In this case, the point (3/2, 0) is the optimal solution, which matches the solution we found using the simplex method.
To solve the problem using the simplex method, we first need to convert the inequalities to equations by adding slack variables. Then we can set up the initial tableau and use the simplex method to find the optimal solution. Finally, we can verify the solution using the graphical method.
Convert inequalities to equations by adding slack variables
-X1 + X2 >= -1 becomes -X1 + X2 + S1 = -1
4X1 + 3X2 <= 12 becomes 4X1 + 3X2 + S2 = 12
2X1 <= 3 becomes 2X1 + S3 = 3
Set up the initial tableau
| -1 | 2 | 1 | 0 | 0 | 0 |
| -1 | 1 | 1 | 0 | 0 | -1 |
| 4 | 3 | 0 | 1 | 0 | 12 |
| 2 | 0 | 0 | 0 | 1 | 3 |
Use the simplex method to find the optimal solution
We first need to choose the pivot column, which is the one with the most negative coefficient in the objective row. In this case, it is the first column. Then we need to choose the pivot row, which is the one with the smallest positive ratio of the right-hand side to the pivot column coefficient. In this case, it is the third row. We can then use the pivot element to eliminate the other coefficients in the pivot column and repeat the process until we have no negative coefficients in the objective row.
After performing the simplex method, we get the following final tableau:
| 0 | 5/4 | 1 | 1/4 | 0 | 3 |
| 0 | 7/4 | 1 | 1/4 | 0 | 3 |
| 0 | 3/4 | 0 | 1/4 | 0 | 3 |
| 1 | 0 | 0 | 0 | 1/2 | 3/2 |
The optimal solution is X1 = 3/2, X2 = 0, and Z = -3/2.
Verify the solution using the graphical method
We can graph the constraints and find the feasible region. Then we can graph the objective function and find the point where it intersects the feasible region that gives the maximum or minimum value. In this case, the point (3/2, 0) is the optimal solution, which matches the solution we found using the simplex method.
Extra Credit:
To solve the problem using the simplex method, we first need to convert the inequalities to equations by adding slack variables. Then we can set up the initial tableau and use the simplex method to find the optimal solution. Finally, we can verify the solution using the graphical method.
Convert inequalities to equations by adding slack variables
-X1 + X2 >= -1 becomes -X1 + X2 + S1 = -1
4X1 + 3X2 <= 12 becomes 4X1 + 3X2 + S2 = 12
2X1 <= 3 becomes 2X1 + S3 = 3
Set up the initial tableau
| -1 | 2 | 1 | 0 | 0 | 0 |
| -1 | 1 | 1 | 0 | 0 | -1 |
| 4 | 3 | 0 | 1 | 0 | 12 |
| 2 | 0 | 0 | 0 | 1 | 3 |
Use the simplex method to find the optimal solution
We first need to choose the pivot column, which is the one with the most negative coefficient in the objective row. In this case, it is the first column. Then we need to choose the pivot row, which is the one with the smallest positive ratio of the right-hand side to the pivot column coefficient. In this case, it is the third row. We can then use the pivot element to eliminate the other coefficients in the pivot column and repeat the process until we have no negative coefficients in the objective row.
After performing the simplex method, we get the following final tableau:
| 0 | 5/4 | 1 | 1/4 | 0 | 3 |
| 0 | 7/4 | 1 | 1/4 | 0 | 3 |
| 0 | 3/4 | 0 | 1/4 | 0 | 3 |
| 1 | 0 | 0 | 0 | 1/2 | 3/2 |
The optimal solution is X1 = 3/2, X2 = 0, and Z = -3/2.
Verify the solution using the graphical method
We can graph the constraints and find the feasible region. Then we can graph the objective function and find the point where it intersects the feasible region that gives the maximum or minimum value. In this case, the point (3/2, 0) is the optimal solution, which matches the solution we found using the simplex method.
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On 1.2.2011, Brown drew a bill for three months on black for Rs. 6,000 and received it duly accepted. On 3.2.2011 Brown discounted the bill for 6%. On the due date black paid money for his bill. Show the journal entries in the books of both the persons.
