Actuarial math

A 10-year $100 par value bond bearing a 10% coupon rate payable semiannually, and redeemable at 105, is bought to yield 8% convertible semiannually. Find the price. Verify that all four formulas produce the same answer.

3) The tire diameter using the aspect ratio is; 29 inches

4) The aspect ratio of a standard print paper is; 22:17

The aspect ratio is defined as the relationship between the width and height.

Now, in this case, the aspect ratio must be in whole numbers and not decimals and as such we can solve as follows;

3) We are told that the aspect ratio of a tire is 50%.

Thus, the aspect ratio is; 50/100 = 1/2 or 1:2.

Now, we are told that width 10 inches, and the rim diameter is 19 inches, then it means that the tire diameter = 10/2 + 19 + 10/2 = 29 inches

4) The standard print paper is 11" by 8.5. This can also be written as;

11 by 17/2 or 11 * 2 : 17

= 22:17

This then gives us the aspect ratio.

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The two polynomials whose difference is 2x^2 + x + 4 are

6x² + 3x + 5 and 4x² + 2x + 1.

Polynomial is an equation written as the sum of terms of the form kx^n.

where k and n are positive integers.

We have,

The difference between the two polynomials is 2x² + x + 4.

The two polynomials are:

6x² + 3x + 5 _____(1)

4x² + 2x + 1 ______(2)

The difference between (1) and (2)

6x² + 3x + 5 - 4x² - 2x - 1

2x² + x + 4

Thus,

The two polynomials are 6x² + 3x + 5 and 4x² + 2x + 1.

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It has been (mathematically) proven that there is no polynomial time algorithm for any np-complete problem. the given statement is false.

In order to solve previously unsolvable issues, Leonid Khachiyan developed a polynomial-time method, in which the number of processing steps increases as a power of the number of variables rather than exponentially.

Determining whether NP-complete problems are tractable or intractable remains one of the most crucial concerns in theoretical computer science since it is unknown if any polynomial-time algorithms will ever be developed for them.

So, it is unknown if any polynomial-time algorithms will ever be developed for them.

By applying different constraints, a linear function is maximized or reduced in linear programming, a mathematical modelling approach. In commercial planning, industrial engineering, and—to a lesser extent—the social and physical sciences, this method has proven helpful for directing quantitative decisions.

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Answer:

The sequence is not arithmetic. In an arithmetic sequence, the difference between consecutive terms is constant. However, in this sequence, the difference between the terms is not constant. For example, the difference between the first and second terms is 1^5 - 2^5 = -3, while the difference between the second and third terms is 2^5 - 3^5 = -6. Since the difference between consecutive terms is not constant, this sequence is not arithmetic. Therefore, the common difference is NONE.

Answer:

No, the sequence is not arithmetic.

d = NONE

Step-by-step explanation:

An Arithmetic Sequence has a common difference between each term (the difference between each term is the same).

Given sequence:

[tex]1^5, 2^5, 3^5, 4^5, 5^5, ... = 1, 32, 243, 1024, 3125, ...[/tex]

Calculate the difference between each term:

[tex]1 \underset{+31}{\longrightarrow} 32 \underset{+211}{\longrightarrow} 243 \underset{+781}{\longrightarrow} 1024 \underset{+2101}{\longrightarrow} 3125[/tex]

As the difference between each term is not constant, the sequence is not an arithmetic sequence.

Therefore, the common difference, d = NONE.

The equation of a hyperbola can be written in the form:

(x-h)^2/a^2 - (y-k)^2/b^2 = 1

where (h, k) is the center of the hyperbola, a is the distance from the center to the vertices on the x-axis, and b is the distance from the center to the vertices on the y- axis.

In this case, the vertices of the hyperbola are (6,4) and (0,4), and the center of the hyperbola is at (3,4). The distance from the center to the vertices on the x- axis is 3, and the distance from the center to the vertices on the y- axis is 0.

Plugging these values into the equation for a hyperbola, we get:

(x-3)^2/3^2 - (y-4)^2/0^2 = 1

This equation is not defined, because the value of b is 0. This means that the hyperbola has a horizontal orientation, and its vertices are on the y-axis.

