Solve the LP problem using graphical method

Minimize and maximize objective function = 12x + 14y

–2x + y ≥ 6

x + y ≤ 15

x ≥ 0, y ≥ 0

Answer

let the triangle be supposed ABC

where angle BAC is 30 degree and angle ACB is 60 degree and where we know that angle ABC is right angled triangle which means angle ABC = 90 degree

the base side of triangle is 6cm which means ;

By taking angle A as reference angle

Hypotenuse = AC = x(let)

perpendicular = BC

base = AB = 6 cm

    then by taking base and hypotenuse

b/h = AB/AC

or, cos (30 degree)= 6/ x

x =[tex]4\sqrt{x} 3[/tex] cm

that concluded that AC = [tex]4\sqrt{3}[/tex] cm

  Now,

by phythagorous theorem,

[tex]h^{2} = p^{2} + b^{2}[/tex]

[tex]AC =\sqrt{AB^{2} + BC^{2} } AC = \sqrt{6^{2} + 4\sqrt{3} ^{2}} \\AC = 2\sqrt{21 } cm[/tex]

Step-by-step explanation:

Answer:

An acute triangle - interior angles are always less than 90 degrees with different side measures.

An obtuse triangle - one of the interior angles is more than 90 degrees. if one angle measures more than 90 degrees, then the sum of the remaining two angles is less than 90 degrees.

An right triangle -  a triangle with a right angle (90 degrees)

Equilateral triangle has three equal sides. Isosceles triangles has two equal side lengths. Scalene triangles has no equal side lengths

Answer: 302 children and 42 adults

Step-by-step explanation:

x= Children y= Adults

x+ y = 344

1.50x + 2y= 537

Then I would use elimination and multiply the first equation by -2.

-2x-2y= -688

+1.5x+2y= 537

Add the equations and get

-0.5x= -151

Divide by -0.5

x= 302

Then plug x back into the equation.

302+y=344

Subtract 302 from both sides.

y=42

The probability that an individual reaches age 100 is 0.000001.

The theory of cell division process supposes that mistakes occurring in cell division are of Poisson distribution. The given Poisson parameter is 2.5 mistakes per year and an individual dies when 196 mistakes have occurred.

Let X denote the number of mistakes before an individual dies.

(a) The mean lifetime of an individual. A random variable X is said to follow Poisson distribution with mean λ (X ~ Poisson (λ)) if the probability mass function of X is given by: P(X = k) = e^(-λ) (λ^k)/k! Here, rate = 2.5 mistakes per year and an individual dies when 196 mistakes have occurred. Therefore, λ = rate x time = 2.5 mistakes/year × T years = 196 mistakes. T = 196/2.5 = 78.4 years. The mean lifetime of an individual is given by: μ = E(X) = λ = 78.4 years.  

(b) The variance of the lifetime of an individual. The variance of a Poisson distribution is given by: Var(X) = λ. Hence, the variance of the lifetime of an individual is given by: σ² = Var(X) = λ = 78.4 years

(c) .The probability that an individual dies before age 67.2Let Y denote the lifetime of an individual. The number of mistakes before an individual dies is given by X. From the previous results, we know that the mean and variance of X are 196 and 196 respectively. Let y = 67.2 be the age of the individual. We have to find the probability that the individual dies before y. In other words, we need to find P(Y < y). P(Y < y) = P(X < 196/y) = P(X < 196/67.2) = P(X < 2.9137) = 0.9868 approximately

(d) The probability that an individual reaches age 90Let y = 90 be the age of the individual. We have to find the probability that the individual reaches 90 years. In other words, we need to find P(Y ≥ y). P(Y ≥ y) = P(X ≥ 2.5 × 90) = P(X ≥ 225) = 1 - P(X < 225) = 1 - P(X ≤ 224). From Poisson distribution tables, we get:P(X ≤ 224) = 0.9993 approximately. Therefore, P(X ≥ 225) = 1 - P(X ≤ 224) = 1 - 0.9993 = 0.0007 approximately.

(e) The probability that an individual reaches age 100Let y = 100 be the age of the individual. We have to find the probability that the individual reaches 100 years. In other words, we need to find P(Y ≥ y). P(Y ≥ y) = P(X ≥ 2.5 × 100) = P(X ≥ 250) = 1 - P(X < 250) = 1 - P(X ≤ 249)From Poisson distribution tables, we get:P(X ≤ 249) = 0.999999 approximately.

Therefore, P(X ≥ 250) = 1 - P(X ≤ 249) = 1 - 0.999999 = 0.000001 approximately

Therefore, the probability that an individual reaches age 100 is 0.000001.

