For one cook, the invariant distribution is given by π_n = (1/2)ⁿ for n ≥ 0. For two cooks and a maximum of 5 customers, the fraction of arriving customers that leave immediately in equilibrium is approximately 0.361.
a.) For one cook, we can solve for the invariant distribution using the balance equations. For n ≥ 1, we have λπ_n = μπ_(n-1), where λ = 20 (arrival rate) and μ = 10 (service rate). Solving these equations, we find π_n = (1/2)ⁿ for n ≥ 0.
b.) For two cooks, we use a similar approach but with a maximum of 5 customers. Let ρ = λ/(2μ) = 1/2. We calculate the probabilities of the states 0, 1, 2, 3, 4, and 5 using the Erlang loss formula:
π_0 = 1/(1 + 2ρ + 2ρ² + 2ρ³ + 2ρ⁴ + ρ⁵),
π_n = 2ρⁿπ₀ for n = 1, 2, 3, 4,
π_5 = ρ⁵π₀.
The fraction of arriving customers that leave immediately is given by π_5, which is approximately 0.361.
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Solve the system of equations 4x+5y=-1 and -5x-8y=10 by combining the equations.
The solution of the equation are as follows:
x = 6 and y = -5
How to solve the system of equation?4x + 5y = -1
-5x - 8y = 10
Therefore,
20x + 25y = -5
-20x - 32y = 40
-7y = 35
y = -5
Hence,
4x + 5(-5) = -1
4x - 25 = -1
4x = -1 + 25
4x = 24
x = 24 / 4
x = 6
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Which represents the solution set to the inequality 5.1(3 + 2.2x) > –14.25 – 6(1.7x + 4)?
x < –2.5
x > 2.5
(–2.5, ∞)
(–∞, 2.5)
The solution to the inequality is (–2.5, ∞)
How to solve the inequality?The inequality is given as:
5.1(3 + 2.2x) > –14.25 – 6(1.7x + 4)
Open the brackets
15.3 + 11.22x > –14.25 – 10.2x - 24
Collect the like terms
11.22x + 10.2x > -15.3 - 14.25- 24
Evaluate the like terms
21.42x > -53.55
Divide both sides by 21.42
x > -2.5
This can also be represented as (–2.5, ∞)
Hence, the solution to the inequality is (–2.5, ∞)
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Answer:
c
Step-by-step explanation:
on edge
Select the correct answer from the drop-down menu.
The expression
is not equivalent to (1 - sin²(x)) tan(-x).
J
e
C
✔
The expression that is not equivalent to (1 - sin²(x)) tan(-x) is D. (cos²x - 1)(cot -x).
How to illustrate the expression?It should be noted that (cos²x - 1)(cot -x). is not equivalent to the given expression.
This is illustrated as:
(cos²x - 1)(cot -x)
= (-sin²x) × (-cot x)
= sin²x × cosx/sin x = sinxcosx
In conclusion, the correct option is D.
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The expression _______ is not equivalent to (1 − sin2(x)) tan(-x).
a. (1 - cos^2(x)) cot(-x)
b. (cos^2(x) - 1) cot(x)
c. (sin^(x) - 1) tan(x)
d. (cos^2(x) - 1) cot(-x)
A basketball player has a 0.689 probability of making a free throw. If the player shoots 18 free throws, what is the probability that she makes no more than 11 of them
It is determined while using the binomial distribution that there is still a 1.145=114.5% chance that she produces no more than 11 of them.
Calculating the probabilityThere are just two possible results for each throw. Either she succeeds or she fails. The binomial probability distribution is employed to answer this issue since the probability of completing a shot is regardless of all other throws.
