write an algebraic equation for the followingn problem and then solve for it. Hilltop dormitory just purchased a new soft drink machine on sale for $480. The sale price was 75% of the original price. What was the original price of the soft drink machine?

Answer: 13 1/2

Step-by-step explanation:

There are a few methods to complete this problem, but here is one of the simplest methods by separating the parts:

First, we rewrite our equation with parts separated

7 + 1/4 + 6 + 1/4

Next, we solve the whole number parts

7 + 6 = 13

Solving the fraction parts and reducing the result to the lowest terms

1/4 + 1/4 = 2/4

2/4 = 1/2

Combining the whole and fraction parts

13 + 1/2 = 13 1/2

Hope this helps! Please feel free to ask any questions if needed :)

The trigonometric identities for the terms in the left hand side of the equation indicates that we get;

tan(θ)/(1 - tan(θ)) = sin(θ)/(cos(θ) - sin(θ))

-cot(θ)/(1 - cot(θ)) = cos(θ)/(cos(θ) - sin(θ)), therefore;

Trigonometric identities are equations involving trigonometric functions that are true for all values in the interval of the input variable.

The specified equation can be expressed as follows;

tan(θ)/(1 - tan(θ)) - cot(θ)/(1 - cot(θ)) = (cos(θ) + sin(θ))/(cos(θ) - sin(θ))

The trigonometric identity indicates that the left hand side of the equation can be presented as follows;

tan(θ) = sin(θ)/cos(θ)

cot(θ) = cos(θ)/sin(θ)

(sin(θ)/cos(θ))/(1 - (sin(θ)/cos(θ))) - (cos(θ)/sin(θ))/(1 - (cos(θ)/sin(θ)))

(sin(θ)/cos(θ))/(1 - (sin(θ)/cos(θ))) = (sin(θ)/cos(θ)) × (cos(θ) - (sin(θ))/cos(θ)))

(sin(θ)/cos(θ)) × (cos(θ) - (sin(θ))/cos(θ))) = -(cos²(θ) - 1 - cos(θ)·sin(θ))/(2·cos²(θ) - 1)

-(cos²(θ) - 1 - cos(θ)·sin(θ))/(2·cos²(θ) - 1) = -(cos²(θ) - 1 - cos(θ)·sin(θ))/(cos²(θ) - 1 + cos²(θ))

= -(-sin²(θ) - cos(θ)·sin(θ))/(cos²(θ) - sin²(θ)) = (sin²(θ) + cos(θ)·sin(θ))/(cos²(θ) - sin²(θ))

(sin²(θ) + cos(θ)·sin(θ)) = sin(θ)·(sin(θ) + cos(θ))

(cos²(θ) - sin²(θ)) = (cos(θ) - sin(θ)) × (cos(θ) + sin(θ))

sin(θ)·(sin(θ) + cos(θ))/((cos(θ) - sin(θ)) × (cos(θ) + sin(θ)))

sin(θ))/(cos(θ) - sin(θ))

Similarly;

-(cos(θ)/sin(θ))/(1 - (cos(θ)/sin(θ))) = cos(x)·(cos(x) + sin(x))/((cos(θ) - sin(θ)) × (cos(θ) + sin(θ)))

cos(x)·(cos(x) + sin(x))/((cos(θ) - sin(θ)) × (cos(θ) + sin(θ))) = cos(x)/((cos(θ) - sin(θ)))

Therefore; (sin(θ)/cos(θ))/(1 - (sin(θ)/cos(θ))) - (cos(θ)/sin(θ))/(1 - (cos(θ)/sin(θ))) = sin(θ)/(cos(θ) - sin(θ)) + cos(x)/((cos(θ) - sin(θ)))

(sin(θ)/cos(θ))/(1 - (sin(θ)/cos(θ))) - (cos(θ)/sin(θ))/(1 - (cos(θ)/sin(θ))) = (sin(θ) + cos(θ))/(cos(θ) - sin(θ)) = (cos(θ) + sin(θ))/(cos(θ) - sin(θ))

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Answer:

To simplify the expression 2√(2u) - 7√(3v), we can combine like terms.

Since both terms have square roots, we can look for any common factors within the square roots. In this case, we can factor out the square root of 2 from the first term and the square root of 3 from the second term:

2√(2u) - 7√(3v) = 2√2 √u - 7√3 √v

Now, we can simplify further by combining the coefficients outside the square roots:

2√2 √u - 7√3 √v = 2√2u - 7√3v

Therefore, the simplified form of 2√(2u) - 7√(3v) is 2√2u - 7√3v.