To find the equation of the hyperbola, we need to rewrite the equation in a different form. We can do this by using the fact that the distance between the center of the hyperbola and a point on the hyperbola is given by:

√ ((x-h)^2 + (y-k)^2) = a*√ (1 + (y-k)^2/b^2)

Substituting in the values for h, k, and a, we get:

√ ((x-3)^2 + (y-4)^2) = 3*√ (1 + (y-4)^2/0^2)

This equation can be simplified to:

√ ((x-3)^2 + (y-4)^2) = 3*√((y-4)^2/0^2)

The value of b is still 0, so the equation for the hyperbola is:

(x-3)^2 = 9*(y-4)^2

This is the equation for a horizontal hyperbola with vertices (3,4) and (9,4) and asymptotes y = 1/3x + 3 and y = -1/3x + 5.

A hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces which is known as connected components, that are mirror images of each other and resemble two infinite bows.

Therefore, the correct answer is as given above

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The solution of the system of equations by the graphical methods yields  x equals 9 and y is 6 i.e, 9 hours of landscaping and 6 hours of waiting tables.

Bella worked x hours working landscaping at $10 per hour.

Total amount earned = 10x

Bella worked y hours working on clearing tables at $ 15 per hour

Amount earned = 15 y

x and y are variables as given

She worked a total of 15 hours and she gets $180 for the combined efforts.

The system of equation can be represented as:

x + y = 15

10x + 15y = 180

Now we will solve the two equations graphically:

So from the attached graph we can see that the point (9,6) is the required solution.

Hence she worked landscaping for 9 hours and clearing tables for 6 hours.

Let us solve by substitution to verify the results:

x + y = 15

or, x = 15 - y

Now 10x + 15y = 180

or, 10 (15-y) + 15y = 180

or, 150 - 10y +15y = 180

or, 5y = 180 - 150

or, y = 6

At y = 6 , x = 15 - 6 = 9

Hence (9,6) is the solution.

System of equations also known as simultaneous equations can also be solved by the elimination method.

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The horizontal and vertical asymptotes for the function f(x) =6/(-2x +1) is equal to y = 0 and x = 1/2 respectively.

Graph is attached.

As given in the question,

Given function is :

f(x) =6 /(- 2x +1)

f(x) = y

Degree of the numerator is zero

Degree of the denominator is 1.

Degree of denominator is greater than the degree of numerator.

Function represent the horizontal asymptotes for y =0

To get the vertical asymptotes equate denominator equals to zero

(- 2x +1) = 0

⇒2x = 1

⇒x = 1/2

Vertical asymptotes is given by x =1/2.

Graph is attached.

Therefore , the horizontal and vertical asymptotes for the given function is equal to y=0 and x =1/2 respectively.

The above question is incomplete, the complete question is:

Graph the rational function. f(x) =6 /(- 2x +1) Start by drawing the vertical and horizontal asymptotes. Then plot two points on each piece of the graph. Finally, click on the graph-a-function button X.

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Answer:

0.65 pens

2.25 rubber

Step-by-step explanation

You can make two equations with the information given and solve them simultaneously to find x and y. Assume x is pens and y is rubber.

The total income from the machines over the first 5 years is $185.89

From the question, we have the following parameters that can be used in our computation:

f(t) = 160e0.03t

Where

Rewrite the function properly

so, we have the following representation

f(t) = 160e⁰.⁰³ⁿ

The above function is an exponential function

So: In the first 5 years, we have the value of t to be 5

This is represented as follows

t = 5

Substitute the known values in the above equation, so, we have the following representation

f(5) = 160e⁰.⁰³ ˣ ⁵

Evaluate the product of the exponents

f(5) = 160e⁰.¹⁵

The products above give

f(5) = 185.89

Hence, the total income is $185.89

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The graph that shows the linear relationship of the table is attached below.

To determine the graph of a linear relationship, do the following:

Given the table which represents a linear relationship above, use two pairs of values, (0, 0) and (2, 1) to find the slope:

Slope = change in y / change in x = (1 - 0) / (2 - 0)

Slope = 1/2.

The y-intercept (b) = 0.

Thus, this means that the graph of the linear relationship will have a slope of 1/2 and crosses the y-axis at 0.

Thus, the graph is shown below.