Binomial probability distribution-
[tex]P(X=x) = C_{n,x} .p^{x}(1-p)^{n-x}[/tex]
[tex]C_{n,x} = n!/x! (n-x)![/tex]
where,
the no. of success= x
the no. of trials = n
the probability of a success on one trial = p
The probability of throwing not more than 11 will be:
P(X<11) = P(X=0) + P(X=1) +P(X=2) + P(X=3) +P(X=4) +P(X=5)+P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)+P(X=11)
Where,
[tex]P(X=x) = C_{n,x} .p^{x}(1-p)^{n-x}[/tex]
[tex]P(X=0) = C_{18,0} .(0.689)^{0}(0.311)^{18}[/tex]≈0
[tex]P(X=1) = C_{18,1} .(0.689)^{1}(0.311)^{17}[/tex]≈0
[tex]P(X=2) = C_{18,2} .(0.689)^{2}(0.311)^{16}[/tex]≈0
[tex]P(X=3) = C_{18,3} .(0.689)^{3}(0.311)^{15}[/tex]≈0
[tex]P(X=4) = C_{18,4} .(0.689)^{4}(0.311)^{14}[/tex]≈0
[tex]P(X=5) = C_{18,5} .(0.689)^{5}(0.311)^{13}[/tex]= 0.0003
[tex]P(X=6) = C_{18,4} .(0.689)^{6}(0.311)^{12}[/tex]=0.0016
[tex]P(X=7) = C_{18,7} .(0.689)^{7}(0.311)^{11}[/tex]=0.0062
[tex]P(X=8) = C_{18,8} .(0.689)^{8}(0.311)^{10}[/tex]=0.0188
[tex]P(X=9) = C_{18,9} .(0.689)^{9}(0.311)^{9}[/tex]=0.0463
[tex]P(X=10) = C_{18,10} .(0.689)^{10}(0.311)^{8}[/tex]=0.9232
[tex]P(X=11) = C_{18,11} .(0.689)^{11}(0.311)^{7}[/tex]=0.1488
So,
P(X<11) = P(X=0) + P(X=1) +P(X=2) + P(X=3) +P(X=4) +P(X=5)+P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)+P(X=11)
=0+0+0+0+0+0.0003+0.0016+0.0062+0.0188+0.0463+0.9232+0.1488 =1.145
Therefore, she makes 1.145=114.5% probability, no more than 11 of them.
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Which numbers are integers? Check all that apply.
4
Negative 1 and one-third
-10
2.5
-4
0.ModifyingAbove 13 with Bar
YOOOO QUICK I REALLY NEED THIS PLEASEEE
The numbers : 4,-10 and -4 are integers.
A number without a decimal or fractional element is known as an integer, which can be both positive and negative, including zero.
The Latin word "integer" signifies "whole" or "intact." Thus, fractions and decimals are not included in integers.
All whole numbers and negative numbers are considered integers. This means that if we combine negative numbers with whole numbers, a collection of integers results.
"Negative 1 and one-third" includes a fractional part, so it is not an integer.
"2.5" and "0.ModifyingAbove 13 with Bar" contain a decimal, so it is also not an integer.
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A basin can hold 8542ml of water.the basin can hold 0.458l more water than a fish tank. how much water can the fish tank hold?express your answer in litres.
Answer:
8.048 liters
Step-by-step explanation:
Convert 8542 ml to liters: (8542 ml)(1 liter/1000 ml) = 8.542 liters
Now subtract 0.458 liters to find the amount that a fish tank can hold:
8.542 liters
-0.458 liters
8.048 liters is the fish tank capacity
PLEASE HELP!!! I'LL MARK AS BRAINLIEST TO THE FIRST PERSON THAT CAN ANSWER!!!
When two exponents with the same base are multiplied together, this reflects the _______ of powers property.
B(x) = 0. 06x^2 - 0. 2x^3, find the dosage at which the resulting blood presure is maximized
The dosage at which the resulting blood pressure is maximized is x at 0.2 and the maximum dosage is 0.0008.
In the question,
The function is [tex]B(x) = 0. 06x^2 - 0. 2x^3[/tex]
To find the maximum or minimum, take the derivative and set it equal to zero.
⇒ [tex]B'(x) = 2(0. 06)x - 3(0. 2)x^{2}[/tex]
Setting it equal to zero, we get
⇒ [tex]0.12x - 0.6x^{2}=0[/tex]
⇒ 0.6x (0.2-x) = 0
⇒ x = 0 or x = 0.2
Now substitute x = 0.2 in B(x), we get
⇒ [tex]B(0.2) = 0. 06(0.2)^{2} - 0. 2(0.2)^{3}[/tex]
⇒ B(0.2) = 0.0024 - 0.0016
⇒ B(0.2) = 0.0008
To know B(0.2) is maximum, let us find the values for x = 1 and x = 0.01.
For x = 1,
⇒ [tex]B(1) = 0. 06(1)^{2} - 0. 2(1)^{3}[/tex]
⇒ B(1) = 0.06-0.2
⇒ B(1) = -1.04
For x = 0.01,
⇒ [tex]B(0.01) = 0. 06(0.01)^{2} - 0. 2(0.01)^{3}[/tex]
⇒ B(0.01) = 0.000006 - 0.0000002
⇒ B(0.01) = 0.0000058
Thus, x = 0.2 is the maximum.
