There are 26¹⁰ - 4 × (25)¹⁰ + 2 × (24)¹⁰ + 22¹⁰ exponential generating Function to make an arrangement of Pennies, nickels, dimes and quarters with at least one penny and an odd number of quarters.
Exponential generating Function:
An exponential generator provides a way to encode a sequence as the coefficients of a power series. This encoding helps in many ways.
Therefore, According to the Question:
The exponential generating function is,
G(x) = (x¹/1! + x²/2! + ...)(x¹/1! + x³/3! + x⁵/5! +...)(x°/0! + x¹/1! + x²/2! + ...)²
where the first factor describes the pennies, the second describes the quarters, and the last describes nickels and dimes.
We want the coefficient of Xˣ /r!.
Using exponential identities we get,
G(x) = [tex]e^{x} - 1 (\frac{e^{x} - e^{-x} }{2} ) (e^{x} )^2[/tex]
= [tex]\frac{1}{2} (e^{4x} -e^{2x} -e^{3x} + e^{x}[/tex]
and the coefficient is
[tex]\frac{1}{2} (4^r -2^r -3^r +1)[/tex]
The generating function is
G(x) = (x¹/1! + x²/2! + ...)⁴(x°/0! + x¹/1! + x²/2!+...)²² = ()⁴,
and we again want the coefficient of X¹⁰/10!.
Multiplying we get,
G(x) = [tex](e^{4x} - 4e^{3x} + 2e^{2x} + 1)e^{22x}[/tex]
= [tex]e^{26x} - 4e^{25x} + 2e^{24x} +e^{22x}[/tex]
so the coefficient of X¹⁰/10! is
26¹⁰ - 4 × (25)¹⁰ + 2 × (24)¹⁰ + 22¹⁰
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Please Help on Thissss
Answer:
ok first you'll have to subtract
6 from 22 = 16
so now easy
you'll say
16+3 and you'll have your answer
Answer:
11
Step-by-step explanation:
ΘΘ + 6 = 22
Factor on the left.
2(Θ + 3) = 22
Divide both sides by 2.
Θ + 3 = 11
if a1 = 2 and an = -5an-1 +3 then what's the value of a5
HELP PLEASE
Answer:
10
Step-by-step explanation:
a1=2
a=2
an=-5an-1+3
plug in
(2)5=10
how do you graph data?
A right triangle is shown, where the horizontal leg has a measure of 42.7 centimeters(cm) and the hypotenuse has a measure of 52.5cm. Determine the length, in centimeters, of the vertical leg. Show your work for partial credit.
30.54cm is the length of the vertical leg.
What is Trigonometry?Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.
In a right angled triangle the horizontal leg has a measure of 42.7 centimeters(cm).
The hypotenuse has a measure of 52.5cm.
We need to find the measure of the vertical leg.
We can find this by using pythagoras theorem.
In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides
52.5²=42.7²+x²
2756.25=1823.29+x²
Subtract 1823.29 on both sides
2756.25-1823.29=x²
932.96=x²
Taking square root on both sides
x=30.54
Hence, the length of the vertical leg is 30.54cm.
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A. The graph is incorrect and the line should go through the point (0.6, 1).
B. The x-axis and y-axis are labeled incorrectly.
C. The graph does not show a proportional relationship.
D. The equation c = 0.6w is correctly represented by the graph.
Answer:
D
Step-by-step explanation:
An easy way to check is to look for a number that is on an exact corner on the graph for example 3. 1.8 divided by 3 is 0.6
2.
a.
The tax rate at this restaurant is 9.5%.
Date: Sep 12th
Time: 6:04 PM
Server: # 27
Bread Stix
Ravioli Bites
Cheesecake
Subtotal
Sales Tax
Total
9.50
10.50
4.95
24.95
Date: Sep 12th
Time: 7:12 PM
Server: #27
Garden Salad
Broccoli Bites
Subtotal
Sales Tax
Total
1.61
Another person's subtotal is $24.95. How much will their sales tax be?
a) The sales tax of the person with a subtotal of $24.95 is of: $2.37.
b) The subtotal of a person with a sales tax of $1.61 is of: $16.95.