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Answer:

x is 44 Hope that helps.

a. The blue line represents g(x) = 250,000csc(π30x).

b. g(5) = 25,000 km. This is the distance of the comet from Earth after 5 days.

c. The minimum distance between the comet and Earth is 250,000 km, which occurs when x = 0. This corresponds to the constant csc(π30x).

d. The equation has a vertical asymptote at x = 30/π.

a. Graph g(x):

To graph g(x), we can plot several points on the interval [0,35] and then connect them to form a graph of the function. For example, we can plot the points (0,250,000), (5,50,000), (10,25,000), (15,16,667), (20,12,500), (25,10,000), (30,8,333) and (35,7,143). Connecting these points will give us a graph of the function g(x).

b. Evaluate g(5):

The value of g(5) can be found by plugging in x = 5 into the equation: g(5) = 250,000 csc(π30*5) = 50,000 km. This means that after 5 days, the comet will be 50,000 km away from Earth.

c. Minimum distance between the comet and Earth:

The minimum distance between the comet and Earth can be found by taking the derivative of the equation and setting it to 0.

d. Vertical Asymptotes:

The vertical asymptotes of g(x) occur at x = 0 and x = 30. This means that the comet approaches an infinitely large distance from Earth at x = 0 and x = 30.

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By graphing, the solution to the system, y = 5x + 2 and y = x - 2 is: A. (-1, -3)

The solution to a system of equations is the coordinates of an ordered pair that will make both of the equations to be true. This coordinates is determined by graphing.

To do this, plot both lines on a coordinate plane, then determine the coordinates of the point of intersection of both lines.

Given the system of equations,

y = 5x + 2

y = x - 2

The graph that shows both equations are attached below where the red line is y = 5x + 2, and y = x - 2 is the blue line.

The point of intersection is at (-1, -3). Therefore, the solution is:

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We had written an algorithm, using pseudo code.

//Define the required variables.

total_number_of_steps = 0

number_of_steps = 0

//robot stands up to walk.

stand

//the robot moves forward till the wall is not reached.

while (wall is not reached)

//the robot senses the wall with the help of arms.

raise arms

sense_the_wall

if (wall is not there)

//if the wall is not found, the robot lowers the arms

lower arms

//if wall is not detected, the robot moves forward.

step_forward

//increment the number of steps forward.

number_of_steps=number_of_steps+1

//increment the total number of steps.

total_number_of_steps=total_number_of_steps+1

else

exit

end if

end while

//else if the wall is found, lower the arms

lower arms

//the robot turns 90 degrees right once.

turn_right_90

//the robot turns 90 degrees right again.

turn_right_90

//till the number of steps does not reach 0, move

for (number_of_steps! = 0)

step

//increment the total number of steps.

total_number_of_steps=total_number_of_steps+1

//the robot moves backwards on detecting the wall, so

//decrement the number of steps.

number_of_steps = number_of_steps-1

//end the for loop.

end for

//the robot turns 90 degrees left once.

turn

//the robot turns 90 degrees left again.

turn

//the robot sits down.

sit

//display the total number of steps taken by robot

display total_number_of_steps

The last line ends the pseudocode

Stop

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A set-to-set function f is a relation that assigns one element of the set A to each element of the other set B. The set A is the domain of the function (also called the set of inputs) and the set B contains the range (also called the set of outputs).

Let's denote f as exponential function with the base 'a':

                                  f(x) = a ˣ

where, a > 0 and a ≠ 1 and x is any real number.

1. Not an exponential function:

We have the following equation:

[tex]y = 100 (1 - 12)^{t}[/tex]

We can write the equation as:

[tex]y = 100 (-11)^{t}[/tex]

Let's denote f as exponential function with the base 'a':

                                  f(x) = a ˣ

where, a > 0 and a ≠ 1 and x is any real number.

Therefore: a must be more than 01.

In this problem, a = -11, therefore this is not an exponential function.

2. Growth.:

The function:

[tex]y= 0.1(1.25)^{t}[/tex]

The above equation is an exponential function because is a function of the form

[tex]f(t) = ka^t[/tex]

where, a > 0 and k is constant

Therefore, k = 0.1 and  a = 1.25 . So, a is greater than 1.

3. Growth:

The function:

[tex]y = {(1 - 0.03)12}^2t[/tex]

The above equation can be written as:

[tex]y = 11.64^{2t}[/tex]

which  is an exponential function because is a function of the form

[tex]f(t) = a^{bt}[/tex]

where, a > 0 and b is constant.