The calculated measure of m∠PNM is 87 degrees

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

The circle

Where, we have

PM = 360 - 64 - 122

Evaluate

PM = 174

Next, we have

m∠PNM = 1/2 * 174

So, we have

m∠PNM = 87

Hence, the measure of m∠PNM is 87 degrees

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The best statement that describes the amount of water in the pool is that it is decreasing during the time interval 4. That is option 3.

To determine the time interval of the water in the pool the following is carried out.

At 0 hour, the water is at the level of = 40 gallons

At 1 hour, the water decreased to = 25 gallons.

At 2 hours, the water decreased to = 10 gallons.

The water remained at a constant level till 4 hours.

Therefore, the amount of water in the pool is that it is decreasing during the time interval 4.

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Answer:

b) [tex]MN = 6\frac{3}{4}[/tex]

Step-by-step explanation:

In ΔPQN and ΔLMN,

∠PQN = ∠LMN (given)

∠PNQ = ∠LNM (vertically opposite angles)

Since two angles in the triangles are same, the third angle must alos be same

Therefore,

∠QPN = ∠MLN

By AAA property, the two triangles are similar

Therefore,

[tex]\frac{PQ}{LM} =\frac{QN}{MN} =\frac{PN}{LN} \\\\\implies \frac{8}{6} =\frac{9}{MN} =\frac{PN}{LN} \\\\\implies \frac{8}{6} =\frac{9}{MN}\\\\\implies MN =\frac{9*6}{8}\\\\\implies MN =\frac{54}{8}\\\\\implies MN =\frac{27}{4}\\\\\implies MN =6\frac{3}{4}[/tex]

Answer:

The speed of the plane in the still air is 420 miles/hour

The speed of the wind 60 miles/hour

Step-by-step explanation:

Let the speed of the plane with the wind be v

Let the speed of the plane against the wind be u

Now, speed = distance/time

With the wind,

v = (1440 miles)/(3 hours) = 480 miles/hour

v = 480 miles/hour

Against the wind,

u = (1440 miles)/(4 hours) = 360 miles/hour

u = 360 miles/hour

Now, let the speed of plane be p, and speed of wind be w,

Now, with the wind, the speed is 480 mph,

so,

speed of plane + speed of wind = 480 mph

p + w = 480  (i)

and against the wind, the speed is 360 mph,

so,

speed of plane - speed of wind = 360mph

p-w = 360 (ii)

adding equations (i) and (ii), we get,

p+w + p-w = 480 + 360

2p = 840

p = 840/2

p = 420 miles/hour

Then, the speed of the wind will be,

p + w = 480,

420 + w = 480

w = 480 - 420

w = 60 miles/hour

The speed of the plane in still air is calculated to be 420 mph, and the speed of the wind is calculated to be 60 mph by solving the two simultaneous equations obtained from the time, rate, and distance relationship.

This problem is about the rate, time, and distance relationships. The rate at which the airplane travels in still air is r (unaffected by wind), and the speed of the wind is w. When the plane flies with the wind, it is 'assisted' and therefore travels faster - at a speed of (r + w); against the wind, it travels slower - at a speed of (r - w).

From the problem, we know that:

By solving these two equations, we get the following:

Adding these two gives 2r = 840 => r = 420 mph (the speed of the plane in still air), and subtracting gives 2w = 120 => w = 60 mph (the speed of the wind).

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The formal null hypothesis, denoted as H₀, for the comparison of means in this context would be: H₀: μ₁ = μ₂ = μ₃

In words, the null hypothesis states that the population means of English, Spanish, and Korean language proficiency levels (represented by μ₁, μ₂, and μ₃, respectively) are all equal.

The null hypothesis assumes that there is no significant difference between the means of the three language proficiency levels. It suggests that any observed differences in the sample means are due to random sampling variability and not representative of true differences in the population means.

When conducting hypothesis testing, the null hypothesis is initially assumed to be true, and the alternative hypothesis (H₁) is formulated as the opposite or complementary statement. In this case, the alternative hypothesis would be:

H₁: At least one of the population means of English, Spanish, and Korean language proficiency levels is different.

To determine whether to reject the null hypothesis, statistical tests such as analysis of variance (ANOVA) or t-tests can be used to compare the means of the language proficiency levels. If the test results provide sufficient evidence to reject the null hypothesis, it suggests that there are significant differences between the means of the language proficiency levels.

