Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of
14. Use the empirical rule to determine the following.

(a) What percentage of people has an IQ score between 86 and 114?

(b) What percentage of people has an IQ score less than 86 or greater than 114?

(c) What percentage of people has an IQ score greater than 114?

The domain of  fof is all real numbers.

(f.f)(x) = f(f(x)) = f(6x+3) = 6(6x+3) + 3 = 36x + 21. The domain of fof is all real numbers.

(a) (f.g)(x) = f(g(x)) = f(x^2) = 6(x^2) + 3 = 6x^2 + 3. The domain of fog is all real numbers.

(b) (g.f)(x) = g(f(x)) = g(6x+3) = (6x+3)^2 = 36x^2 + 36x + 9. The domain of gof is all real numbers.

(d) (g.g)(x) = g(g(x)) = g(x^2) = (x^2)^2 = x^4. The domain of gog is all real numbers.

Therefore, the correct answer is OB. The domain of fog is all real numbers.

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Answer:

60 is a composite number, that has twelve factors. They are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. 

Step-by-step explanation:

The solution is:

Number of Adult tickets sold = 420

Number of Students tickets sold = 360

Given as ,

Number of adult ticket + Number of Students tickets = 780  

Each Adult ticket cost = $8

Each Student ticket cost = $3

Total of ( Adult ticket cost + Student ticket cost) = $ 4,440

i.e

A + S = 780         ....1

And 8A + 3S = 4440         .....2

Solve eq 1 and 2

Or, 8A + 3S = 4440

    3A + 3S = 2340

Or, (8A + 3S) - (3A + 3S) = (4440 - 2340)

or,  5A = 2100

So, A = 420    

Now put this A value in equation 1

So 420 + S = 780

Or, S = 780 - 420

So , S = 360  

Hence The Number of Adult tickets sold = 420

And     The Number of Student tickets sold = 360

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Answer:

[tex]3 \: {ft}^{2} [/tex]

Step-by-step explanation:

Given:

A rhombus

The diagonals would be AH and MT

AH = 2 ft

MT = 3 ft

Find: A (area) - ?

[tex]a = \frac{ah \times mt}{2} = \frac{2 \times 3}{2} = \frac{6}{2} = 3 \: {ft}^{2} [/tex]

Hence the correct option is C. for the fourth term of the expansion of

[tex](a - b)^{10}[/tex].

Either the process of expanding or the condition of expanding. An expanded object, surface, or component. the amount by which anything extends; its degree, extent, or size. an expansion, growth, or development, especially in a company's operations.

An expression that has been raised to any finite power can be expanded using the binomial theorem. A useful expansion technique with applications in  probability theory or probability  and algebra is the binomial theorem. A binomial expression is an algebraic expression with two terms that are not the same.

According to The binomial theorem  to determine the fourth term in the expansion of[tex](a - b)^{10}[/tex], Therefore,

[tex](a - b)^{10}[/tex] = [tex]C(10,0)*a^{10}*b^0 + C(10,1)*a^9*b^1 + C(10,2)*a^8*b^2 + C(10,3)*a^7*b^3 + ...[/tex]

where C(n,r) denotes the variety of options for selecting item(s) r from a set of item(s) n.

So, the term r = 3 equal to the fourth term in the given expansion:

[tex]C(10,3)*a^7*b^3 = (10!)/(3!7!)*a^7*b^3[/tex]

[tex]C(10,3)*a^7*b^3\\ = (10 *9 * 8*7!) /(3*2 * 1*7!)*a^7*b^3\\ = 120*a^7*b^3 C(10,3)*a^7*b^3 \\=(10 *9 * 8) /(3 *2 *1)*a^7*b^3 =\\ =120*a^7*b^3[/tex]

hence [tex]120*a^7*b^3[/tex] is the fourth term in the given expansion of [tex](a - b)^{10}[/tex]

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The exact value of cos( α+ β) is (-24/85)(√7 + 2).

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities are used to simplify expressions involving trigonometric functions, to solve trigonometric equations, and to prove other mathematical results.