How to obtain the amounts?The amounts are obtained applying the proportion of 9.5%, which represents the sales tax of this restaurant.
The sales tax of the person with a subtotal of $24.95 is obtained with the multiplication of the subtotal by the sales tax of 9.5% = 0.095 as follows, hence:
Sales tax = 0.095 x 24.95 = $2.37.
The subtotal of a person with a sales tax of $1.61 is obtained considering that $1.61 represents 9.5% = 0.095 of the subtotal x, due to the tax rate, as follows:
0.095x = 1.61
x = 1.61/0.095
x = $16.95 -> which is the subtotal of the person.
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Determine if the sequence below is arithmetic or geometric and determine the common difference/ratio in simplest form.
22,16,10
The given sequence is a arithmetic sequence with the common difference is -6.
What is an arithmetic sequence?An arithmetic sequence is a sequence of numbers where the differences between every two consecutive terms is the same. The general form an arithmetic sequence is an=a+(n-1)d
The given sequence is 22, 16, 10.
Here,
Common difference = Second term - First term
16-22=-6 and 10-16=-6
Since the common difference between the terms is same, the given sequence is arithmetic sequence.
Hence, 22, 16, 10 is a arithmetic sequence with the common difference is -6.
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An architect is making a model version of your house. She
plans to use a 1:25 scale, which means every inch on the
model equates to 25" on your house. If a part of the model
measures 7", what would the corresponding measurement
be on your house?
Answer:
If the model is made using a 1:25 scale, then every inch on the model corresponds to 25 inches on the actual house. Therefore, if a part of the model measures 7 inches, the corresponding measurement on the actual house would be 7 * 25 = <<7*25=175>>175 inches.
Step-by-step explanation:
The sum of two numbers is 58 and the difference is 18. What are the numbers?
Answer And step-by-step explanation: The sum of x and y is 58. In other words, x plus y equals 58 and can be written as equation A:
x + y = 58
The difference between x and y is 18. In other words, x minus y equals 18 and can be written as equation B:
x - y = 18
Now solve equation B for x to get the revised equation B:
x - y = 18
x = 18 + y
Then substitute x in equation A from the revised equation B and then solve for y:
x + y = 58
18 + y + y = 58
18 + 2y = 58
2y = 40
y = 20
Now we know y is 20. Which means that we can substitute y for 20 in equation A and solve for x:
x + y = 58
x + 20 = 58
X = 38
Summary: The sum of two numbers is 58 and their difference is 18. What are the two numbers? Answer: 38 and 20 as proven here:
Sum: 38 + 20 = 58
Difference: 38 - 20 = 18
Hope this is helpful if it is can I be marked With Crown?
Answer:
x=38 y=20
Step-by-step explanation:
x+y=58
x-y=18
combine
2x=76
x=38
38+y=58
-38 -38
y=20
Describe in detail how can you use problem-solving in personal, professional, or mathematical problems?.
In order to solve the challenges we encounter every day, Problem solving strategies are needed. To tackle a specific problem, we must first comprehend it, then create a plan for resolving it, and finally, put the plan into action.
The techniques that a person uses to solve difficulties are referred to as problem-solving strategies. A person may experience difficulties at their place of employment or home. A problem-resolution strategy must be in place in order to address such issues.One must first understand the issue. Then, potential solutions to the issue must be identified, and the best one among the available options must be picked. The chosen approach would subsequently be put into practice.Mathematical problem-solving techniques are also necessary. One must first comprehend the issue at hand before attempting to resolve it in order to tackle a particular mathematical problem.To sum up, everyone must use problem-solving techniques to overcome the obstacles we confront.
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f(x)=2x^2+7x-4. solve using quadratic fomula
The given function is solved to obtain x = -4, 0.5.
What is the quadratic formula?Quadratic formula is used to find the roots of a quadratic function f(x) = ax² + bx + c as x = (-b ± √(b² - 4ac))/2a.
