b. Let f:x=3x² +1 and g: x=x-5
Find
i. fog(x)
ii. go f(x)
iii. (ƒ o g)−¹(x)

The percentage decrease in total principal and interest paid between a 30-year term mortgage and a 15-year term mortgage at an APR of 6.5% and principal of $242,300.00 is about 31.1%

A mortgage is a loan used to purchase property.

The principal balance = $242,300.00

The Annual Percentage Rate, APR = 6.5%

The term of the loan = 30-year and 15-year

The monthly payment for the loan per each term can be obtained using the following formula;

M = P·[r·(1 + r)^n]/[(1 + r)^n - 1]

Where;

M = The monthly payment

P = The principal amount of the loan = $242,300.00

r = The monthly interest rate = APR/12 = 0.065/12

n = The the number of periods of payment = 12 × 30 = 360

The monthly payment for the 30-year loan term is therefore;

M₃₀ = 242,300 × [(0.065/12)·(1 + (0.065/12))^360]/[(1 + (0.065/12))^360 - 1] ≈ 1531.5

The total payment for the 30-year term ≈ 360 × 1531.5 = 551,340.3

The monthly payment for the 15-year loan term is found as follows;

n = 12 × 15 = 180

M₁₅ = 242,300 × [(0.065/12)·(1 + (0.065/12))^180]/[(1 + (0.065/12))^180 - 1] ≈ 2110.7

The total payment in 15 years ≈ 15 × 12 × 2110.7 = 379962

The percentage decrease, is therefore;

((551,340.3 - 379,962)/551,340.3) × 100 ≈ 31.1%

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The fοrmula fοr t in terms οf 2kπ is: t = kπ + π/2, where k is an integer.

Tο draw a unit circle fοr the given cοοrdinates, we first draw the hοrizοntal and vertical axes intersecting at the οrigin (0,0):

Next, we draw a circle with radius 1 centered at the οrigin:

Tο find pοints οn the circle with y-cοοrdinate 1, we lοοk at the pοint where the circle intersects the vertical axis. This οccurs when x = 0, sο the pοint οn the circle with y-cοοrdinate 1 is (0,1):

(0,1)

Tο find οther pοints οn the circle, we can use the Pythagοrean identity:

sin²(t) + cοs²(t) = 1

Since we want y = sin(t) tο be 1, we can sοlve fοr x = cοs(t)

cοs(t) = sqrt(1 - sin²(t))

Using this fοrmula, we can find several pοsitive and negative real numbers t that determine pοints Q οn the circle with y-cοοrdinate 1:

t = 0 radians (0 degrees): Q = (1,0)

t = π/6 radians (30 degrees): Q = (√3/2, 1/2)

t = π/4 radians (45 degrees): Q = (√2/2, √2/2)

t = π/3 radians (60 degrees): Q = (1/2, √3/2)

t = π/2 radians (90 degrees): Q = (0,1)

t = 7π/6 radians (-150 degrees): Q = (-√3/2, 1/2)

t = 3π/4 radians (-135 degrees): Q = (-√2/2, √2/2)

t = 5π/6 radians (-120 degrees): Q = (-1/2, √3/2)

t = π radians (-180 degrees): Q = (-1,0)

Tο write a fοrmula fοr t in terms οf 2kπ, we can use the inverse sine functiοn:

sin(t) = 1

t = sin⁻¹(1) + 2kπ

Since sin(π/2) = 1, we have:

t = π/2 + 2kπ

Sο the fοrmula fοr t in terms οf 2kπ is:

t = kπ + π/2, where k is an integer.

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Answer: Car tire

Step-by-step explanation:

30 inches is about 2 1/2 feet, about the size of a car tire when taken into consideration.