Hence we can conclude that the dosage at which the resulting blood pressure is maximized is x at 0.2 and the maximum dosage is 0.0008.
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1. Reflect Rectangle ABCD across the y-axis, then
translate it using the following rule:
(x,y) → (x-4, y - 3).
A B
D
с
-4 -2
4
2
ТУ
O
-2-
-4-
2
4
X
√x
A' (
B'(
C'(
D'(
A"(
B"(
C"(
D"(
)
)
)
)
PLS HELP ME WITH THIS
Answer:
See attached image
Step-by-step explanation:
Right triangles \boxed{1}
1
start box, 1, end box, \boxed{2}
2
start box, 2, end box, and \boxed{3}
3
start box, 3, end box are given with all their angle measures and approximate side lengths.
Use one of the triangles to approximate the ratio \dfrac{WY}{WX}
WX
WY
start fraction, W, Y, divided by, W, X, end fraction.
Choose 1 answer:
Considering right-angled triangle XYW, an approximate ratio WY/WX is equal to: A. 0.34.
How to approximate the ratio WY/WX?Considering right-angled triangle XYW, we would apply the law of cosine to approximate the ratio WY/WX. Mathematically, the law of cosine is given by:
cos(θ) = Adj/Hyp
Where:
Adj is the adjacent side of a right-angled triangle.Hyp is the hypotenuse of a right-angled triangle.θ is the angle.Substituting the given parameters into the formula, we have;
cos(θ) = WY/WX
cos(70) = 3.4/10
0.34 = 0.34.
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Complete Question:
Right triangles 1-2 and 3 are given with all their angle measures and approximate side lengths.
Use one of the triangles to approximate the ratio WY/WX
A. 0.34
B. 0.94
C. 1.06
D. 2.76
Zoe says an equilateral triangle is always an acute triangle, but an acute triangle is never an equilateral triangle. Which statement explains whether Zoe is correct or not?
A. Zoe is not correct because equilateral triangles are acute triangles with three sides of equal length. Acute triangles can sometimes be equilateral triangles.
B. Zoe is not correct because equilateral triangles have three acute angles. Acute
triangles have three acute angles, so acute triangles are always equilateral triangles.
C. Zoe is correct because equilateral triangles have three sides of equal length, and acute triangles have three sides of different lengths.
D. Zoe is correct because equilateral triangles have three acute angles. Acute triangles have one acute angle, so an acute triangle cannot be an equilateral triangle.
HELP!
Answer:
Step-by-step explanation:
Discussion
She is correct.
An equilateral triangle is a specific acute triangle. An acute triangle can never be an equilateral triangle because it has 3 unequal angles. Your best choice is likely C, but none of the choices are first rate choices.
Answer: C
Ralph Warren purchased 27 shares of stock at 16 3/8 per share. He paid a $27.50 brokerage fee. He later sold all 27 shares at 17 5/8 and paid a $28.75 brokerage fee. (36) What was his total cost for the stock including his brokerage fee? (37) What did he receive from the sale of the stock after he paid the brokerage fee? (38) Did he have a capital gain or loss? (39) How much was the gain or loss? (40) What was the net change from 16 3/8 to 17 5/8?
Answer:
Ralph Warren purchased 27 shares of stock at 16 3/8 per share. He paid a $27.50 brokerage fee. He later sold all 27 shares at 17 5/8 and paid a $28.75 brokerage fee. (36) What was his total cost for the stock including his brokerage fee? (37) What did he receive from the sale of the stock after he paid the brokerage fee? (38) Did he have a capital gain or loss? (39) How much was the gain or loss? (40) What was the net change from 16 3/8 to 17 5/8?
Step-by-step explanation:
How many palindromes greater than 10000 and less than 100000 are multiples of 18?
Let [tex]n=abcba[/tex] be such a number. If 18 divides [tex]n[/tex], then both 2 and 9 divide [tex]n[/tex].
To be divisible by 2, we must have [tex]a\in\{2,4,6,8\}[/tex]. Meanwhile we can have [tex](b,c)\in\{0,1,2,\ldots,9\}^2[/tex].
To be divisible by 9, we the sum of the digits of [tex]n[/tex] must itself be divisible by 9, or
[tex]2a+2b+c=9k[/tex]
for some integer [tex]k[/tex].