Some of the most common trigonometric identities include the Pythagorean identity, which relates the sine and cosine functions to the unit circle and the tangent function; the sum and difference formulas, which relate the trigonometric functions of the sum or difference of two angles to the trigonometric functions of the individual angles; and the double angle and half angle formulas, which relate the trigonometric functions of twice or half an angle to the trigonometric functions of the original angle.

We can use the trigonometric identity cos(α + β) = cos(α)cos(β) - sin(α)sin(β) to find the value of cos(α + β), given the values of sin(α), sin(β), and the quadrants in which α and β lie.

We are given that sin(α) = 15/17 and that a lies in quadrant I, so we can use the Pythagorean identity to find cos(α):

cos(α) = √(1 - sin²(α)) = √(1 - (15/17)²) = √(112/289) = (4/17)√7

We are also given that sin(β) = 4/5 and that ẞ lies in quadrant II. Since sin(ẞ) is positive and cos(β) is negative in quadrant II, we can use the Pythagorean identity to find cos(β):

cos(β) = -√(1 - sin²(β)) = -√(1 - (4/5)²) = -3/5

Now we can substitute these values into the formula for cos(α + β):

cos(α + β) = cos(α)cos(β) - sin(α)sin(β)

= (4/17)√7 * (-3/5) - (15/17) * (4/5)

= (-12/85)√7 - (12/17)

= (-24/85)(√7 + 2)

Therefore, the exact value of cos( α+ β) is (-24/85)(√7 + 2).

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Answer:

f(n) = -20 + 4(n - 1)

Step-by-step explanation:

The explicit formula for an arithmetic sequence in function notation is [tex]f(n) = f(1) + d(n - 1)[/tex], where:

We need to find the values of f(1) (the first term) and 'd' (the common difference). The first term in the arithmetic sequence is -20 because it's the beginning number. The common difference is 4 because we add 4 to each term to get the next term (-20 + 4 = -16, -16 + 4 = -12, -12 + 4 = -8...). Now, we can plug in the values of f(1) and 'd' to get the formula of this arithmetic sequence: f(n) = -20 + 4(n - 1).

Ernesto will get $202.38 in US dollars owing to the remaining amount of €150.

We will use the unitary method to know the amount returned to Ernesto when he got home. It means finding the value of unit entity to know the larger or small amount.

As we see, €630 is equal to $850

So performing unit conversion, €1 will be = 850/630

Now, the value of €150 in dollars will be - (850/630)×150

Cancel the zero common in numerator and denominator and perform multiplication and division as per the numbers

The value of €150 in US dollars = $202.38

Hence, the value in US dollars will be $202.38.

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Mark needs a ladder that is 13 feet long in order to reach the second floor windows of the building.

Length is a measurement of distance or space from one point to another. It is typically measured in units such as meters, feet, or inches. Length is a scalar quantity, meaning it has magnitude but no direction. Length is a fundamental property of physical objects and can be used to measure the size of an object or the distance between two points.

To properly calculate the ladder length Mark needs, we must use the Pythagorean Theorem. The Theorem states that for the triangle formed by the ladder and the building, the square of the hypotenuse (the ladder length) is equal to the sum of the squares of the other two sides. Therefore, the ladder length Mark needs is equal to the square root of the sum of the squares of 5 feet and 12 feet, which is equal to the square root of 169, or 13 feet. Therefore, Mark needs a ladder that is 13 feet long in order to reach the second floor windows of the building.

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We have used the calculation below to sow that the definition to show that lim (x,y)→(3,4) f(x, y) = 3 / 26

In order to show that lim (x,y)→(3,4) f(x, y) = 3 / 26, we need to show that for any ε > 0, there exists a δ > 0 such that if 0 < √((x-3)² + (y-4)²) < δ, then |f(x,y) - 3/26| < ε.

Using the given function, we have:

|f(x,y) - 3/26| = |x/(x²+y²+1) - 3/26|

= |(26x - 3(x²+y²+1)) / 26(x²+y²+1)|.

To simplify this expression, we can use the fact that x² + y² ≥ 0 for any x and y. Therefore,

26(x²+y²+1) ≥ 26(0+0+1) = 26.