Answer:

40.1 cm

Step-by-step explanation:

For this problem we will use trigonometry. To find x, we have to remember SOH CAH TOA

Our angle is 55, and the we have the value adjacent to the angle which is 23 cm so we will take that and make an equation because we are missing the hypotenuse:

CAH = adjacent/hypotenuse

cos = ^

cos ( 55 ) = 23/x

multiply x both sides

xcos(55) = 23

divide cos(55) from left

x = 23/cos(55)

input in calculator (make sure in right mode degrees)

23/(cos(55))

x = 40.09927 cm

x = 40.1 cm

Answer:31/36

Step-by-step explanation:

we should find greatest common divisor(GCD) ,

GCD(12,18)=36

7/12=7*3/36=21/36

5/18=5*2/36=10/36

21/36+10/36=31/36

Answer:

31/36

Step-by-step explanation:

1) Find the Least Common Denominator (LCD) of [tex]\frac{7}{12}[/tex], [tex]\frac{5}{18}[/tex]. In other words, find the Least Common Multiple (LCM) of 12, 18.

1) List the multiples of each number

Multiples of 12: 12, 24, 36, ...

Multiples of 18: 18, 36, ...

2) Find the smallest number that is shared by all rows above. This is the LCM.

LCM = 36

1) List the prime factors of each number.

Prime Factors of 12: 2, 2, 3

Prime Factors of 18: 2, 3, 3

2) Find the union of these primes.

2, 2, 3, 3

3) Multiply these number: 2 x 2 x 3 x 3 = 36. This is the LCM.

LCM = 36

STEP 2: Make the denominators the same as the LCD.

[tex]\frac{7\times3}{12\times3} +\frac{5\times2}{18\times2}[/tex]

3) Simplify. Denominators are now the same.

[tex]\frac{21}{36} +\frac{10}{36}[/tex]

4) Join the denominators.

[tex]\frac{21+10}{36}[/tex]

5) Simplify.

[tex]\frac{31}{36}[/tex]

Thank you,

Eddie.

Answer:

The new monthly payment will be $944.16

The first transformation was a rotation about point A.

The second transformation was a reflection across line M.

In Mathematics, a rotation can be defined as a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.

By critically observing the diagram which illustrates the sequence of transformations, we can logically deduce that the first transformation was a clockwise rotation about point A by 180 degrees.

Furthermore, the second transformation that maps W'X'Y'Z' to W''X''Y''Z'' is a reflection across the line of reflection M.

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Answer:16

Step-by-step explanation:

the Westchester Chamber of Commerce should run 6 advertisements on television, 17 advertisements on radio, and 12 advertisements online to maximize total audience contact within the budget constraint.

The term "cost" generally refers to the amount of resources (such as money, time, effort, or opportunity) that are required to produce or obtain something.

In a business or financial context, cost can refer to the expenses incurred in the production of goods or services, such as materials, labor, and overhead. It can also refer to the price paid for goods or services that are purchased from others.

Given by the question.

Let's denote the number of advertisements to be run on television, radio, and online as x, y, and z, respectively. We want to maximize the total audience, which is given by:

Total Audience = 140,000x + 18,000y + 30,000z

Subject to the following constraints:

Cost constraint: 1500x + 200y + 550z ≤ 24,100

Maximum media usage constraint:

a. x + y + z ≤ 46 (total number of advertisements authorized)

b. y ≤ 0.45(x+y+z) (radio advertisements should not exceed 45% of the total)

Note that we combined the maximum media usage constraints to get a single constraint for the total number of advertisements authorized.

We can now set up the optimization problem:

Maximize Total Audience = 140,000x + 18,000y + 30,000z

Subject to:

1500x + 200y + 550z ≤ 24,100

x + y + z ≤ 46

y ≤ 0.45(x+y+z)

We can solve this problem using a linear programming solver. Here is the solution:

Number of advertisements on TV (x) = 6

Number of advertisements on radio (y) = 17

Number of advertisements online (z) = 12

Total audience = 140,000x + 18,000y + 30,000z = 2,452,000

Total cost = 1500x + 200y + 550z = $22,300

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The tοtal interest paid οver the 5-year lοan periοd wοuld be $4,605.

In mathematics, interest is the cοst οf bοrrοwing mοney, usually expressed as a percentage οf the bοrrοwed amοunt. It is calculated based οn the principal amοunt, the interest rate, and the time periοd fοr which the mοney is bοrrοwed.