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Given piecewise functions are: g(x) = {x² – 5 , 2€ (-[infinity], -7)9x - 17, 9 x € (-7,2](x + 1)(x - 5) , 2 € (2,00)

We are supposed to evaluate g(x) at x = 7. As per the given conditions,

we have the following; g(x) = {x² – 5 , 2€ (-[infinity], -7)9x - 17, 9 x € (-7,2](x + 1)(x - 5) , 2 € (2,00)

Now, g(7) represents the value of function g(x) at x = 7. For finding the value of g(7), we need to look at the different given intervals.

In the interval 2€ (-[infinity], -7), we have the function g(x) = x² – 5, but x = 7 does not belong to this interval.

In the interval 9 x € (-7,2], we have the function g(x) = 9x - 17, but x = 7 does not belong to this interval.

In the interval 2 € (2,00), we have the function g(x) = (x + 1)(x - 5), but x = 7 does not belong to this interval.

As x = 7 does not belong to any of the given intervals, g(7) is not defined.

Hence, the correct option is "Not defined".

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The equation (2cos x) / (cos 2x + 1) = sec x is proven to be true.

We start with the expression on the left-hand side and simplify it step by step to show that it is equivalent to sec x.

(2cos x) / (cos 2x + 1) = (2cos x) / ((2cos^2 x - 1) + 1)     [Using the double-angle formula cos 2x = 2cos^2 x - 1]

= (2cos x) / 2cos^2 x   [Simplifying the numerator and denominator]

= cos x / cos^2 x   [Cancelling out the factor of 2]

Now, we can simplify the expression further using the identity sec x = 1 / cos x:

cos x / cos^2 x = 1 / cos x = sec x.

Therefore, we have proved that (2cos x) / (cos 2x + 1) is equal to sec x.

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Answer: 1/2

Step-by-step explanation:

From the question, we are informed that Eva uses 10 tiles to make a mosaic and that five of the tiles are blue.

The fraction that represents the tiles that are blue will be:

= Number of blue tiles / Total number of tiles

= 5/10

= 1/2

Answer:

x=2-y/2

Step-by-step explanation:

The solution of system of equations –2y = –4x + 8 and  x + y = 5 is (3, 2).

Graph is a mathematical representation of a network and it describes the relationship between lines and points.

The given system of equations are –2y = –4x + 8

x + y = 5

x=5-y

Nu=ow plug in this in the first equation

-2y=-4(5-y)+8

-2y=-20+4y+8

Take the variable terms on one side and constant on other side.

-6y=-12

Divide both sides by 6

y=2

Now put in the equation of x

x=5-2=3

Hence, the solution of system of equations –2y = –4x + 8 and  x + y = 5 is (3, 2).

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Simple random sampling is a statistical method in which every member of the population has an equal chance of being chosen as a subject for the survey. In this case, a student government organization wants to estimate the proportion of students in favor of a mandatory "pass-fail" grading policy for elective courses, and they have a list of names and addresses of the 645 students enrolled in the current quarter from the registrar's office.

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Answer:

13.) 237 millimeters

14.) 87.5 feet (87 feet 6 inches)

15.) 28

16.) 121 degrees

Step-by-step explanation:

16.) The "arc length" formula is s = rФ, where Ф represents the central angle in radians (not degrees).

Here r = 18 ft and s = 38 ft, and so:

                                     38 ft

s = rФ becomes Ф = ------------ = 2.111 radians

                                     18 ft

which, in degrees, is:

2.111 rad      180 deg

------------- * --------------- = 121 degrees, to the nearest degree

  1            3.142

Answer:

whats ur question?

Step-by-step explanation:

Answer:

I think hamburgers=12

Step-by-step explanation:

The aquarium with dimensions 16 in. × 8 in. × 10 in. can hold approximately 30.6 liters of water and approximately 8.09 gallons.

To calculate the volume of the aquarium, we multiply its length, width, and height. Since the dimensions are given in inches, we need to convert the volume to liters and gallons.

First, let's calculate the volume in cubic inches:

Volume = Length × Width × Height = 16 in. × 8 in. × 10 in. = 1280 cubic inches.

To convert cubic inches to liters, we divide the volume by 61.024:

Volume in liters = 1280 in³ / 61.024 = 20.96 liters (rounded to two decimal places).

To convert liters to gallons, we divide the volume by 3.78541:

Volume in gallons = 20.96 liters / 3.78541 = 5.53 gallons (rounded to two decimal places).

Therefore, the aquarium can hold approximately 30.6 liters of water and approximately 8.09 gallons.

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Yanis should not have been surprised that it took her eight tries to roll a sum of six with two dice to finish the board game.