Using this inequality, we can write:

|f(x,y) - 3/26| = |(26x - 3(x²+y²+1)) / 26(x²+y²+1)|

≤ |(26x - 3(x²+y²+1)) / 26|

≤ |26x / 26| + |3(x²+y²+1) / 26x²+26y²+26|

= |x| + 3(x²+y²+1) / (26(x²+y²+1))

= |x| + 3 / 26 (1 + 1 / (x²+y²+1)).

Now, we choose δ = ε / (1 + 3/26). Then, if 0 < √((x-3)² + (y-4)²) < δ, we have:

|x| ≤ √((x-3)² + (y-4)²) < δ,

and also

1 < 1 + 1 / (x²+y²+1) ≤ 1 + 1 / δ².

Therefore, |f(x,y) - 3/26| ≤ δ + 3 / 26 (1 + 1 / δ²)

< ε / (1 + 3/26) + 3 / 26 (1 + δ²)

≤ ε / (1 + 3/26) + 3 / 26 (1 + ε² / (1 + 3/26)²).

Since the last term does not depend on x and y, we can choose ε sufficiently small such that ε / (1 + 3/26) + 3 / 26 (1 + ε² / (1 + 3/26)²) < ε. Therefore, we have shown that lim (x,y)→(3,4) f(x, y) = 3 / 26.

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Answer:

10 students.

Step-by-step explanation:

Considering that 60% of 25 is 15, all we need to do is find 40% of 25

Multiply 25 by 0.4=10.

Hope this helps!

Answer:

10

Step-by-step explanation:

x = students who don't play basketball

[tex]x=25(\frac{100-60}{100} )=25(\frac{40}{100} )=25(0.4)=10[/tex]

Hope this helps.

Answer:

True

Step-by-step explanation:

divide 21 feet by 4 feet to convert to inches, that is

21 feet ÷ 4 feet = 21 ÷ 4 = 5.25 inches

Answer: The line passing through the point with vector 2a + 3b and parallel to c can be written as:

r = (2a + 3b) + t c

where t is a scalar parameter.

To find where this line intersects the plane, we can substitute the line equation into the plane equation and solve for t and s:

a - b + t(a + b - c) + s(a + c - b) = (2a + 3b) + t c

Expanding both sides and collecting terms, we get:

[(1 + t + s) a + (-1 - t + s) b - tc + ts c] = (2a + 3b)

Equating the coefficients of a, b, and c on both sides, we get the following system of equations:

1 + t + s = 2

-1 - t + s = 3

-t + ts = 0

Solving for t and s, we get:

t = -1

s = 1

Substituting t = -1 into the line equation, we get:

r = (2a + 3b) - c

Therefore, the position vector of the line where it intersects the plane is 2a + 3b - c.

Answer:

(310 − 10) ÷ 15 = s

Step-by-step explanation:

Number of students s:

After s students have had 15 candies each there are 10 left, so

the total number of pieces given to the students = 300 .

We therefore have the equation:

s = (310- 10) / 15

which can be written as

(310 − 10) ÷ 15 = s.

The value of given quadrant is cos(5π/3) = -1/2 and sin(5π/3) = -√3/2.

Trigonometry is a branch of mathematics concerned with relationships between angles and ratios of lengths. There are six trigonometric ratios, namely sine, cosine, tangent, cotangent, secant, and cosecant of a reference angle.

The reference angle 5π/3 is π/3.

To find the quadrant where 5π/3 is, we can divide the angle by π/2. Since 5π/3 is between 4π/3 and 2π, it is in the third quadrant.

We can use the unit circle or trigonometric identities to find the exact values ​​of sin(5π/3) and cos(5π/3).

In the unit circle, we start from the positive x-axis (cosine) and rotate clockwise 5π/3 radians, which corresponds to  π/3 radians counterclockwise. This takes us to (-1/2, -√3/2).

Therefore cos(5π/3) = -1/2 and sin(5π/3) = -√3/2.

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The solution of the inequality whereby the function f in the inequality increases over the domain is; -8 < x ≤ -5; 1 ≤ x < 4

An inequality is a mathematical statement comparing two expressions using the inequality symbols, ≠, <, >, ≤, and ≥.