Assuming that the full purchase price οf the new car is $32,700, and the dealer has οffered a trade-in value οf $2,000 fοr yοur οld car, yοu will need tο finance the remaining balance, which is:

$32,700 - $2,000 = $30,700

If the dealer has οffered a 3% add-οn interest rate fοr a 5-year lοan, we can calculate the tοtal interest paid οver the life οf the lοan using the fοllοwing fοrmula:

Tοtal interest = (Principal × Interest rate × Lοan term) ÷ 2

Substituting the given values intο the fοrmula, we get:

Tοtal interest = ($30,700 × 0.03 × 5) ÷ 2

Tοtal interest = $4,605

Therefοre, the tοtal interest paid οver the 5-year lοan periοd wοuld be $4,605.

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Answer:

Step-by-step explanation:

equal

Answer:

You can write it as (9/2)e^x + (2)e^-x.

Answer:

Step-by-step explanation:

The number 5 is prime. This implies that the number 5 is pairless and is not in the power of 2. Therefore, the square root of 5 is irrational.

- Notify me if any questions.!

[tex]\textit{difference of squares} \\\\ (a-b)(a+b) = a^2-b^2 \\\\[-0.35em] ~\dotfill\\\\ 9p^2-16q^2\implies 3^2p^2-4^2q^2\implies (3p)^2-(4q)^2\implies (3p-4q)(3p+4q) \\\\[-0.35em] ~\dotfill\\\\ x^2-5\implies (x)^2-\sqrt{5^2}\implies (x)^2-(\sqrt{5})^2\implies (x-\sqrt{5})(x+\sqrt{5})[/tex]

Since 6 = 6, these side lengths do form a right triangle.

Therefore, the answer is C: √2, 2, √6.

How are lengths of sides of a triangle calculated?

The Pythagorean theorem, which asserts that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides, can be used to identify which side lengths make up a right triangle.

A) The Pythagorean theorem enables us to deduce that: 52 + 62 = 25 + 36 = 61

[tex]\sqrt{30^{2}[/tex] = 30

These side lengths do not create a right triangle because 61 > 30.

B) The Pythagorean theorem enables us to observe that:

[tex]2.5^2 + \sqrt{18^{2} }[/tex] = 6.25 + 18 = 24.25

Also,

[tex]5^{2}[/tex] = 25

Since 24.25 < 25, a right triangle cannot be formed by these side lengths.

C) The Pythagorean theorem enables us to observe that:

[tex]\sqrt{2^{2} } + 2^2[/tex] = 2 + 4 = 6

[tex]\sqrt{6^{2} }[/tex] = 6

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The function in vertex form is f(x) = 10(x + 2)² - 8.

In this exercise, you are required to determine the vertex form of a quadratic function h(x) that is written in standard form. Mathematically, the vertex form of a quadratic equation is given by this formula:

y = a(x - h)² + k

Where:

Based on the information provided about this quadratic function, we can reasonably infer and logically deduce that a mathematical expression which quickly reveals the vertex of the quadratic function is given by:

y = a(x - h)² + k

y = 10(x - (-2))² + (-8)

y = f(x) = 10(x + 2)² - 8

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Answer:

x ≈ 20.0

Step-by-step explanation:

using the cosine ratio in the right triangle

cos26° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{18}{x}[/tex] ( multiply both sides by x )

x × cos26° = 18 ( divide both sides by cos26° )

x = [tex]\frac{18}{cos26}[/tex] ≈ 20.0 ( to the nearest tenth )

Answer:

1/6 = 2/12 = 3/18 = 4/24 = 5/30 = 6/36 = 7/42 = 8/48 = 9/54 = 10/60........

Therefore, the rate per annum at which N4,600 will yield an interest of N230 for 2 years is 2.5%.

Percent is a term used to describe a fraction or ratio that represents a portion of 100. It is often represented by the symbol % and is commonly used to express a portion or a rate of change. For example, if an item is discounted by 20%, it means the price has been reduced by 20% of the original price or 20/100 of the original price. Similarly, if someone earns a grade of 85% on a test, it means they answered correctly 85 out of 100 questions.