So, a = 11.64 and b = 4 . Since a is greater than 1.

4. Decay:

The function:

[tex]y = 426(0.98)^t[/tex]

This is an exponential function because is a function of the form

[tex]f(t) = ka^t[/tex]

where, a > 0 and k is constant

So, k = 426 and a = 0.98. Since, a is less than 1.  

5. Growth:

The function:

[tex]y = 2050(12)^t[/tex]

This is an exponential function because is a function of the form

[tex]f(t) = ka^t[/tex]

where, a > 0 and k is constant.

So, k = 2050 and a = 12  . Since, a is greater than 01.  

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The equation of slope intercept form is y = 3/2x + 12 if a line passes through the point (-6,3) and has a slope of 3/2.

The line with m as the slope, m and c as the y-intercept is the graph of the linear equation y = mx + c. The values of m and c are real integers in the slope-intercept form of the linear equation. The slope, m, is a measure of how steep a line is. The equation of a straight line that passes through a particular point and is inclined at a specific angle to the x-axis can be found using the point slope form. Every point on a line must satisfy the equation for the line in order for it to exist. This implies that a line is represented by a linear equation in two variables.

Given

A line passes through the point (-6,3)

and has a slope of 3/2

Slope intercept form

y = mx + b

Equation,

y - 3 = 3/2(x - (-6))

y - 3 = 3/2(x+6)

2y - 6 = 3x + 18

2y = 3x + 24

y = 3/2x + 12

The equation of slope intercept form is y = 3/2x + 12 if a line passes through the point (-6,3) and has a slope of 3/2.

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The volume of the pentagonal prism as per the given values is obtained as 2.5 cm³.

A pentagonal prism has two pentagonal surfaces on the top and the bottom. It has ten vertices, 15 edges and 7 faces. The number of rectangular surfaces are five.

The expression for the volume of a regular pentagon is given as V =  5/2 × abh

Where h is the height and a and b are the length of the sides.

Now, the value of a, b and h as per the question is given as,

a = b = h = 1.

Then, the volume can be calculated as,

V = 5/2 × 1 ×1 × 1
   = 2.5

Hence, the volume of the regular pentagonal prism  is given as 2.5 cm³.

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The degree to turn before walking 673 paces is 59 degrees

From the question, we have the following parameters that can be used in our computation:

Lengths = 989 paces, 673 paces and 861 paces

These paces can be represented as

x = 989

y = 673

z = 861

The measure of the required angle (angle Z) i.e.  the degree to walk can be calculated using the following cosine rule

z² = x² + y² - 2xy * cos(Z)

Substitute the known values in the above equation, so, we have the following representation

861² = 989² + 673² - 2 * 989 * 673 * cos(Z)

Evaluate the exponents, the products and the summation

So, we have the following representation

741321 = 1431050 - 1331194 * cos(Z)

Evaluate the like terms

- 1331194 * cos(Z) = -689729

Divide both sides by  - 1331194

cos(Z) = 0.51812808651

Take the arc-cos of both sides

Z = 58.7

Approximate

Z= 59 degrees

Hence, the measure of the angle is 59 degrees

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Answer:

x = - 3

Step-by-step explanation:

2(x - 6) + 2 = 4x - 4 ← distribute parenthesis on left side and simplify

2x - 12 + 2 = 4x - 4

2x - 10 = 4x - 4 ( subtract 4x from both sides )

- 2x - 10 = 4x - 4 ( add 10 to both sides )

- 2x = 6 ( divide both sides by - 2 )

x = - 3

Answer:

SOLUTION

2(X-6)+2=4X-4

open the blanket by multiplication

2x-12+2=4x-4

2x-10=4x-4

collect like term

2x+4x=-4+10

6x=6

divide both side by 6

x=1

Answer:

B, F

Step-by-step explanation:

[tex]2x^2(x-4)^2=0 \\ \\ \implies x^2=0, (x-4)^2=0 \\ \\ \implies x=0,4[/tex]

The population standard deviation [tex]\sigma[/tex] is 6.697.

The average distance between each data item and the mean is known as the mean absolute deviation.

The gap between each data value and the mean in absolute terms is represented by this average.

The given data set can be arranged as 4, 7, 8, 10, 12, 12, 15, 18, 23, 26.