The characteristics of the function indicates;

f(4·x - 3) ≥ f(2 - x²)

The domain of the function, Df = (-8, 4)

The function is increasing over the domain indicates;

4·x - 3 ≥ 2 - x²

4·x - 3 - (2 - x²) ≥ 0

x² - 4·x - 3 - 2 ≥ 0

x² - 4·x - 5 ≥ 0

(x - 1)·(x + 5) ≥ 0

The positive value of the right hand side of the inequality indicates that; x ≤ -5, or x ≥ 1

The domain of the function therefore indicates that the solution of the inequality is therefore;

x ∈ (-8, -5] ∪ [1, 4), which is; -8 < x ≤ -5; 1 ≤ x < 4

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The elasticity function, considering the demand function, is given as follows:

E(p) = 5p.

The elasticity function is obtained considering the demand function, as follows:

E(p) = -pD'(p)/D(p).

In which:

The demand function for this problem is given as follows:

D(p) = e^(-5p).

The derivative of the demand function is given as follows:

D'(p) = -5e^(-5p).

Hence the elasticity function is given as follows:

E(p) = -pD'(p)/D(p).

E(p) = -p x -5e^(-5p)/e^(-5p)

E(p) = 5p.

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Answer:

To find the growth factor, we need to first convert the percentage increase to a decimal:

3.3% = 0.033

The growth factor is then calculated as:

1 + percentage increase as a decimal = 1 + 0.033 = 1.033

Therefore, the growth factor over this time is 1.033. This means that the population of Alaska increased by a factor of 1.033 since 2010.

The result suggests that the mean wake time might have really reduced since the values barely fall above 100 min as in before treatment with a high degree of confidence. thus , the drug is effective.

Confidence interval is written in the form as;

(Sample mean - margin of error, sample mean + margin of error)

The sample mean represent x , it is the point estimate for the population mean.

Margin of error = z × s/√n

Where s = sample standard deviation = 21.8

n = number of samples = 17

Now the population standard deviation is unknown and the sample size is small, hence, we would use the t distribution to find the z score

then the degree of freedom, df for the sample.

df = n - 1 = 17 - 1 = 16

Since confidence level = 99% = 0.99, α = 1 - CL = 1 – 0.99 = 0.01

α/2 = 0.01/2 = 0.005

Therefore the area to the right of z0.005 is 0.005 and the area to the left of z0.005 is 1 - 0.005 = 0.995

the t distribution table, z = 2.921

Margin of error = 2.921 × 21.8/√17

= 15.44

The confidence interval for the mean wake time for a population with drug treatments will be; 90.3 ± 15.44

The upper limit is 90.3 + 15.44 = 105.74 mins

The lower limit is 90.3 - 15.44 = 74.86 mins

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Answer: (0, 3)

Step-by-step explanation:

You need to put the equation in the circle formula.  In order to do that, you need to complete the square.

(x-h)² +(y-k)² =  r²    where (h,k) is the center

Your equation:

x²+y²-6y-16 = 0   bring over constants to other side

x²+y²-6y = 16    complete the square by taking the y term (6)  divide by 2 and square that number to add to both sides

[tex](\frac{6}{2} )^{2}[/tex]  = 3² = 9   this is the completion of square for the y quadratic part

add to both sides of equation

x²+y²-6y+9= 16 +9

x²+y²-6y +9= 25   factor the y quadratic part

        you get (y-3)(-3) = (y-3)²

x²+(y-3)² = 16    we have put it in the format above (x-h)² +(y-k)² =  r²

        there is no x so  h=0   k=3

so the center is (0,3)

Answer: It would take 4 years without a claim for Option 2 to break even with Option 1. After 4 years, the savings from the lower premium on Option 2 would offset the higher deductible, resulting in lower total cost.

Step-by-step explanation: To calculate the break-even point, we need to determine the point at which the savings from the lower premium on Option 2 offset the higher deductible.

Option 1:

Annual Premium = $775

Deductible = $500

Option 2:

Annual Premium = $650

Deductible = $1000

Let x be the number of years without a claim.

For Option 1, the total cost over x years would be:

Total Cost = $775x + $500

For Option 2, the total cost over x years would be:

Total Cost = $650x + $1000

To find when the two options break even, we need to set these two equations equal to each other and solve for x:

775x + 500 = 650x + 1000

125x = 500

x = 4

Therefore, it would take 4 years without a claim for Option 2 to break even with Option 1. After 4 years, the savings from the lower premium on Option 2 would offset the higher deductible, resulting in lower total cost.