It's important to note that this is a generic explanation of the formal null hypothesis in the context of comparing means. The specific null hypothesis may vary depending on the research question and the variables being compared.

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Furthermore, changes in velocity do not always lead to shifts in the price level or nominal GDP.

The quantity theory of money suggests that changes in the money supply may result in changes in the price level if there is no change in velocity and real GDP. Changes in velocity are related to changes in the amount of money required to finance transactions. A shift in velocity can occur due to several factors, including financial innovation and changes in regulations, technology, and consumer preferences.

However, such changes may not necessarily be related to shifts in the money supply.According to the quantity theory of money, MV = PY, where M refers to the money supply, V refers to the velocity of money, P refers to the price level, and Y refers to real GDP. Thus, a change in the money supply is anticipated to result in an equal change in the nominal GDP, given that velocity and real GDP remain unchanged.

A shift in velocity can be brought about by a number of factors, including financial innovation and changes in regulations, technology, and consumer preferences. Velocity is defined as the number of times per unit of time that the monetary unit is used for transactions in the economy.

When velocity changes, the amount of money required to finance transactions also changes.The quantity theory of money asserts that an increase in the money supply will raise the price level and nominal GDP, holding constant the velocity and real GDP. Nonetheless, changes in velocity are often associated with changes in financial innovation and regulations, consumer behavior and technology, rather than changes in the money supply.

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The correct statement is;

False,  because x = [tex]cos^-^1({\frac{-1}{2} )[/tex] is in the interval [tex]( -\frac{\pi }{2} , \frac{\pi }{2} )[/tex] that is, [tex]cos^-^1({\frac{-1}{2} )[/tex] = [tex]\frac{2\pi }{3}[/tex]. So all solutions of cos x = -1/2 will be  x = 2π/3 + 2nx and x = 4π/3 + 2nx.

Option D

From the information given, we have that;

All solutions are cos x = -1/2 are given by x = 4x/2 + 2πx

There are multiple solutions to the equation cos x = -1/2, and they are denoted by the values x = 2π/3 + 2nx and x = 4π/3 + 2nx

Such that n is an integer.

This is so because the cosine function repeats itself every 2π, or its period. We therefore multiply the general answer by multiples of 2 to obtain all solutions.

The formula x = 4x/2 + 2πx implies that each solution can be reached by x being multiplied by 4/2 and 2πx being added.

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The expression is simplified to x+8/x- 4

First, we need to know that algebraic expressions are defined as expressions that are made up of terms, variables, constants, factors and coefficients.

These expressions are also made up of arithmetic operations.

From the information given, we have that;

[tex]\frac{x^2 + 2}{x^2 - 9x + 20} - \frac{42 - 3x}{x^2 - 9x + 20}[/tex]

Now, find the lowest common multiple, we have;

x² + 2 - (42 - 3x)/x² - 9x + 20

collect the like term and expand the bracket, we have;

x² + 3x - 40/x² - 9x + 20

x² + 8x - 5x - 40/x² - 5x - 4x + 20

(x-5)(x+8)/(x-5)(x -4)

x+8/x- 4

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Answer:

7, 2, -3, -8

Step-by-step explanation:

y = -5x + 2  Substitute in -1 for x

y = -5(-1) + 2

y = 5 + 2

y = 7

y = -5x + 2  Substitute in 0 for x

y = -5(0) + 2

y = 0 + 2

y = 2

y = -5 + 2 substitutes in 1 for x

y = -5(1) + 2

y = -5 + 2

y = -3

y = -5x + 2  Substitute in 2 for x

y = -5(2) + 2

y = -10 + 2

y = -8

Helping in the name of Jesus.

Rounding to the nearest cent, the value of the investment at the end of 5 years for the different compounding methods are:

Annually compounded: $31,730.82

Semi-annually compounded: $31,992.85

Quarterly compounded: $32,136.11

Monthly compounded: $32,232.45

Daily compounded: $32,265.83

To calculate the value of the investment at the end of 5 years, we can use the compound interest formula:

A = P * (1 + r/n)^(n*t)

Where:

A is the final amount (value of the investment)

P is the principal amount (initial investment)

r is the interest rate (in decimal form)

n is the number of compounding periods per year

t is the number of years

Given:

Principal amount (P): $20,700

Interest rate (r): 9% per year (0.09 in decimal form)

Number of compounding periods per year (n): It is not specified in the question. We will assume it to be annually.