Mean[tex](\overline{x})[/tex] = (4 + 7 + 8 + 10 + 12 + 12 + 15 + 18 + 23 + 26)/10.

Mean[tex](\overline{x})[/tex] = 135/10 = 13.5.

We know variance[tex](\sigma)[/tex] = [tex](\[ \sum_{i=1}^{n}x_i^2 \]/n) - \overline{x}^2[/tex].

= (4² + 7² + 8² + 10² + 12² + 12² + 15² + 18² + 23² + 26²)/10 - (13.5)².

= (16 + 49 + 64 + 100 + 144 + 144 + 225 + 324 + 529 + 676)/10 - 182.25.

= 44.85

And we also know the standard deviation is [tex]\sqrt{\sigma}[/tex] = [tex]\sqrt{44.85}[/tex] = 6.697

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The procedure is correct is  1 + tan²(theta) = sec²(theta)

What is  Fraction?

Any number of equal parts is represented by a fraction, which also represents a portion of a whole. A fraction, such as one-half, eight-fifths, or three-quarters, indicates how many components of a particular size there are when stated in ordinary English.

And

cos²(theta) = 1 - sin²(theta)

Are both valid identities

Keisha and the David.

As per the question Kisha and the David found the same value of the cosine theta as given by the  Sine theta = Negative Start Fraction 8 Over 17 End Fraction. The Keisha solution and the Davis solution for the solution were tan-squared (theta) + 1 = sec-squared (theta).

Thus the answer is both procedures are correct.

As they both wanted to find the solutions to the numerical problems they made two different methods.

Such as the Sine theta = Negative Start Fraction and the tan-squared (theta) + 1 = sec-squared (theta).

Hence both options are the same and true.

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Answer:0.1875

Step-by-step explanation: you would take the miles and divide it my the minutes which would be 3 divided by 16 which equals 0.1875

The expression (4c − 3d)(3c + d) is equivalent to 12c² - 5cd - 3d²

An expression is combination of variables, numbers and operators.

(4c − 3d)(3c + d) is the given expression.

Four times c minus three times d into three times f c plus d.

In the given expression c and d are variables.

Plus and minus are operators.

We need to find the equivalent expression of (4c − 3d)(3c + d).

(4c − 3d)(3c + d)

4c(3c+d)-3d(3c + d)

Apply the distributive property.

12c² + 4cd - 9cd - 3d²

Add the like terms

12c² - 5cd - 3d²

Hence, 12c² - 5cd - 3d² is the equivalent expression of (4c-3d)(3c+d).

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Answer:

Answer is y = 11

Step-by-step explanation:

8 = y - 3

or, 8 + 3 = y - 3 + 3

or, 11 = y

or, y = 11

The value of down payment will be;

⇒ $705

A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.

Given that;

Monique buys a $4700 air conditioning system using an installment plan that requires 15% down.

Now,

Since, Monique buys a $4700 air conditioning system using an installment plan that requires 15% down.

Hence, The value of down payment = 15% of $4700

                                                        = 15/100 × 4700

                                                        = 15 × 47

                                                        = 705

Thus, The value of down payment = $705

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The best estimate of the solution to the system of linear equations is the point are (x,y) will be (0.76, 4.30).

A linear equation in two variables is one that is written in the form axe + by + c=0, where a, b, and c are real numbers and the coefficients of x and y, i.e. a and b, are not equal to zero.

Given:

as x,y are variables,

-3x + y = 2 ..............(1)

- 4x + 7y = 1 ...........(2)

We know that,

The solution of the system of linear equations is the intersection point of both graphs

Multiply equation (1) by 7 on both sides and then subtract it from eq (2)

- 4x + 7y = 1 ...........(2)

-21x + 7y = 14 ..............(1)

⇒- 17 x = -13

⇒ 17 x = 13

⇒ x =[tex]\frac{13}{17}[/tex]

⇒ x = 0.76

Now on putting value of x  = 0.76 in eq (1) we get

- 3x + y = 2 ............ (1)

⇒- 3(0.76) + y = 2

⇒y = 4.30

The best estimate of the solution to the system of linear equations is the point (x,y) will be (0.76, 4.30).

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Answer: 160 inches

Step-by-step explanation:

To put lights around the window, you will need a total of 32 + 48 + 32 + 48 = <<32+48+32+48=160>>160 inches of lights. You will need this much light to go completely around the window.