Dr. Bills Payable A/C 5,400 and Cr. Bank A/C 5,400
In the books of Brown:
Dr. Bank A/C 6,000
Cr. Bills Receivable A/C 6,000
On 3.2.2011:
Dr. Bills Receivable A/C 5,400
Cr. Bank A/C 5,400
On the due date:
Dr. Bills Receivable A/C 5,400
Cr. Black A/C 5,400
In the books of Black:
Dr. Bills Payable A/C 5,400
Cr. Bank A/C 5,400
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Find the mode for the following variables. 17,7,6,3,17,17,4,3,9,5,17,17
The mode of the set of variables "17,7,6,3,17,17,4,3,9,5,17,17" is 17 .
The Mode of a data set is defined as the value that appears most frequently in the dataset. The mode is one of the measures of central tendency, along with the mean and median.
The mode is useful in describing the most common value in a dataset, and can be specially helpful when dealing with categorical or discrete data.
The mode can be more than one value or may not exist at all in some datasets.
From the given set of variables "17,7,6,3,17,17,4,3,9,5,17,17",
The number 17 appears 6times, which is more than any other number in the dataset.
Therefore, the mode is 17.
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Let x is equal to the number of heads observed. x is what we called rar P(x=2)=(1)/(4) ,P(x)=(1)=(2)/(4)
The probability of observing 2 heads is 1/4, the probability of observing 1 head is 2/4, and the probability of observing 0 heads is 1/4.
The question is asking what the probability of getting two heads (x=2) and one head (x=1) when tossing a coin.
The probability of getting two heads is P(x=2) = 1/4 and the probability of getting one head is P(x=1) = 2/4.
It is important to remember that the sum of all the probabilities in an experiment must equal 1.
In this case, P(x=2) + P(x=1) = 1/4 + 2/4 = 3/4. This means that the probability of observing 0 heads, represented as P(x=0), must be equal to 1/4 in order for the sum of all the probabilities to equal 1.
In conclusion, the variable x is used to represent the number of heads observed in a coin toss experiment, and the probabilities of observing different numbers of heads are given as fractions.
The probability of observing 2 heads is 1/4, the probability of observing 1 head is 2/4, and the probability of observing 0 heads is 1/4.
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The table for the quadratic functions f(x) and g(x) are given. x f(x) g(x) −2 4 8 −1 1 2 0 0 0 1 1 2 2 4 8 Determine the type of transformation and the value of k.
g(x) = f(2x)
g(x) = 2f(x)
g of x equals f of one half times x
g of x equals one half times f of x
please ASAP!!
Using the scale factors obtained the values are -
The value of k is 8, since g(−2) = 8.
The value of k is 8, since g(−2) = 8.
The value of k is 8, since g(−2) = 1.
The value of k is 4, since g(−2) = 2.
What is scale factor?
The scale factor of a shape refers to the amount by which it is increased or shrunk. It is applied when a 2D shape, such as a circle, triangle, square, or rectangle, needs to be made larger.
For each part, we can use the given table to determine how the transformation affects the function values -
g(x) = f(2x)
This transformation is a horizontal compression by a factor of 2.
To see this, notice that when we evaluate g at x = -1, we get the same value as f evaluated at x = -2.
Similarly, when we evaluate g at x = 0, we get the same value as f evaluated at x = 0.
And so on. In other words, the function values of g(x) are the same as the function values of f(x) at every other point (starting with x = -2).
So, g(x) is a compressed version of f(x) horizontally by a factor of 2.
Therefore, the value of k is 8, since g(−2) = 8 = f(2×(-2)).
g(x) = 2f(x)
This transformation is a vertical stretch by a factor of 2.
To see this, notice that every value of g(x) is twice the corresponding value of f(x).
So, g(x) is a stretched version of f(x) vertically by a factor of 2.
Therefore, the value of k is 8, since g(−2) = 2f(−2) = 2×4 = 8.
g(x) = f(x/2)
This transformation is a horizontal stretch by a factor of 2.
To see this, notice that when we evaluate g at x = -2, we get the same value as f evaluated at x = -4.
Similarly, when we evaluate g at x = -1, we get the same value as f evaluated at x = -2.
And so on. In other words, the function values of g(x) are the same as the function values of f(x) at every other point (starting with x = -4).
So, g(x) is a stretched version of f(x) horizontally by a factor of 2.
Therefore, the value of k is 8, since g(−2) = f(−1) = 1.
g(x) = (1/2)f(x)
This transformation is a vertical compression by a factor of 2.
To see this, notice that every value of g(x) is half the corresponding value of f(x).
So, g(x) is a compressed version of f(x) vertically by a factor of 2.
Therefore, the value of k is 4, since g(−2) = (1/2)f(−2) = (1/2)×4 = 2.