Using the formula, we can calculate the value of the investment for different compounding methods:

Annually compounded:

A = 20700 * (1 + 0.09/1)^(1*5) = 20700 * (1.09)^5 ≈ $31,730.82

Semi-annually compounded:

A = 20700 * (1 + 0.09/2)^(2*5) = 20700 * (1.045)^10 ≈ $31,992.85

Quarterly compounded:

A = 20700 * (1 + 0.09/4)^(4*5) = 20700 * (1.0225)^20 ≈ $32,136.11

Monthly compounded:

A = 20700 * (1 + 0.09/12)^(12*5) = 20700 * (1.0075)^60 ≈ $32,232.45

Daily compounded:

A = 20700 * (1 + 0.09/365)^(365*5) ≈ $32,265.83

Rounding to the nearest cent, the value of the investment at the end of 5 years for the different compounding methods are:

Annually compounded: $31,730.82

Semi-annually compounded: $31,992.85

Quarterly compounded: $32,136.11

Monthly compounded: $32,232.45

Daily compounded: $32,265.83

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he slope of the line passing through the points (-6, -8) and (2, 8) is 2.

The slope of the line passing through two given points can be calculated by using the formula:m = (y₂ - y₁)/(x₂ - x₁)Where m is the slope of the line, (x₁, y₁) and (x₂, y₂) are the two given points on the line. Given a coordinate plane with a straight line passing through the points (negative 6, negative 8) and (2, 8), we can find the slope of the line using the above formula as follows:We have the two points: (x₁, y₁) = (-6, -8) and (x₂, y₂) = (2, 8)Let's substitute these values into the formula to find the slope:m = (y₂ - y₁)/(x₂ - x₁)m = (8 - (-8))/(2 - (-6))m = (8 + 8)/(2 + 6)m = 16/8m = 2

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Haleema would have approximately £3947.46 in her account after 2 years and 11 months, considering the monthly compounding interest of 5%.

To calculate the amount of money in Haleema's account after 2 years and 11 months, we need to consider the monthly compounding interest on the account.

The interest rate is given as 5% per year, which means the monthly interest rate is (5%/12) = 0.4167%.

Let's calculate the final amount using the compound interest formula: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years.

In this case, P = £3700, r = 0.004167, n = 12 (monthly compounding), and t = 2.917 (2 years and 11 months).

Plugging these values into the formula, we get:

A = £3700(1 + 0.004167/12)^(12*2.917)

A ≈ £3947.46

The amount of money in Haleema's account after 2 years and 11 months would be approximately £3947.46.

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The correct version of the sentence would be A Conference attendee's need to check in by 9:00 a.m. or risk forfeiting his or her reservation.

The sentence that contains at least one error is:

O A Conference attendee's need to check in by 9:00 a.m. or risk forfeiting their reservation.

The error in this sentence is the incorrect use of "their" to refer to a singular possessive noun "attendee's."

To maintain subject-verb agreement, it should be "his or her" instead of "their." Therefore, the correct version of the sentence would be:

A Conference attendee's need to check in by 9:00 a.m. or risk forfeiting his or her reservation.

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Answer:

Step-by-step explanation:

[tex]\frac{26y^3-8y}{2y} (4y+1)\\ \\=\frac{2y(13y^2-4}){2y} (4y+1)\\\\=(13y^2-4) (4y+1)\\\\= 13(4)y^3 -4(4y) + 13y^2 -4\\\\= 52y^3 -15y + 13y^2 -4\\\\= 52y^3 + 13y^2 -15y -4[/tex]

Answer:

x = 11

Step-by-step explanation:

3x+1 = 2x+12 (subtract 2x from both sides)

x+1 = 12 (subtract 1 from both sides)

x = 11

So x is equal to 11

To find all points on the x-axis that are 16 units away from the point (5, -8), we can use the distance formula. The distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

In this case, the y-coordinate of the point (5, -8) is -8, which lies on the x-axis. So, any point on the x-axis will have a y-coordinate of 0. Let's substitute the given values and solve for the x-coordinate.

d = √((x - 5)² + (0 - (-8))²)

Simplifying:

d = √((x - 5)² + 64)

Now, we want the distance d to be equal to 16 units. So, we set up the equation:

16 = √((x - 5)² + 64)

Squaring both sides of the equation to eliminate the square root:

16² = (x - 5)² + 64

256 = (x - 5)² + 64

Subtracting 64 from both sides:

192 = (x - 5)²

Taking the square root of both sides

√192 = x - 5

±√192 = x - 5

x = 5 ± √192

Therefore, the two points on the x-axis that are 16 units away from the point (5, -8) are:

Point 1: (5 + √192, 0)

Point 2: (5 - √192, 0)

In summary, the points on the x-axis that are 16 units away from the point (5, -8) are (5 + √192, 0) and (5 - √192, 0).