The expression b² - 4ac is called the determinant D.
When D = 0, roots are real and equal.
For D > 0, roots are real and unequal and for D < 0, roots are complex conjugates.
The given quadratic function is f(x) = 2x² + 7x - 4.
It can be solved by using quadratic formula as follows,
2x² + 7x - 4 = 0
x = (-7 ±√(-7²- 4 × 2 × -4))/(2 × 2)
= (-7 ± 9)/(2 × 2)
= -4, 0.5
Hence, the value of x found by using quadratic formula are -4 and 0.5 respectively.
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is 7.33 a terminating or repeating decimal?
Answer: 7.33 is a terminating decimal. 7.333333... or 7 1/3 is a repeating decimal.
Step-by-step explanation:
Write the word sentence as an equation. 8 is the quotient of a number g and 3.
the sooner the better
a rectangle is 7 cm tall and 14 cm wide. if it is reduced to a height of 1cm , then how wide will it be
With the help of ratio, we know that when the length will be 1cm then the width of the rectangle will be 2cm.
What is the ratio?In mathematics, a ratio shows how frequently one number appears in another.
For instance, the ratio of oranges to lemons in a fruit plate would be eight to six if there were eight oranges and six lemons.
Oranges make up 8:14 of the total fruit, whereas lemons make up 6:8 of the total fruit.
So, 1st we have a rectangle which:
length 7cm
width 14cm
Just get the ratio of length:width:
7:14
7/14
1/2
1:2
When height becomes 1cm, then according to the ratio the width will be:
1/2 * 1/1 = 1/2cm
Where 2cm is the width when height becomes 1cm
Therefore, with the help of ratio, we know that when the length will be 1cm then the width of the rectangle will be 2cm.
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f(x)=x^2. What is g(x)?
Which two steps will complete the list correctly? 4. Divide the original measurement by the conversion factors. 5. Check for reasonableness. 4. Multiply the acoriginal measure by the conversion factors. 5. Simplify the answer. 4. Multiply the original measure by the conversion factors. 5. Check for reasonableness. 4. Divide the original measure by the conversion factors. 5. Simplify the answer.
Two steps that will complete the list correctly are:
4. The original measurement should be multiplied by the conversion factors.
5. Verify that it makes sense.
What is meant by conversion factor?A conversion factor can be used to switch from one set of units to another by multiplying or dividing a number. When a conversion is necessary, the appropriate conversion factor to an equivalent value must be applied.
Let's use an example of converting from inches to centimeters to address this query.
10 inches should be converted to cm.
The first step is to identify the measure that has to be converted, which in this case is inches.The conversion factors, which are; are written in the second step. 1 inch equals 2.54 centimetersThe third step is to cancel the units; in this instance, inches will be cancelled.The original measurement is multiplied by the conversion factor in the fourth step to give; 10 × 2.54 cm = 25.4 cmThe fifth stage is to assess reasonableness by attempting to determine whether choosing a different number will produce a reasonable result.To know more about units, visit:
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Marta makes 90\%90%90, percent of the free throws she attempts. She is going to shoot 333 free throws. Assume that the results of free throws are independent from each other. Let xxx represent the number of free throws she makes.
Answer: 299
Step-by-step explanation: 333 X .9 = 299.7; Can't make .7 of a free throw.
An economics graduate student in the united states earns an $18,000 stipend per year. the u.s. sees significant inflation. how might the student react?
The reaction of the graduate student on inflation is that a cost-of-living increase to her stipend.
What is inflation?Inflation is the pace of expansion in costs over a given time-frame. Inflation is regularly a wide measure, like the general expansion in costs or the expansion in the typical cost for many everyday items in a country.
According to question:Inflation is term which states there is an expansion in the cost level for any labor and products.
In the unique situation, a set of experiences graduate living in the US got a payment of 18,000 dollar. However, there is a critical expansion that US economy endures. In this way the expense of products and the administrations accessible to individuals increments quickly. Individuals presently need to spend more on everything. Accordingly the alumni understudy requirements to request the typical cost for most everyday items to be remembered for the payment as need might arise to spend more on living.