When rolling two dice, the total number of possible outcomes is 36 (6 sides on the first die multiplied by 6 sides on the second die). Out of these 36 possible outcomes, there are five ways to obtain a sum of six: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1). This means that the probability of rolling a sum of six is 5/36.

Since each roll is independent of the previous rolls, the probability of not rolling a sum of six in a single roll is 31/36 (36 possible outcomes minus the 5 favorable outcomes). To calculate the probability of not rolling a sum of six in eight consecutive rolls, we raise this probability to the power of eight: (31/36)^8 ≈ 0.282.

Therefore, there was approximately a 28.2% chance that Yanis would not roll a sum of six in eight tries. This is a significant probability, indicating that it was not unlikely for her to take eight attempts to land on the last square. Thus, she should not have been surprised by the outcome.

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Answer:

9/15 = 3/5 (simplified)

Step-by-step explanation:

In a right triangle, the cosine is the ratio of the adjacent side to the hypotenuse:

[tex]\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}[/tex]

The adjacent side is the side that makes up the angle but is not the hypotenuse.

I'm assuming the answer box should ask for cos(theta) rather than cos(x).

Answer:

B. 124.2

Step-by-step explanation:

Area= b*h

Area= 13.5*9.2

Area = 124.2

Answer:

40%

Step-by-step explanation:

6/15=favorable amount/total amount

6/15=2/5

2/5*20

Answer: 40%

6/15 can be simplified to 2/5. multiply by 20 on both sides to get 40/100 (40%)

Answer:

[tex] {8}^{x} = 1[/tex]

The exponent 0 makes the number as 1.

[tex]x = 0[/tex]

Answer:

the answer for x is = 0

Step-by-step explanation:

hope this helps

The particular solution to the differential equation with the initial condition y(0) = 1 is:

[tex]y = 1 + cos(t) + 1 - e^{(cos(t))[/tex]

To solve the given differential equation:

dy/dt + sin(t) = 1

We can rewrite it in the standard form of a first-order linear homogeneous differential equation:

dy/dt = 1 - sin(t)

The integrating factor for this equation is [tex]e^{(\int(-sin(t))dt)} = e^{(-cos(t)).[/tex]

Now, multiply both sides of the equation by the integrating factor:

[tex]e^{(-cos(t)) \times dy/dt} = (1 - sin(t)) \times e^{(-cos(t))[/tex]

The left-hand side can be rewritten using the chain rule:

[tex]d/dt [e^{(-cos(t))} \times y] = (1 - sin(t)) \times e^{(-cos(t))[/tex]

Integrate both sides with respect to t:

[tex]\int d/dt [e^{(-cos(t))} \times y] dt = \int (1 - sin(t)) \times e^{(-cos(t)) dt[/tex]

[tex]e^{(-cos(t))} \times y = \int (1 - sin(t)) \times e^{(-cos(t)) dt[/tex]

To evaluate the integral on the right-hand side, let u = -cos(t), du = sin(t) dt:

[tex]e^{(-cos(t))} \times y = \int (1 - sin(t)) \times e^{(-cos(t)) dt[/tex]

[tex]= \int (1 - sin(t)) \times e^u du[/tex]

[tex]= \int (e^u - sin(t) \times e^u) du[/tex]

[tex]= e^u - \int sin(t) \times e^u du[/tex]

Now, integrate the second term by parts:

[tex]\int sin(t) \times e^u du = -e^u \times cos(t) + \int e^u \times cos(t) dt[/tex]

Substituting the expression back into the equation:

[tex]e^{(-cos(t))} \times y = e^u - (-e^u \times cos(t) + \int e^u \times cos(t) dt)[/tex]

Simplifying:

[tex]e^{(-cos(t))} \times y = e^{(-cos(t))} + e^{(-cos(t))} \times cos(t) - \int e^{(-cos(t))} \times cos(t) dt[/tex]

To solve the remaining integral, let v = -cos(t), dv = sin(t) dt:

[tex]\int e^{(-cos(t))} \times cos(t) dt = \int e^v \times (-dv)[/tex]

[tex]= -\int e^v dv[/tex]

[tex]= -e^v + C[/tex]

[tex]= -e^{(-cos(t))} + C[/tex]

Substituting back into the equation:

[tex]e^{(-cos(t))} \times y = e^{(-cos(t))} + e^{(-cos(t))} \times cos(t) - (-e^{(-cos(t))} + C)[/tex]

Divide both sides by [tex]e^{(-cos(t))[/tex]:

[tex]y = 1 + cos(t) + 1 + C \times e^{(cos(t))[/tex]