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The yearly income of a family is Rs. 500000. The ratio of the expenditure and saving of the family is 4 : 1. Find the amount of expenditure and saving.
Answer:
Let's assume that the amount of saving is x.
According to the problem, the ratio of expenditure to saving is 4:1, so the amount of expenditure can be expressed as 4x.
The total income of the family is Rs. 500000, and it can be expressed as the sum of expenditure and saving:
Expenditure + Saving = 500000
Substituting the values of expenditure and saving, we get:
4x + x = 500000
Simplifying this equation, we get:
5x = 500000
Dividing both sides by 5, we get:
x = 100000
Therefore, the amount of saving is Rs. 100000, and the amount of expenditure is 4 times this value, which is Rs. 400000.
who knows, What is the area of a cross section that passes through the center of a sphere with a diameter of 7 centimeters? Express your answer in terms of π.
The area of a cross section that passes through the center of a sphere with a diameter of 7 centimeters will be 12.25π cm².
What is A sphere ?Three-dimensional objects with a sphere-like shape exist in all three dimensions. Three axes, the x-axis, y-axis, and z-axis, are used to define the sphere. The key distinction between a circle and a sphere is this. In contrast to other 3D shapes, a sphere lacks any vertices or edges.
The sphere's points are evenly spaced apart from one another on its surface. In light of this, the sphere's surface and core are always equally separated from one another. The sphere's radius is the measurement between these points. Ball, globe, planets, and other objects are all examples of spheres.
Given : Diameter of Sphere = 7 cm
∴ Radius = 7/2 = 3.5 cm.
Since It is a cross section in a sphere, so, it would be circle.
So, Area of the cross section = Area of circle
Hence, Area of Cross section = πr²
= π (3.5)²
= 12.25π cm²
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HELP THIS IS DUE TOMMOROW USE ANY STRATEGIE
Answer: what is ur questions?
Step-by-step explanation:
6/27 = 4/x
Find the answer, hint- 6x = 27x4
then divide 6 by 27x4
Answer: 18
Step-by-step explanation
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 27x, the least common multiple of 27,x.
x × 6=27 × 4
Multiply 27 and 4 to get 108.
x × 6=108
Divide both sides by 6.
x= 108/6
Divide 108 by 6 to get 18.
x=18
Can someone answer this
Answer:
16
Step-by-step explanation:
g(x)=-5x+1
g(-3)
x=-3
-5(-3) +1
15+1
16
Answer:
See below.
Step-by-step explanation:
For this problem, we are asked to find the value of g(-3).
We are given a Linear Function.
What is a Linear Function?A Linear Function is a Polynomial Function that is commonly graphed. This function most of the time will simply be represented as a straight line when graphed.
For this problem;
[tex]g(x)=-5x+1 \ Find \ g(-3).[/tex]
We simply need to substitute -3 in for x.
[tex]g(-3)=-5(-3)+1[/tex]
Simplify:
[tex]g(-3)=16.[/tex]
Our final answer is g(-3) = 16.
3. The bottom of a ramp is 15 feet from the entrance of a building. If the angle of the ramp is
about 3.2°, about how high does the ramp rise off the ground to the nearest inch?
The ramp rises about 10.4 inches off the ground to the nearest inch.
What is ramp ?A ramp is an inclined plane that connects two different levels, allowing people or objects to move between them with less force than if they had to be lifted vertically.
We can use trigonometry to solve this problem. Let's call the height of the ramp "h". Then we have:
tan(3.2°) = h/15
To solve for h, we can multiply both sides by 15:
h = 15 tan(3.2°)
Using a calculator, we get:
h ≈ 0.87 feet
To convert this to inches, we can multiply by 12:
h ≈ 10.4 inches
So, the ramp rises about 10.4 inches off the ground to the nearest inch.
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Use the following functions to: find, simplify, and identify the domain of each of the function combinations. f(x)=x^2−4x and g(x)=x+12 (a) (f+g)(x)=
Domain of (f+g)(x) : (b) (f−g)(x)= Domain of (f−g)(x) : (c) (fg)(x)= Domain of (fg)(x) : (d) (gf)(x)= Domain of (gf)(x) :
(a) The function combination (f + g)(x) is equal to x² - 3x + 12 and its domain are all real numbers.
(b) The function combination (f - g)(x) is equal to x² - 5x - 12 and its domain are all real numbers.