Thus, graduate student argue on the a cost-of-living increase to her stipend.
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m is a two-digit integer. the two-digit integer n is obtained by reversing the digits of m. the difference between m and n is one eleventh (1/11) of the sum of m and n. find the value of m n.
The values will be of two digit either 92 or 29, 74 or 47 etc.
Let M = AB(A..B = 0…9) and N = BA. where A and B are numbers from 0 to 9.
The subtraction rule requires the last digit of the difference to be the Difference of A-B. To explain further, for example 44, take the last digit 4. This can be expressed as the difference of 5-1, 8-4, 4-0. (meaning 1st digit is A and 2nd digit is B) = 88 - 44 = 44, similarly 99-55 = 44, 44-0 = 44. So we can represent 44 according to the rules above.
For numbers greater than 100, the last digit can be represented as the sum of the numbers 0 through 9 and the numbers greater than 5. To explain this, let's analyze 121. The last digit is 1.
digit can be represented as the sum of the numbers 0 through 9 and the numbers greater than 5. To explain this, let's analyze 121.
The last digit is 1. Since the number is greater than 100, it must be expressed as the sum of single digit numbers that sum to 11,
so 9+2.8+3.7. + 4.6+5, then applying any of these gives 92+29 = 121, 83+38 = 121, 74+47+121, 65+56=121. So 121 is removed.
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Shen and Linda are driving separate cars to New York. They begin the trip with full gas tanks. The amount of gas (in gallons) remaining in the tank of each car depends on the number of miles driven, as shown below.
The interpretation of the straight line graph of the gallons of gas to distance in miles are as follows;
(a) After 100 miles, the equation for the graph and the graph indicates that Linda's car will have more gas
Linda's car will have 1 gallon more gas than Shen's car.
(b) Linda and Shen will have the same amount of gas in their cars after 150 miles
The graph indicates that Shen's car will have less gas than Linda's car before reaching 150 miles
What is a straight line graph?A straight line graph is a graph of a linear equation.
The points on the graphs are;
Linda; (0, 14), (150, 8)
Shen; (0, 11), (150, 8)
The equation for the amount of gas remaining in Linda's car can be found as follows;
Slope of the line = (8 - 14)/(150 - 0) = -6/150 = -1/25 = -0.004
Equation of the line is therefore;
y - 14 = (-1/25)(x - 0)
y = -0.04·x + 14The slope of the graph of the equation of Shen's car = (8 - 11)/(150 - 0) = -0.02
The equation of the line is; y - 11 = -0.02·x
y = -0.02·x + 11After 100 miles we get;
Gas left in Linda's car = -0.04 × 100 + 14 = 10
Gas left in Shen's car = -0.02 × 100 + 11 = 9
Therefore, Linda's car will have more gas after 100 miles
Her car will have 10 gallons of gas
(b) From the graph, at the point x = 150, the tanks of Linda and Shen will contain the same amount of gas
If the miles driven is less than 150 miles, the graph indicates that Shen's car will have less gas than Linda's car
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The sun of the values in the solution set is
7e +63 < 12e
Answer:
Step-by-step explanation:
4m-3/4=-9+4m/84m-3/4=-9+4m/84m-3/4=-9+4m/84m-3/4=-9+4m/84m-3/4=-9+4m/84m-3/4=-9+4m/84m-3/4=-9+4m/84m-3/4=-9+4m/8
A set of exam scores is normally distributed with a mean = 76 and standard deviation = 8.
Use the Empirical Rule to complete the following sentences.
68% of the scores are between __ and __
.
95% of the scores are between __ and __
.
99.7% of the scores are between __ and __
.
The required sentences can be completed as follows:
68% of the scores are between 68 and 84.
95% of the scores are between 60 and 92.
99.7% of the scores are between 52 and 100.
To complete the first sentence, we need to find the range of scores that includes 68% of the values.