The largest value of [tex]2a+2b+c[/tex] is 2•8 + 2•9 + 9 = 43, so we must have [tex]k\in\{1,2,3,4\}[/tex].
I'm not sure what the best way to get the final count may be, but there are 44 such numbers. It's rather tedious to do by hand.
• If [tex]k=1[/tex], then [tex]2a+2b+c=9[/tex], and we can do this in 4 ways.
For example,
2•2 + 2•0 + 5 = 9 [tex](n = 20502)[/tex]
• If [tex]k=2[/tex], then [tex]2a+2b+c=18[/tex] and can be done in 16 ways.
2•2 + 2•3 + 8 = 18 [tex](n = 23832)[/tex]
• If [tex]k=3[/tex], then [tex]2a+2b+c=27[/tex] and can be done in 18 ways.
2•2 + 2•7 + 9 = 27 [tex](n = 27972)[/tex]
• If [tex]k=4[/tex], then [tex]2a+2b+c=36[/tex] and can be done in 6 ways.
2•6 + 2•8 + 8 = 36 [tex](n = 68886)[/tex]
Mark each statement as true or false. Suppose A is an n n matrix. a. If an n × n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable. False b. If A is diagonalizable, then A is also diagonalizable False c. If there is a basis of R n consisting of eigenvectors of A, then A is diagonalizable. True d. A is diagonalizable if and only if A has n eigenvalues, counting multiplicity. False e. If A is diagonalizable, then A is invertible. False
The correct option for the matrix will be:
FalseTrueTrueFalseFalseHow to explain the matrix?
a) If an n x n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable.
FalseIt could have repeated eigenvalues as long as the basis of each eigenspace is equal to the multiplicity of that eigenvalue.
b) If A is diagonalizable the A2 is diagonalizable
TrueIf A is diagonalizable then there exists an invertible matrix
c) If Rn has a basis of eigenvectors of A, then A is diagonalizable.
Trued) A is diagonalizable if and only if A has n eigenvalues, counting multiplicity.
Falsee) If A is diagonalizable, then A is invertible.
FalseIt’s invertible if it doesn’t have a zero as eigenvalue but this doesn’t affect diagonalizable.
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A rectangular closet has a perimeter of 18 feet and an area of 20 square feet. What are the dimensions of the closet.
Answer:
The width can either be 4 or 5 feet.
Step-by-step explanation:
Assuming the length is x and the width is y.2x+2y=18xy=202y=18−2xy=9−xx(9−x)=209x−x2=200=x2−9x+200=(x−5)(x−4)x=5and4 The width can either be 4 or 5 feet
i need help finding all of this stuff
Answer:
Axis Of Symmetry = -2
Vertex = (1,-2)
Min (0,-2)
Domain = All Real #'s
Range = Y is greater than or equal to -2
There are Z fish in a aquarium. 1/4 of the fish are angelfish. How may are angelfish?
Step-by-step explanation:
since there are z fishes
the number of angelfish = 1 × z
4
= 1 z
4
Select the correct answer from each drop-down menu.
The hexagonal seating section in an auditorium has a small entrance at the front. A walkway leading from the entrance to the front of the stage is
30 ft long. What is the approximate area of the entrance and seating section combined?
30 ft
Each side of the hexagon is about oft long. The area of the seating section is about
ft². The area of the entrance and seating section combines is about
ft².
ft². The area of the entrance is about
Each side of the hexagon is about 105.219 ft².
The area of the seating section is about 631.315 ft². The area of the entrance and seating section combines is about 46.764ft².The area of the entrance is about 678. 078ft².What is the hexagon about?Note that:
A F: 30 ft long.
A O = 1/2 A F = 3ft
Δ C J X = 0 X = 1/2 (A F - A O)
= 1/2 (30-3)
= 13.5ft
Sin 60° =[tex]\frac{0x}{cx}[/tex]
cx = [tex]\frac{ox}{sn 60}[/tex]
= 15.558ft = CJ (Sides)
Hence: Each side of the hexagon is: 1/2 CJ x OX = 105.219 ft².
The area of the seating section is:
6 ( Each side of the hexagon )
= 6 x 105.219 ft².
631.315 ft².
The area of the entrance and seating section combines is:
CJ x AO = 46.764ft².
The area of the entrance is:
631.315 ft²+ 46.764ft².
= 678. 078ft².
Hence, Each side of the hexagon is about 105.219 ft².