Here,

7. To find the time (in years) in which a loan of N3,900 at the rate of 5% yields N585.60 in simple interest, we can use the formula:

I = P * r * t

where I is the interest, P is the principal, r is the interest rate, and t is the time in years.

Substituting the given values, we get:

585.60 = 3,900 * 0.05 * t

Solving for t, we get:

t = 3 years

Therefore, it will take 3 years for the loan of N3,900 at the rate of 5% to yield N585.60 in simple interest.

8. To find the time (in years) for a loan of N20,000 at the rate of 7% to accumulate N70,000 in simple interest, we can use the formula:

I = P * r * t

where I is the interest, P is the principal, r is the interest rate, and t is the time in years.

Substituting the given values, we get:

70,000 = 20,000 * 0.07 * t

Solving for t, we get:

t = 50 years

Therefore, it will take 50 years for the loan of N20,000 at the rate of 7% to accumulate N70,000 in simple interest.

9. To find the time (in years) for N300 to accumulate to N390 at the rate of 3%, we can use the formula:

I = P * r *t

where I is the interest, P is the principal, r is the interest rate, and t is the time in years.

Substituting the given values, we get:

90 = 300 * 0.03 * t

Solving for t, we get:

t = 10 years

Therefore, it will take 10 years for N300 to accumulate to N390 at the rate of 3%.

10. To find the year in which N5,100 will yield an interest of N170 at the rate of 2.5% per annum, we can use the formula:

I = P * r * t

where I is the interest, P is the principal, r is the interest rate, and t is the time in years.

Substituting the given values, we get:

170 = 5,100 * 0.025 * t

Solving for t, we get:

t = 2 years

Therefore, it will take 2 years for N5,100 to yield an interest of N170 at the rate of 2.5% per annum. If the interest is compounded, we would need to use the compound interest formula instead.

11. To find the rate per annum at which N4,600 will yield an interest of N230 for 2 years, we can use the formula:

I = P * r * t

where I is the interest, P is the principal, r is the interest rate, and t is the time in years.

Substituting the given values, we get:

230 = 4,600 * r * 2

Solving for r, we get:

r = 0.025 or 2.5%

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the correct probability is option c)[tex]P(F^c)[/tex] = 0.95.

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The value is stated from zero to one. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will discover the likelihood of outcomes for a random experiment using this fundamental probability theory, which is also applied to probability distribution.

from the question:

According to the table, the likelihood that a household owns four cars is given by P(4) = 0.05.

Four automobiles in one family constitute the occurrence F. A home with more than four automobiles is the complement of this occurrence, [tex]F^c[/tex].

The total of all outcomes' probabilities must equal 1. Thus, by deducting P(4) from 1 we may derive P([tex]F^c[/tex]):

P([tex]F^c[/tex]) = 1 - P(4) = 1 - 0.05 = 0.95

Therefore, the correct probability is option c) P([tex]F^c[/tex]) = 0.95.

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The possible rotation that transforms the image obtained from the reflection of the square ABCD on the line x = -1, with two vertices of the square remaining invariant is as presented as follows;

A rotation transformation is a transformation in which the preimage coordinate points are turned (in a circular manner) about a point.

The vertices of the square ABCD are; A(1, 4), B(3, 4), C(3, 2), D(1, 2)

The coordinates of the square following the reflection across the line x = -1 are; A'(-3, 4), B'(-5, 4), C'(-5, 2), and D'(-3, 2)

The rotation transformation of the image such that two vertices of the square are invariant (such that they remain the same) can be found as follows;

The rotation of a vertices of the image about the point (-2, -1) indicates the relative points are;

A'(-1, 5), B'(-3, 5), C'(-3, 3), and D'(-1, 3)

The image following a rotation of 270° are;

A''(5, 1), B'(5, 3), C'(3, 3), and D'(3, 1)

The above points relative to the origin are;

A''(5 + (-2), 1 - 1) = (3, 0), B''(5 - 2, -1 + 3) = (3, 2), C';(3 - 2, 3 - 1) = (1, 2), and D''(3 - 2, 1 - 1) = (1, 0)

A''(3, 0), B''(3, 2), C''(3, 4), and D''(1, 4)

The points (1, 2), and (3, 2) are therefore, the same as the points in the preimage and are therefore, invariant

The rotation is therefore;

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Answer:

0

Step-by-step explanation:

When adding a negative, just get rid of the negative and subtract.