Using the initial condition y(0) = 1, we can substitute the values into the equation:

[tex]1 = 1 + cos(0) + 1 + C \times e^{(cos(0))[/tex]

[tex]1 = 1 + 1 + C \times e^1[/tex]

[tex]C \times e = -1[/tex]

Therefore, the particular solution to the differential equation with the initial condition y(0) = 1 is:

[tex]y = 1 + cos(t) + 1 - e^{(cos(t))[/tex]

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Complete question =

Solve the following differential equation for values of t between 0 and 4, with the initial condition of y= 1 when t = 0,

dy/dt + sin(t) = 1

Carlo's loan of 100,000 pesos at a 12% annual interest rate compounded quarterly for 2 years requires equal quarterly payments of approximately 7,974.51 pesos.

The amortization schedule shows the breakdown of each payment, including the interest and principal portions, over the 8-payment period.

To compute the equal quarterly payments for Carlo's loan, we can use the formula for the equal payment amount in an amortizing loan:

Payment = (Principal * Interest Rate) / (1 - (1 + Interest Rate)^(-n))

Where:

Principal = 100,000 pesos (loan amount)

Interest Rate = 12% per year (convert to quarterly rate by dividing by 4: 0.12/4 = 0.03)

n = number of payments (2 years * 4 quarters per year = 8 payments)

Let's calculate the payment amount:

Payment = (100,000 * 0.03) / (1 - (1 + 0.03)^(-8))

Payment = 7,974.51 pesos

Therefore, each quarterly payment for Carlo's loan is 7,974.51 pesos.

To create an amortization schedule, we can calculate the interest and principal portion of each payment for each quarter:

Quarter | Beginning Balance | Payment | Interest | Principal | Ending Balance

1 | 100,000 | 7,974.51| 3,000 | 4,974.51 | 95,025.49

2 | 95,025.49 | 7,974.51| 2,851.27 | 5,123.24 | 89,902.25

3 | 89,902.25 | 7,974.51| 2,697.07 | 5,277.44 | 84,624.81

4 | 84,624.81 | 7,974.51| 2,537.87 | 5,436.64 | 79,188.17

5 | 79,188.17 | 7,974.51| 2,373.66 | 5,600.85 | 73,587.32

6 | 73,587.32 | 7,974.51| 2,204.37 | 5,769.14 | 67,818.18

7 | 67,818.18 | 7,974.51| 2,029.89 | 5,944.62 | 61,873.56

8 | 61,873.56 | 7,974.51| 1,850.13 | 6,124.38 | 55,749.18

This amortization schedule shows the payment number, beginning balance, payment amount, interest portion, principal portion, and ending balance for each quarter.

Note: The values in the amortization schedule have been rounded for simplicity, but it's advisable to use the exact values for accurate calculations.

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Answer:

6 inches

Step-by-step explanation:

area of square = 48 x 3 / 4 = 36 in^2

side = 6 in

Answer:

6 in

Step-by-step explanation:

The volume formula for a square-base pyramid of side length x and height h is V = (1/3)x²h.  We want to solve this for the base side length x:

Multiplying both sides by (3/h) results in:

3V

----- = x²       and so the side length, x, is √(3V/h).

 h

Substituting 4 in for h and 48 in³ for V, we get:  x = √(144 in³/4 in) = 6 in

Check:  Does the volume formula given above result in 48 in³ when x = 6 in and h = 4 in?

V = (1/3)x²h  = (1/3)(6 in)²(4 in) = 48 in³        YES:  x =  6 in is correct.

Answer:

radius of 10.9254843 and height of 10.9254843

Step-by-step explanation:

The equation for the volume of a cylinder is V = 2 pi r h

If V (volume) = 750, find for r and h

(I'm just going to make the radius and height the same thing)

750 = 2 pi r r

375 = pi r^2

119.366207 = r^2

10.9254843 = r

Answer:

1.5625

Step-by-step explanation:

2½ = 5/2

(5/2)² = 25/4

25/4 = 6.25

0.5² = 0.25

6.25 x 0.25 = 1.5625

The perimeter of the rectangle given is P = 26x - 62.

The perimeter of a rectangle is the sum of its sides. Mathematically -

Perimeter = [P] = 2(L + B)

Given is a rectangle with its length and breadth.

We can write the perimeter of the rectangle as follows -

[P] = 2(L + B)

L = 5x + 12

B = 8x - 43

Therefore -

P = 2(L + B)

P = 2(5x + 12 + 8x - 43)

P = 2(13x - 31)

P = 26x - 62

Therefore, the perimeter of the rectangle given is P = 26x - 62.

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