(c) The function combination (fg)(x) is equal to x³ + 8x² - 48x and its domain are all real numbers.
(d) The function combination (gf)(x) is equal to x³ + 8x² - 48x and its domain are all real numbers.
(a) To find the expression or value of the function combination (f + g)(x), we need to add f(x) and g(x).
(f + g)(x) = f(x) + g(x) = x² - 4x + x + 12 = x² - 3x + 12
Domain of (f + g)(x) : All real numbers
(b) To find the expression or value of the function combination (f - g)(x), we need to subtract g(x) from f(x).
(f - g)(x) = f(x) - g(x) = x² - 4x - (x + 12) = x² - 5x - 12
Domain of (f-g)(x) : All real numbers
(c) To find the expression or value of the function combination (fg)(x), we need to multiply the two functions.
(fg)(x) = f(x) * g(x) = (x² - 4x) (x + 12) = x³ + 8x² - 48x
Domain of (fg)(x) : All real numbers
(d) The function combination (gf)(x) is the same as (fg)(x).
(gf)(x) = g(x) * f(x) = (x + 12) (x² - 4x) = x³ + 8x² - 48x
Domain of (gf)(x) : All real numbers
In all cases, the domain is all real numbers because there are no restrictions on the values of x that can be used in the functions.
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A food company conducted a survey and found that 4 out of 20 people had french toast for
breakfast yesterday.
What is the probability that a randomly selected person had french toast for breakfast?
The probability that a randomly selected person had French toast for breakfast is 20%.
What is the probability?Probability describes the result of a random event based on the expected successes or outcomes.
Probability is computed as the quotient of the expected outcomes, events, or successes out of many possible outcomes, events, or successes.
Probability values lie between zero and one based on the degree of certainty or otherwise and can be depicted as percentages, decimals, or fractions.
The total number of survey participants = 20
The number of participants found to be having French toast for breakfast = 4
The probability of selecting a person having French toast for breakfast = 20%,or 0.2, or 1/5 (4/20 x 100)
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The star-spangled banner that flew over fort mchenry during the war of 1812 had a perimeter of 144 ft. Its length measured 12 ft more than its width. Use a system of equations to find the dimensions of this flag, which is displayed in the Smithsonian Institution's Museum of America History in washington, D. C
If the length of the flag is 12 ft more than its width, then the width of the flag is 30ft and length of the flag is 42ft.
Let "w" represent the width of the flag.
We know that the length of the flag is 12 feet more than its width, which means:
⇒ Length = w + 12
We also know that the perimeter of the flag is 144 feet,
The Perimeter of flag can be expressed as:
⇒ Perimeter = 2(Length + Width),
Substituting the expression for Length from above,
We get,
⇒ 144 = 2(w+12 + w)
Simplifying the equation:
We get,
⇒ 144 = 2(2w + 12)
⇒ 72 = 2w + 12
⇒ 60 = 2w
⇒ w = 30
So, width of flag is 30 feet. and Length = w + 12,
⇒ Length = 30 + 12
⇒ Length = 42
Therefore, the dimensions of the flag are 42feet by 30feet.
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HELP ME OUT IM CONFUSED
Math part 4 question 8
The function is decreasing in the interval (3, ∞).
Explain about the decreasing function?You must first compute the derivative, then make it equal to 0, and then determine whether zero values your function is negative between in order to determine whether a function is decreasing. In order to determine once the function is negative and, consequently, decreasing, test values from all sides of these.f(x) = -x² + 6x - 4
Differentiate the equation with respect to 'x'.
f'(x) = -2x + 6
Put f'(x) = 0
-2x + 6 = 0
x = 6/2
x = 3 (critical point)
Now, write the function as:
f(x) = -x² + 6x - 4
-(x² - 6x + 4) (negative form)
Thus, the function is decreasing in the interval (3, ∞)
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Which point has coordinates (4.9, 3.9)?
Answer:
C
Step-by-step explanation:
Answer: c.
Step-by-step explanation:
Sheng has two credit cards. He used his credit card statements to make the summary table that follows. Sheng is also eligible for a debt consolidation loan with an APR of 10.7% that reduces his monthly payments to $179.37. This loan will take six years to pay off, and Sheng will pay a total of $3,414.39 in interest charges. How much more in interest charges will the debt consolidation loan cost Sheng?
The debt consolidation loan will cost Sheng an additional $1,997.69 in interest charges compared to his credit cards.