The area of the seating section is about 631.315 ft². The area of the entrance and seating section combines is about 46.764ft².The area of the entrance is about 678. 078ft².Learn more about hexagon from
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Teboho has to sell a certain number of cars each month. For every car he sells over and above this number he earns 17% commission on the price of the car before VAT is added. He sold three cars on which he earned commission. The VAT included prices were R271 800,00; R316 800,00 and R99 300,00. Calculate:
1.1 The prices of the cars before VAT was added
1.2 The commission he earned on the cars.
The commission earned are R41123.17, R47931.67 and R15024.09
The prices of the cars before VAT was addedThe VAT included prices are given as:
R271 800,00; R316 800,00 and R99 300,00.
The VAT rate is 12.36%.
So, the price before VAT is calculated as:
VAT included = Price without VAT * (1 + 12.36%)
This gives
VAT included = Price without VAT * 1.1236
So, we have:
Price without VAT = VAT included/1.1236
For the three cars, we have:
Price without VAT = R271 800.00/1.1236 = R241901
Price without VAT = R316800.00 /1.1236 = R281951
Price without VAT = R99300.00/1.1236 = R88377
Hence, the prices of the cars before VAT was added are R241901, R281951 and R88377
The commission he earned on the cars.This is calculated as:
Commission = Price * 17%
So, we have:
Commission = R241901 * 17% = R41123.17
Commission = R281951 * 17% = R47931.67
Commission = R88377 * 17% = R15024.09
Hence, the commission earned are R41123.17, R47931.67 and R15024.09
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x-15/3 = 18x. How to solve this question? I will mark the first answerer as brainliest.
Explain how you can find the constant of proportionality from a graph representing a proportional relationship when it shows a point with an x-value of 1 and if it doesn’t show an x-value of 1.
For a proportional relationship, the constant is found dividing all values of y by each respective value of x.
What is a proportional relationship?A proportional relationship is a function in which the output variable is given by the input variable multiplied by a constant of proportionality, that is:
y = kx
In which k is the constant of proportionality.
The constant can be represented as follows:
[tex]k = \frac{y}{x}[/tex]
Hence the constant is found dividing all values of y by each respective value of x.
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write [tex]sec^{2} x=\frac{1}{cos^2x}[/tex] in two equivalent forms.
write sec²x = 1/cos²x in two equivalent forms
sec²x = 1/cos²x
cos²x/cos²x = 1
sec²x = 1 + tan²x
The number of people in the world was 2.5 billion in 1950. The population has grown by 1.2% each year since then. Create a equation to model this situation using P to represent population (in billions) and y to represent years.
The equation which correctly models the situation in discuss is; P = 2.5(1.012)^y.
Which equation best models the situation?If follows from the task content that the situation given is that The number of people in the world was 2.5 billion in 1950.
Afterwards, The population has grown by 1.2% each year since then. Hence it follows that the function is an exponential one whose factor is; (100+1.2)%= 1.012.
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PLEASE HELP!!!! I don’t understand this topic please can someone help explain how to work out the attached question on similar area. Will mark brainliest!
Step-by-step explanation:
Look, you're taking these kinds of questions too serious and that's why you believe they're hard, now pay close attention:
the areas are defined in cm² but the radius is defined in cm, so you need find the positive root of cm².
[tex] \sqrt{ \frac{5500 {cm}^{2} }{220 {cm}^{2} } } = \frac{r}{5cm} = = > \\ 5 = \frac{r}{5} = = = > r = 25[/tex]
and the question wants the base area:
[tex]s = \pi {r}^{2} = = = > s = 3.14 \times {(25cm)}^{2} = = > 1963.5 {cm}^{2} [/tex]
A 12-ounce cup of juice contains 70 percent fruit and 30 percent water. If combined with 16 ounces of juice that contain 10 percent fruit and 90 percent water, what portion of the mixture is fruit?