We know that 68% of the scores fall within 8 points of the mean, so the range of scores that includes 68% of the values is 76 - 8 to 76 + 8, or 68 to 84.
Therefore, the first sentence can be completed as follows:
68% of the scores are between 68 and 84.
To complete the second sentence, we need to find the range of scores that includes 95% of the values.
We know that 95% of the scores fall within 16 points of the mean, so the range of scores that includes 95% of the values is 76 - 16 to 76 + 16, or 60 to 92.
Therefore, the second sentence can be completed as follows:
95% of the scores are between 60 and 92.
To complete the third sentence, we need to find the range of scores that includes 99.7% of the values.
We know that 99.7% of the scores fall within 24 points of the mean, so the range of scores that includes 99.7% of the values is 76 - 24 to 76 + 24, or 52 to 100.
Therefore, the third sentence can be completed as follows:
99.7% of the scores are between 52 and 100.
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Chase looks around at an assembly. He notices his younger sister to his right and his older brother 16 feet ahead of him. If Chase's brother and sister are 20 feet apart, how far apart
are Chase and his younger sister?
The 16 feet distance from Chase to his older brother and the 20 feet distance from his older brother to his younger sister, using Pythagorean, Theorem, indicates that the distance from Chase to his younger sister in the assembly is 12 feet
What is Pythagorean Theorem?Pythagorean Theorem states how the length of the sides of a right triangle are related, that is the square of the longest side of a right triangle is equivalent to the sum of the square of the other two sides.
The location of Chase, his younger sister, and his older brother forms a right triangle.
The distance from Chase to his older brother = A leg of the right triangle
The distance from Chase to his sister = A second leg of the right triangle
The distance from his older brother to his sister = The hypotenuse side of a right triangle
From the question, we get;
Distance between Chase and his older brother = 16 feet
Distance between Chase's older brother and his sister = 20 feet
According to Pythagorean Theorem, we get;
Distance between Chase and his younger sister = √(20² - 16²) = 12
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What is the area of the shaded perimeter?
The required area of the shaded perimeter is given as 28 square units.
What are trigonometric equations?These are the equation that contains trigonometric operators such as sin, cos.. etc.. In algebraic operations.
Here,
As shown in the figure the shaded part is a parallelogram,
The area of the parallelogram is given as = base × height
Substitute the value in the above equation,
Area = 4sin45 × 7 / sin45 [since base / 4 = sin45 and 7 / height = sin45]
Area = 4/√2 × 7 / [1 / √2]
Area = 28
Thus, the shaded perimeter's needed area is specified as 28 square units.
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What is 1 angle of the square?
A 1 angle of the square Class 1 is a type of four-sided polygon. It is a regular polygon, meaning that all of its sides have the same length and all of its angles are equal.
There are only right angles in a square. Having the opposing sides parallel defines a square as a quadrilateral. Only one dimension, the side length, may be used to describe a square. Each angle is 90° in length and has an equal number of sides. To help you comprehend the topic better, let's talk more about square angles and work through a few instances. A square Class 1 is a type of four-sided polygon. It is a regular polygon, meaning that all of its sides have equal length and all of its angles are equal. The sides of a square Class 1 are all of the same length, and the angles between them are all 90 degrees. This type of polygon is also known as a regular square.
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A computer system costs $275.
The sales person receives a 2%
commission on the sale. How
much in commission does the
salesperson make from the sale
of the computer system?
Answer: $5.50
Step-by-step explanation:
$275 x 2% = 550
550/100 = 5.5
The salesperson makes $5.50 in commission.
A rectangular storage container without a lid is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $15 per square meter. Material for the sides costs $9 per square meter. Let w denote the width of the base. Find a function in the variable w giving the cost C (in dollars) of constructing the box. C(w) = Find the derivative of C. C'(w) = Find the cost (in dollars) of materials for the least expensive such container. (Round your answer to the nearest cent.)