Using confidence interval concepts, it is found that the critical value is zₐ/2 = 1.62

Here, we have,

For a confidence level of , the critical value of z is z with a p-value of:

1+a/2

The p-value is found looking at the z-table.

In this problem, confidence level of 89.48%, thus  the p-value of the z-score is:

1+0.8948/2

=0.9474

Looking at the z-table, this value is zₐ/2 = 1.62.

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Composing f(x)=2x-8 and g(x)=1/2x+4 results in the identity function, f(g(x)) = x.

To compose the two functions, we substitute g(x) into f(x) in place of x:

f(g(x)) = 2(g(x)) - 8

= 2(1/2x + 4) - 8

= x + 8 - 8

= x

Therefore, composing the two functions results in the identity function, f(g(x)) = x.

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Answer:

(a) Star's surface power.

(b) Star's interior power.

(c) Radius = 6 × 10¹⁰ cm

Step-by-step explanation:

Since the star is a sphere, we can calculate its volume and surface area by using the following formulas:

[tex]\boxed{\begin{minipage}{4 cm}\underline{Volume of a sphere}\\\\$V=\dfrac{4}{3} \pi r^3$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]   [tex]\boxed{\begin{minipage}{4 cm}\underline{Surface area of a sphere}\\\\$\vphantom{\dfrac12}SA=4 \pi r^2$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]

Given the star has a radius of about 7 × 10⁸ cm and its interior generates 3 × 10⁻⁷ watts of power per cubic centimeter, to calculate the power that the interior of the star would generate, multiply the volume of the star (in cubic centimeters) by 3 × 10⁻⁷:

[tex]\begin{aligned} \implies \textsf{Interior power}&=V \times 3 \times 10^{-7}\\&=\dfrac{4}{3} \pi (7 \times 10^8)^3\times 3 \times 10^{-7}\\&=4\pi(7^3 \times 10^{24}) \times 10^{-7}\\&=4\pi(343) \times 10^{24} \times 10^{-7}\\&=1372\pi \times 10^{24-7}\\&=1372\pi \times 10^{17}\\&=4310.265...\times 10^{17}\\&\approx 4.31 \times 10^3 \times 10^{17}\\&\approx 4.31 \times 10^{3+17}\\&\approx 4.31 \times 10^{20}\; \sf watts\end{aligned}[/tex]

Similarly, given each square centimeter of the star's surface shines 6,000 watts of power into space, to calculate how much power the star's surface would shine into space, multiply the surface area (in square centimeters) by 6,000:

[tex]\begin{aligned} \implies \textsf{Surface power}&=SA \times 6000\\&=4 \pi (7 \times 10^8)^2 \times 6000\\&=4 \pi(7^2 \times 10^{16}) \times 6000\\&=24000 \pi(49) \times 10^{16}\\&=1176000 \pi \times 10^{16} \\&=1.176\pi \times 10^6 \times 10^{16}\\&=1.176\pi \times 10^{6+16}\\&=1.176\pi \times 10^{22}\\&\approx 3.69 \times 10^{22}\; \sf watts\end{aligned}[/tex]

.

As the exponent on base 10 increases, the standard notation gets larger. Therefore, the power the star's surface shines into space is greater.

Given the star has a radius of about 5 × 10¹² cm and its interior generates 3 × 10⁻⁷ watts of power per cubic centimeter, to calculate the power that the interior of the star would generate, multiply the volume of the star (in cubic centimeters) by 3 × 10⁻⁷:

[tex]\begin{aligned} \implies \textsf{Interior power}&=V \times 3 \times 10^{-7}\\&=\dfrac{4}{3} \pi (5\times 10^{12})^3\times 3 \times 10^{-7}\\&=4\pi(5^3 \times 10^{36}) \times 10^{-7}\\&=4\pi(125) \times 10^{36} \times 10^{-7}\\&=500\pi \times 10^{36-7}\\&=1570.796... \times 10^{29}\\&\approx 1.57 \times 10^3 \times 10^{29}\\ &\approx 1.57 \times 10^{3+29}\\&\approx 1.57 \times 10^{32}\; \sf watts\end{aligned}[/tex]