Write the answer to the nearest hundredth
The portion of the mixture that is fruit to the nearest hundredth is 10.00 ounces
Percentage12 ounce juice:
Water = 30%Fruits = 70%= 70/100 × 12
= 0.7 × 12
= 8.4 ounces
Water = 30%
16 ounces;
Water = 90%Fruits = 10%= 10/100 × 16
= 0.1 × 16
= 1.6 ounces
Portion of mixture that is fruits = 8.4 ounces + 1.6 ounces
= 10.00 ounces
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What is 1 plus 1
Ok so what is 2 times 7
Alright what is 205 minus 306
Add all the answers together and multiply it by 5 then divide it by 1
In a bag of 10 marbles, there are 4 blue, 3 red, 2 green,
and 1 yellow. What is the probability that you draw one
marble that is red, replace it, and draw another marble that
is yellow?
a. 2
-
5
b. 3
100
C. 1
30
O d. 3
10
[tex]P(R \: then \: Y) = \frac{3}{10} \times \frac{1}{10} = \frac{3}{100} [/tex]
[tex]note \: that = \\ order \: matters \: (no \: factorial) \\ replacement \: (equal \: sample \: space)[/tex]
Option BAnswer:
c. 3/100
Step-by-step explanation:
there are 10 marbles
First draw (there are 3 red marbles)
[tex]P=\frac{3}{10}[/tex]
Second draw (there is one yellow marble)
[tex]P=\frac{1}{10}[/tex]
probability of the event:
[tex]P=(\frac{3}{10} )(\frac{1}{10} )=\frac{3}{100}[/tex]
Hope this helps
 What is the sum of the terms of the series 1+3+5+...+15
1+3+5+7+9+11+13+15
=4+12+20+28
=16+48
=64
A farmer uses a lot of fertilizer to grow his crops. The farmer's manager thinks fertilizer products from distributor A contain more of the nitrogen that his plants need than distributor B's fertilizer does. He takes two independent samples of four batches of fertilizer from each distributor and measures the amount of nitrogen in each batch. Fertilizer from distributor A contained 33 pounds per batch and fertilizer from distributor B contained 25 pounds per batch. Suppose the population standard deviation for distributor A and distributor B is three pounds per batch and four pounds per batch, respectively. Assume the distribution of nitrogen in fertilizer is normally distributed. Let µ1 and µ2 represent the average amount of nitrogen per batch for fertilizer A and B, respectively. Which of the following is the correct value of the test statistic?
Based on the calculations, the correct value of the test statistic is equal to 3.2.
How to calculate value of the test statistic?For samples A and B, the hypothesis is given by:
H₀: μ₁ ≤ μ₂
H₁: μ₁ > μ₂
Since both samples have a normal distribution, we would use a pooled z-test to determine the value of the test statistic:
[tex]z = \frac{\bar{x_1} - \bar{x_2} -(\mu_1 - \mu_2) }{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_1^2}{n_1}} }[/tex]
Substituting the given parameters into the formula, we have;
[tex]z = \frac{33 - 25 -(0) }{\sqrt{\frac{3^2}{4} + \frac{4^2}{4}} }\\\\z = \frac{8 }{\sqrt{\frac{9}{4} + \frac{16}{4}} }\\\\z = \frac{8 }{\sqrt{\frac{25}{4} } }\\\\z = \frac{8 }{\frac{5}{2} }}\\\\z = 8 \times \frac{2}{5}[/tex]
z = 3.2.
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A dairyman wishes to mix milk containing 5% butterfat and cream containing 75% butterfat to produce a total mixture of 56 liters. This final mixture should contain 53% butterfat. How much of the milk mixture and how much of the cream mixture should he use
18% of the milk mixture and how much of the cream mixture should he use.
How much of the milk mixture and how much of the cream mixture should he use?A dairyman wants to combine milk with 5 percent butterfat and cream with 75 percent butterfat to create a 56-liter combination. Butterfat should make up 53% of the final combination.
Let x represent the volume of MILK required for the combination, in liters.
Consequently, x/56 is the PROPORTION of milk in the mixture. [because the final mixture has 56 liters in total]
We require 56 - x liters of CREAM in the mixture because we have 56 liters overall in the mixture.
The PROPORTION of cream in the combination is therefore equal to (56 - x)/56.
Our goal is for the final mixture to have 75% butterfat.
Fill in the equation with each of these values to obtain:
50 = (x/60)(5) + ((60 - x)/60) (75)
Add 56 to both sides to get: 3000 = (5)(x) + (56 - x)(75)
The formula is:
Multiply both sides by 56 to get: 3000 = (5)(x) + (56 - x)(75)
Expand: 3000 = 5x + 4200 - 75x
Simplify: 3000 = 4200 - 70x
Subtract 4500 from both sides: -1300 = -70x
Solve: x = (-1300)/(-70) = (1300)/(70) = 130/7
If you don't want to divide 130 by 7, you can evaluate this quickly by first realizing that 30/7 = 18.
Consequently, 130/7 must be a little larger than 18.
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