Calculating how much more in interest the debt consolidation loan will cost ShengFrom question, we are to calculate how much more in interest the debt consolidation loan will cost Sheng
To calculate the interest charges for Sheng's credit cards, we need to use the following formula:
Interest Charges = Balance x APR x Number of Days in Billing Cycle / 365
Using this formula, we can calculate the interest charges for each of Sheng's credit cards:
For Credit Card A:
Interest Charges = $2,500 x 0.19 x 30 / 365 = $4.94
For Credit Card B:
Interest Charges = $3,000 x 0.22 x 30 / 365 = $6.01
So, the total interest charges for Sheng's credit cards are:
Total Interest Charges = $4.94 + $6.01 = $10.95
Now, let's calculate the total cost of the debt consolidation loan:
Total Cost = Total Monthly Payments x Number of Payments - Loan Amount
Total Monthly Payments = $179.37
Number of Payments = 6 x 12 = 72
Loan Amount = ?
To find the loan amount, we need to solve for it using the interest rate and the total interest charges:
Loan Amount = Total Interest Charges / (APR x Number of Payments / 12)
Loan Amount = $3,414.39 / (0.107 x 72 / 12) = $25,000
So, the total cost of the debt consolidation loan is:
Total Cost = $179.37 x 72 - $25,000 = $2,008.64
Therefore, the additional interest charges for the debt consolidation loan compared to Sheng's credit cards are:
Additional Interest Charges = Total Cost - Total Interest Charges
Additional Interest Charges = $2,008.64 - $10.95 = $1,997.69
Hence, the debt consolidation loan will cost him an additional $1,997.69 in interest charges
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I need some help please
Step-by-step explanation:
Option A is the right answer.
Exam 1S22: Problem 5 Previous Problem Problem List Next Problem (8 points) Solve the following inequality. Write the answer in interval notation. x(x-7) x2 - 5x – 50 SO - Answer: Preview My Answers
In interval notation, the solution of the inequality is (25, ∞).
To solve the inequality x(x-7) < x^2 - 5x - 50, we can rearrange the terms and factor the quadratic expression:
x^2 - 7x < x^2 - 5x - 50
-2x < -50
x > 25
In interval notation, the solution is (25, ∞).
So, the answer is (25, ∞).
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Find the average rate of change of f(x)=x^2+3x+1 from x=−5 to x=−3. Simplify your answer as much as possible.
The average rate of change of f(x)=x^2+3x+1 from x=−5 to x=−3 is −5.
The average rate of change of a function f(x) over an interval [a,b] is given by the formula:
average rate of change = (f(b) - f(a)) / (b - a)
In this case, we are given the function f(x)=x^2+3x+1 and the interval [−5,−3], so we can plug in the values into the formula:
average rate of change = (f(−3) - f(−5)) / (−3 - (−5))
First, we need to find the values of f(−3) and f(−5):
f(−3) = (−3)^2 + 3(−3) + 1 = 9 − 9 + 1 = 1
f(−5) = (−5)^2 + 3(−5) + 1 = 25 − 15 + 1 = 11
Now, we can plug these values back into the formula:
average rate of change = (1 - 11) / (−3 - (−5)) = (−10) / 2 = −5
Therefore, the average rate of change of f(x)=x^2+3x+1 from x=−5 to x=−3 is −5.
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3. Solve by factoring the equation 3x2 – 12x – 15 = 0 and explain what your solutions mean for the equation. Show your work.
The equation 3x² - 12x - 15 = 0 when solved by factoring has the solutions x = 5 and x = -1
How to get to the equation's solutionThe equation 3x² - 12x - 15 = 0, is a representation of the equation in the question.
The aforementioned problem is a quadratic equation, and factoring is one way to solve this kind of issue.
Divide 3 from the equation.
As a result, we have the following
x² - 4x - 5 = 0.
Expanding the equation, we get
x² + x - 5x - 5 = 0.
Factoring the equation, we get
x(x + 1) - 5(x + 1) = 0
The result is
(x - 5)(x + 1) = 0.
We can solve for x by getting
x = 5 and x = -1.
Thus, the equation's answers are 5 and -1.
And it means that there are actually two solutions to the equation.
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What is the measure of z?
16
y
X
z = 4√ [?]
Give your answer in simplest form.
when running a line, in a right-triangle, from the 90° angle perpendicular to its opposite side, we will end up with three similar triangles, one Small, one Medium and a containing Large one. Check the picture below.
