide by 2 to isolate 'x'. Division is the opposite of multiplication.

⸻⸻⸻⸻

\boxed{\text{Solving for 'x'...}}\\\\2x - 4 = 20\\------------\\\rightarrow 2x - 4 + 4 = 20 + 4\\\\\rightarrow 2x = 24\\\\\rightarrow \frac{2x=24}{2}\\\\\rightarrow \boxed{x = 12}

⸻⸻⸻⸻

»»————- ★ ————-««  

Hope this helps you. I apologize if it’s incorrect.  

6 0
3 years ago
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