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14. The trigonometric equation (sin47°/cos43°)² + (cos43°/sin47°)² - 4cos²45° = 4

15. In the trigonometric equation 2(cos²θ - sin²θ) = 1, θ = 15°

A trigonometric equation is an equation that contains a trigonometric ration.

14. To find the value of (sin47°/cos43°)² + (cos43°/sin47°)² - 4cos²45°, we proceed as follows

Since we have the trigonometric equation (sin47°/cos43°)² + (cos43°/sin47°)² - 4cos²45°,

We know that sin47° = sin(90 - 43°) = cos43°. So, substituting this into the equation, we have that

(sin47°/cos43°)² + (cos43°/sin47°)² - 4cos²45° = (cos43°/cos43°)² + (cos43°/cos43°)² - 4cos²45°

= 1² + 1² - 4cos²45°

We know that cos45° = 1/√2. So, we have

1² + 1² - 4cos²45° = 1² + 1² - 4(1/√2)²

= 1 + 1 + 4/2

= 2 + 2

= 4

So, (sin47°/cos43°)² + (cos43°/sin47°)² - 4cos²45° = 4

15. If 2(cos²θ - sin²θ) = 1 and θ is a positive acute angle, we need to find the value of θ. We proceed as follows

Since we have the trigonometric equation 2(cos²θ - sin²θ) = 1

We know that cos2θ = cos²θ - sin²θ. so, substituting this into the equation, we have that

2(cos²θ - sin²θ) = 1

2(cos2θ) = 1

cos2θ = 1/2

Taking inverse cosine, we have that

2θ = cos⁻¹(1/2)

2θ = 30°

θ = 30°/2

θ = 15°

So, θ = 15°

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Answer:

x = 9

Q = 64

Step-by-step explanation:

Since QRST is a trapezoid

∠Q = ∠R

9x - 17 = 4x + 28

9x - 4x = 28 + 17

5x = 45

x = 45/5

x = 9

∠Q = 9x - 17

= 9*9- 17

= 81 - 17

= 64

Approximately $10,758.90 must be deposited now in the account to have a balance of $20,000 after 15 years, assuming a 3.5% annual interest rate compounded monthly.

To calculate the amount of money that must be deposited now to have a balance of $20,000 after 15 years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = final balance ($20,000)

P = principal amount (initial deposit we want to find)

r = annual interest rate (3.5% or 0.035 as a decimal)

n = number of times interest is compounded per year (monthly, so n = 12)

t = number of years (15)

Substituting the given values into the formula:

$20,000 = P(1 + 0.035/12)^(12*15)

Next, we can simplify the equation and solve for P:

$20,000 = P(1 + 0.0029166666666667)^(180)

$20,000 = P(1.0029166666666667)^(180)

To isolate P, we divide both sides of the equation by (1.0029166666666667)^(180):

P = $20,000 / (1.0029166666666667)^(180)

P ≈ $10,758.90

Therefore, approximately $10,758.90 must be deposited now in the account to have a balance of $20,000 after 15 years, assuming a 3.5% annual interest rate compounded monthly.

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Answer:

To calculate the average inflation rate between 2006 and 2009, we need to determine the percentage change in the GDP deflator over that period.

The GDP deflator is given by the equation:

GDP deflator = (Nominal GDP) / (Real GDP)

Let's calculate the GDP deflator for both 2006 and 2009:

GDP deflator in 2006 = (1366 billion dollars) / (15338 billion 2012 dollars) = 0.0889

GDP deflator in 2009 = (1692 billion dollars) / (15423 billion 2012 dollars) = 0.1097

Now, we can calculate the percentage change in the GDP deflator:

Percentage change = ((GDP deflator in 2009 - GDP deflator in 2006) / GDP deflator in 2006) * 100

= ((0.1097 - 0.0889) / 0.0889) * 100

= (0.0208 / 0.0889) * 100

= 23.39%

Therefore, the average inflation rate between 2006 and 2009 is approximately 23.39%.

To calculate the average inflation rate between 2009 and 2012, we need to determine the percentage change in the GDP deflator over that period.