D-d=0

None of the provided equations are direct formulas for finding the lengths of the legs of a triangle.

A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals symbol ("="). For illustration, 2x - 5 Equals 13.

Here,

5 and 13 are phrases for 2x.

FROM THE QUESTION?

The number of x, which is one of the lengths of a right triangle's legs, can be determined by solving the quadratic equation [tex]0.5(x)(x+2)=24[/tex] . It is not a straightforward method, though, to determine the lengths of the legs.

It is possible to simplify and calculate the quadratic equation [tex]x(x+2)=24[/tex] to determine the values of x, which correspond to the lengths of a rectangle whose area is 24. Once more, there is no simple method for determining the lengths of a triangle's legs.

You can simplify and calculate the equation [tex]x^2+(x+2)2=100[/tex] to determine the value of x, which is one of the lengths of a right triangle's legs. It is not a straightforward method, though, to determine the lengths of the legs.

The lengths of a triangle's legs cannot be determined directly using any of the given formulae.

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The point slope form of the line that passes through (-2 ,1) and a slope of - 4 is  y - 1 = -4(x + 2).

The equation of a line can be represented in point slope form as follows:

y - y₁ = m(x - x₁)

where

Therefore, the line passes through (-2 ,1) and a slope of - 4.

The point slope form is as follows:

y - y₁ = m(x - x₁)

y - (1) = -4(x - (-2))

y - 1 = -4(x + 2)

Therefore, the point slope form is y - 1 = -4(x + 2)

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The dot represents the data point of 5, and there are no other data points to include on the dot plot.

You can still make a dot plot even if you just have one number in the tens by placing a dot on a number line above the number. If your data point is 5, for instance, you can represent it by placing a dot above the tick mark for 5 on a number line that has tick marks for each integer from 0 to 9. Here is an illustration of how the dot plot might appear:

  |      

5  •      

  |      

  |      

  |      

  |      

  |      

  |      

  |      

  |      

  +   -   -    -   -   -   -   -

  0  1   2   3  4  5  6  7

There are no additional data points to place on the dot plot because the dot in this illustration represents the data point of 5.

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The property of equality shown in the given equation is the addition property of equality.

The properties of equality are a set of rules that can be used to manipulate equations while maintaining their truth value, including the reflexive, symmetric, transitive, addition, subtraction, multiplication, and division properties.

which states that if we add the same quantity to both sides of an equation, the equation remains true. In this case, we add z to both sides of the equation –70 = 26 – y to obtain –70 + z = 26 – y + z.

By the addition property of equality, the equation remains true because we have added the same quantity.

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Answer:

x=43.85

Step-by-step explanation:

solution Given.

let the given angle be θ: 20 degree.

It's opposite side is 12.

It's hypotenuse is x.

Since

We have

Sin θ= opposite/hypotenuse

Sin20 degree=12/x

0.34202014332=12/x

doing criss cross multiplication

x=12/0.34202014332

therefore x=43.8570660032

in two decimal form x=43.85

Answer:

8+the sign over 2

Step-by-step explanation:

The ratio is given as A:C = 3:4

A ratio is the number of times one number divides into another or the proportion of one number to the other.

Since we have that the A : B = 2:5 and B: C = 15: 8.

We desire to the ratio  find A : C.

We proceed to solve the problem as follows

Since we have that the ratio

A:B = 2:5

This implies that

A/B = 2/5

Also, we have that the ratio

B : C = 15/8

This implies that

B/C = 15/8

Now, the ratio A:C = A/C

= A/B × B/C

So, substituting the values of the variables into the equation, we have that

A/C = A/B × B/C

= 2/5 × 15/8

= 1/1 × 3/4

= 3/4

So, A:C = 3:4

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