Similarly, given each square centimeter of the star's surface shines 6,000 watts of power into space, to calculate how much power the star's surface would shine into space, multiply the surface area (in square centimeters) by 6,000:

[tex]\begin{aligned} \implies \textsf{Surface power}&=SA \times 6000\\&=4 \pi (5 \times 10^{12})^2 \times 6000\\&=4 \pi(5^2 \times 10^{24}) \times 6000\\&=4 \pi(25) \times 10^{24} \times 6000\\&=600000\pi \times 10^{24} \\&=6\pi \times 10^5\times 10^{24}\\&=18.8495...\times 10^{5} \times 10^{24}\\ &\approx 1.88 \times 10^1 \times 10^5 \times 10^{24}\\ &\approx 1.88 \times 10^{1+5+24}\\&\approx 1.88 \times 10^{30}\; \sf watts\end{aligned}[/tex]

.

As the exponent on base 10 increases, the standard notation gets larger. Therefore, the power the star's interior generates is greater.

To calculate the radius of the star that makes the power generated in its interior equal to the power leaving its surface, set the volume formula multiplied by 3 × 10⁻⁷ equal to the surface area formula multiplied by 6,000 and solve for r.

[tex]\begin{aligned}\implies \sf Power\;generated\;in\;interior&=\sf Power\;leaving\;surface\\ V \times 3 \times 10^{-7}&=SA \times 6000\\ \dfrac{4}{3} \pi r^3 \times 3 \times 10^{-7}&=4 \pi r^2 \times 6000\end{aligned}[/tex]

Divide both sides by 4πr²:

[tex]\implies \dfrac{1}{3} r\times 3 \times 10^{-7}=6000[/tex]

Multiply the numbers:

[tex]\implies r \times 10^{-7}=6000[/tex]

Divide both sides by 10⁻⁷:

[tex]\implies r =\dfrac{6000}{10^{-7}}[/tex]

[tex]\implies r =6000 \times 10^7[/tex]

Simplify:

[tex]\implies r =6 \times 10^3 \times 10^7[/tex]

[tex]\implies r =6 \times 10^{3+7}[/tex]

[tex]\implies r =6 \times 10^{10}[/tex]

Therefore, the radius of the star for which the power leaving the surface equals the power generated in the interior is 6 × 10¹⁰ cm.

The simplified expression after canceling out the 5a^2b term is (4ba - 12)

To cancel out the term 5a^2b in the expression 5a^2b + 4ba - 15a - 12, you need to multiply ab by 5a.

This is because (ab)(5a) = 5a^2b, which cancels out the 5a^2b term in the numerator.

To maintain the equality of the expression, you need to multiply both the numerator and denominator of the fraction by 5a.

This gives:

(ab - 3)(5a^2b + 4ba - 15a - 12) / (ab - 3)(5a)

Now, the 5a^2b term in the numerator cancels out with the 5a term in the denominator, leaving:

(4ba - 12)

Therefore, the simplified expression after canceling out the 5a^2b term is (4ba - 12)

By long division, we have

           5a + 4

ab - 3 | 5a^2b + 4ba - 15a - 12

           5a^2b - 15a

---------------------------------------------

             4ba - 12

             4ba - 12

-------------------------------------------

             0

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Answer: 0.76 or

sqrt(37)/8

Step-by-step explanation: Since we know cos(theta)=3sqrt(3)/8 we know two side lengths. The adjacent side is 3sqrt(3) and the hypotenuse is 8. So, using pythagorean theorem we can solve for the third length. 8^2-(3sqrt(3))^2= opposite^2.

Step-by-step explanation:

tan A = P/B = 36/27 = 4/3

option B is correct.

hope this helps.

As the linear functions have the same end behavior, the relationship is given as follows:

D. The sign of m and p must be the same. The relationship between the other variables does not matter.

The slope-intercept representation of a linear function is given by the equation presented as follows:

y = mx + b

The coefficients of the function and their meaning are described as follows:

The slope determines if a linear function is increasing or decreasing, hence the only condition needed for two linear functions to have the same end behavior is that the slopes of the two functions have the same signal.

More can be learned about linear functions at https://brainly.com/question/24808124

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