The cost function is C(w) = 300/w + 18w² and the cost of materials for the cheapest such container is $222.0
The function for constructing the cost of the boxFrom the question, we have the following parameters that can be used in our computation:
The length is twice the width;
Volume = 10m³
This means that:
V = 10
l = 2w
The volume of the container is calculated as:
V = lwh
So, we have
v = 2w²h
By substitution, we have
h = v/2w²
So, we have
h = 10/2w²
Divide
h = 5/w²
The surface area of an open rectangular storage is:
A = 2h(l + w) + lw
So, we have
A = 2h(2w + w) + 2w²
A = 2h(3w) + 2w²
Substitute h = 5/w²
A = 2 * 5/w² * (3w) + 2w²
Evaluate
A = 20/w + 2w²
The base costs $15, and the sides cost $9
So, the cost function is:
C(w) = 15 * 20/w + 2w² * 9
Rewrite as:
C(w) = 300/w + 18w²
The least cost of the materialsWe have
C(w) = 300/w + 18w²
Differentiate
C'(w) = -300/w² + 36w
Set to 0
-300/w² + 36w = 0
Rewrite as:
-36w = -300/w²
Cancel out the negatives
36w = 300/w²
Multiply through by ²
36w³ = 300
Divide through by 36
w³ = 8.33
Take cube roots of both sides
w = 2.03
Recall that:
C(w) = 300/w + 18w²
So, we have:
C(2.03) = 300/2.03 + 18(2.03)²
Evaluate
C(2.03) = 222.0
Hence, the cost of the materials is $222.0
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What is the value of x in variable?
Answer: x doesn't really have a value, it is determined by the question and is unknown. For example, an equation like (2x = 6) would be asking for the value of x, but an equation that asks you to simplify would have it different, say 4(2x+8) simplified is 8x+32. As I said before, it depends on what they are asking for, otherwise, it is an unknown value.
Step-by-step explanation: math.exe is running in brain
(c) in a certain country license plates consist of zero or one digit followed by four or five uppercase letters from the roman alphabet. how many license plates have atleast repeated letter?
39,877,760 license plates have at least one repeated letter.
What is probability?
The degree to which something is likely; the probability that something will occur or will be the case.
Here, we have
Given
In a certain country license plates consist of zero or one digit followed by four or five uppercase letters from the roman alphabet.
We have to find how many license plates have at least a repeated letter.
For this,
We will first find how many different license plates the country can produce and how many license plates have no repeated letter.
The country can produce:
10 x 26 x 26 x 26 x 26 x 26 = X
10 x 11,881,376 = X
118,813,760 = X
license plates have no repeated letter:
10 x 26 x 25 x 24 x 23 x 22 = X
10 x 7,893,600 = X
78,936,000 = X
Now, we find license plates have at least repeated letter:
118,813,760 - 78,936,000 = X
39,877,760 = X
Hence, 39,877,760 license plates have at least one repeated letter.
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A lighthouse keeper is in the top of a lighthouse 95 feet above sea level. She notes that the angle of depression to a rock jutting just above the water is 6° . How far is the rock from the lighthouse?
Answer:
808.72 ft
Step-by-step explanation:
We can create a right triangle with the following points:
- the top of the lighthouse
- the base of the lighthouse
- the rock jutting out of the water
I will label the distance from the base of the lighthouse to the rock d.
See the attached image for a detailed model.
Then, solve for the two acute angles of the triangle.
The larger acute angle (∠1) is complementary to 6°, so we can solve by setting their sum to 90°:
6° + m∠1 = 90°
m∠1 = 84°
Because this is a right triangle, the smaller acute angle (∠2) is complementary to the larger acute angle, so we can solve once again by setting their sum to 90°:
m∠1 + m∠2 = 90°
84° + m∠2 = 90°
m∠2 = 6°
We can use either of these angles in combination with the trigonometric ratio tangent to solve for the distance d. I will use tangent of ∠1.
[tex]\tan(84\textdegree) = \dfrac{d}{85 \, \textrm{ft}}[/tex]
[tex]d = 85\tan(84\textdegree)[/tex]
[tex]d \approx 808.72 \, \textrm{ft}[/tex]