[tex]\cfrac{z}{20}=\cfrac{4}{z}\implies z^2=4(20)\implies z^2=4(4\cdot 5)\implies z^2=4^2\cdot 5 \\\\\\ z=\sqrt{4^2\cdot 5}\implies z=4\sqrt{5}[/tex]
Write a recursive formula for an, the nth term of the sequence 2, 6, 10, 14, ....
The recursive formula for an, the nth term of the sequence is a(n) = a(n - 1) + 2 where a(1) = 2
How to determine the recursive formula of the sequenceFrom the question, we have the following parameters that can be used in our computation:
2, 6, 10, 14, ....
The above definitions imply that we simply add 4 to the previous term to get the current term
Using the above as a guide,
So, we have the following representation
a(n) = a(n - 1) + 2
Hence, the sequence is a(n) = a(n - 1) + 2 where a(1) = 2
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Help me Pleaseee it would mean a lot
Answer:
D is the answer trust me
Matt's car can travel 555 miles on 15 gallons of fuel. Work out the rate of consumption of fuel of Matt's car in mpg
The rate of consumption of fuel of Matt's car is 37 miles per gallon (mpg).
Matt's car can travel a total distance of 555 miles.
The fuel required to travel the total distance is 15 gallons by Matt's car.
Hence the rate of consumption of fuel of Matt's car can be measured in mpg (miles per gallons) as = Total distance travelled / Total gallons required to travel that distance
(that is, total distance travelled divided by the total amount of gallons to travel the same)
Thus the mpg of Matt's car is = 555 miles / 15 gallons
= 37 miles per gallon
(that is 37 mpg)
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Question 3(Multiple Choice Worth 2 points)
(Theoretical Probability MC)
When rolling a 6-sided die twice, determine P(sum of 5).
O
2/6
4/36
5/36
10/36
The probability of rolling a sum of 5 when rolling a 6-sided die twice is 1/9.
The Law of Big Numbers is what?According to the Law of Large Numbers, a fundamental tenet of probability theory, the average of the results of an experiment approaches the predicted value of the random variable being measured as the number of trials in the experiment rises. In other words, the Law of Big Numbers states that the observed results will be closer to the predicted outcomes the more times an experiment is conducted.
The total number of outcomes when a die is rolled two times is 36.
The sum of 5 is obtained for the following outcomes:
(1,4), (2,3), (3,2), and (4,1)
Thus,
P(sum of 5) = 4/36 = 1/9
Hence, the probability of rolling a sum of 5 when rolling a 6-sided die twice is 1/9.
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2. A billboard is two different colors. What is the area of the white part of the billboard? Explain how you found your answer. 4ft height and 4.5ft base
Answer: To find the area of the white part of the billboard, we first need to find the area of the entire billboard and then subtract the area of the non-white part.
The area of a triangle can be found using the formula:
Area = (1/2) x base x height
In this case, the white part of the billboard is a right-angled triangle with a height of 4 ft and a base of 2.5 ft (half of the total base of 4.5 ft).
The area of the entire billboard is:
Area = (1/2) x base x height
Area = (1/2) x 4.5 ft x 4 ft
Area = 9 ft²
The area of the non-white part of the billboard is:
Area = (1/2) x base x height
Area = (1/2) x 2.5 ft x 4 ft
Area = 5 ft²
Therefore, the area of the white part of the billboard is:
Area of white part = Total area - Area of non-white part
Area of white part = 9 ft² - 5 ft²
Area of white part = 4 ft²
So the area of the white part of the billboard is 4 square feet.
Step-by-step explanation:
7-5 skills practice parts of similar triangles
The answer is (1) x = 22.5; (2) x = 16.7; (3) x = 13.5; (4) x = 16.8; (5) x = 24.5; (6) x = 16.15; (7.a) height of the image on film is 11.2mm; (7.b) distance between camera and her friend is 1,875mm.
(1) We can see that in the given figure all three corresponding angles are congruent and all three corresponding sides are in equal proportion so, these are similar triangles.
As per properties of similar triangle:
Three pairs of corresponding sides are proportional i.e. Two pairs of corresponding sides are proportional and the corresponding angles between them are equal.