Let's calculate the GDP deflator for both 2009 and 2012:

GDP deflator in 2009 = 0.1097

GDP deflator in 2012 = 1.021

Now, we can calculate the percentage change in the GDP deflator:

Percentage change = ((GDP deflator in 2012 - GDP deflator in 2009) / GDP deflator in 2009) * 100

= ((1.021 - 0.1097) / 0.1097) * 100

= (0.9113 / 0.1097) * 100

= 830.94%

Therefore, the average inflation rate between 2009 and 2012 is approximately 830.94%.

Answer:

m = 25

n = 100

Step-by-step explanation:

1, 1, 4, 8, 9, 27, 16, 64, m, n, ....

In this series,

      First term = 1

     Third term = 1 + 3 = 4

      Fifth term = 4 + 5 = 9

Seventh term = 9 + 7 = 16

So, ninth term = 16 + 9 = 25

                   [tex]\boxed{\bf m = 25}[/tex]

                Second term = 1³ = 1

                 Fourth term = 2³ = 8

                    Sixth term = 3³ = 27

                  Eighth term = 4³ = 64

                   Tenth term = 10³ = 100

                     [tex]\boxed{\bf n = 100}[/tex]

Answer:

this list

Step-by-step explanation:

1×12000=12000

2×6000=12000

3×4000=12000

4×3000=12000

5×2400=12000

6×2000=12000

8×1500=12000

10 ×1200=12000

12 ×1000=12000

The total surface area of the doghouse is 1452 ft²

The area occupied by a three-dimensional object by its outer surface is called the surface area.

The dog house has many surfaces including the roofs . The total surface area is the sum of all the area of the surfaces.

area of the roof part = 2( 13×11) + 2( 12× 10)×1/2

= 286 + 720

= 1006 ft²

surface area of the building

= 2( lb + lh + bh) -bh

= 2( 10× 11 + 10×8 + 11×8) - 10×11

= 2( 110+80+88)-110

= 2( 278) -110

= 556 -110

= 446ft²

The total surface area = 1006 + 446

= 1452 ft²

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Answer:

[tex]\displaystyle[/tex][tex]\displaystyle f'(1)=-\frac{9}{e^3}[/tex]

Step-by-step explanation:

Use Quotient Rule to find f'(x)

[tex]\displaystyle f(x)=\frac{3x^2+2}{e^{3x}}\\\\f'(x)=\frac{e^{3x}(6x)-(3x^2+2)(3e^{3x})}{(e^{3x})^2}\\\\f'(x)=\frac{6xe^{3x}-(9x^2+6)(e^{3x})}{e^{6x}}\\\\f'(x)=\frac{6x-(9x^2+6)}{e^{3x}}\\\\f'(x)=\frac{-9x^2+6x-6}{e^{3x}}[/tex]

Find f'(1) using f'(x)

[tex]\displaystyle f'(1)=\frac{-9(1)^2+6(1)-6}{e^{3(1)}}\\\\f'(1)=\frac{-9+6-6}{e^3}\\\\f'(1)=\frac{-9}{e^3}[/tex]

Answer:

[tex]f'(1)=-\dfrac{9}{e^{3}}[/tex]

Step-by-step explanation:

Given rational function:

[tex]f(x)=\dfrac{3x^2+2}{e^{3x}}[/tex]

To find the value of f'(1), we first need to differentiate the rational function to find f'(x). To do this, we can use the quotient rule.

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Quotient Rule for Differentiation}\\\\If $f(x)=\dfrac{g(x)}{h(x)}$ then:\\\\\\$f'(x)=\dfrac{h(x) g'(x)-g(x)h'(x)}{(h(x))^2}$\\\end{minipage}}[/tex]

[tex]\textsf{Let}\;g(x)=3x^2+2 \implies g'(x)=6x[/tex]

[tex]\textsf{Let}\;h(x)=e^{3x} \implies h'(x)=3e^{3x}[/tex]

Therefore:

[tex]f'(x)=\dfrac{e^{3x} \cdot 6x -(3x^2+2) \cdot 3e^{3x}}{\left(e^{3x}\right)^2}[/tex]

[tex]f'(x)=\dfrac{6x -(3x^2+2) \cdot 3}{e^{3x}}[/tex]

[tex]f'(x)=\dfrac{6x -9x^2-6}{e^{3x}}[/tex]

To find f'(1), substitute x = 1 into f'(x):

[tex]f'(1)=\dfrac{6(1) -9(1)^2-6}{e^{3(1)}}[/tex]

[tex]f'(1)=\dfrac{6 -9-6}{e^{3}}[/tex]

[tex]f'(1)=-\dfrac{9}{e^{3}}[/tex]