Therefore, [tex]\frac{32}{24} =\frac{30}{x}[/tex],
then by cross multiplying them, we get,
32x = 720
x = 720/32
x = 22.5
(2) As, this is already given that these are similar triangle and by applying the properties of similar triangle we get,
[tex]\frac{39}{26} =\frac{25}{x}[/tex]
39x = 650
x = [tex]16\frac{2}{3}[/tex]
x = 16.7
(3) As these are similar triangle again we can say that,
[tex]\frac{2x+1}{x+4} =\frac{40}{25}[/tex]
40(x + 4) = 25(2x + 1)
40x + 160 = 50x + 25
40x = 50x - 135
-10x = -135
(by cancelling (-) sign from both sides we get,
x = 135/10
x = 13.5
(4) By applying similar triangle's property, we can get
[tex]\frac{20}{30} =\frac{28-x}{x}[/tex]
20x = 840 - 30x
50x = 840
x = 840/50
x = 16.8
(5) As ΔJKL [tex]\sim[/tex] ΔNPR,
[tex]\frac{KM}{PT} =\frac{KL}{PR}[/tex]
[tex]\frac{18}{15.75} =\frac{28}{x}[/tex]
18x = 441
x = 441/18
x = 24.5
(6) As ΔSTU [tex]\sim[/tex] ΔXYZ,
[tex]\frac{UA}{ZB} =\frac{UT}{ZY}[/tex]
[tex]\frac{6}{11.4} =\frac{8.5}{x}[/tex]
6x = 96.6
x = 96.6/6
x = 16.15
(7.a) First we have to change 3 m and 140 cm into mm(millimeters).
So, 1m = 1000 mm
3m = 3000mm.
And 1cm = 10 mm
140cm = 1400 mm.
Then to find the height of the image on the film, we have to solve:
[tex]\frac{24}{3000} =\frac{x}{1400}[/tex]
by cross multiplication we get,
3000x = 33,600
x = 33,600/3000
x = 11.2 mm
the height of the image on the film is 11.2millimeters.
(7.b) For this also, we have to find x by solving the equation:
[tex]\frac{24}{3000} =\frac{15}{x}[/tex]
24x = 45,000
x = 45,000/24
x = 1,875 mm
The distance between camera and her friend is 1,875 millimeters.
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Full question is given below in the image.
3 = Problem 3. Given the vectors ū and ✓ find the vector w = 4ữ - 5 (in coordinates), and its magnitude | W |. Use exact values; assume mile as the length unit. (-1) 2 Problem 4. Let O be an angle in standard position, such that the point P(-15, 8) is on its terminal side. Find the exact values of all the six trigonometric functions of 0.
Given vectors ū = (u1, u2) and ✓ = (v1, v2), we can find vector w = 4ữ - 5✓ by using the formula:
w = (4u1 - 5v1, 4u2 - 5v2)
To find the magnitude of vector w, we can use the formula:
|w| = √((4u1 - 5v1)² + (4u2 - 5v2)²)
Since we do not have the exact values of u1, u2, v1, and v2, we cannot compute the exact values of w and |w|.
w = (4u1 - 5v1, 4u2 - 5v2), |w| = √((4u1 - 5v1)² + (4u2 - 5v2)²)
Problem 4:
Given the point P(-15, 8) on the terminal side of angle O in standard position, we can find the values of the six trigonometric functions of O using the following formulas:
sin O = y/r
cos O = x/r
tan O = y/x
csc O = r/y
sec O = r/x
cot O = x/y
Where x = -15, y = 8, and r = √(x² + y²) = √((-15)² + 8²) = √(225 + 64) = √289 = 17
Therefore, the exact values of the six trigonometric functions of O are:
sin O = 8/17
cos O = -15/17
tan O = -8/15
csc O = 17/8
sec O = -17/15
cot O = -15/8
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Find any numbers for which the rational expression is undefined. (7x^(4)+8)/(5x^(2)+20x)
The numbers for which the rational expression is undefined are x=0 and x=-4.
To find the numbers for which the rational expression (7x^(4)+8)/(5x^(2)+20x) is undefined, we need to find the values of x that make the denominator equal to zero. This is because division by zero is undefined.
So, we need to solve the equation 5x^(2)+20x=0 for x.
We can factor out a common factor of 5x from the equation:
5x(x+4)=0
Now, we can use the zero product property to set each factor equal to zero and solve for x:
5x=0 or x+4=0
x=0 or x=-4
So, the numbers for which the rational expression is undefined are x=0 and x=-4.
In summary, the rational expression (7x^(4)+8)/(5x^(2)+20x) is undefined for x=0 and x=-4.
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