Let ​f(x)x and ​g(x). Perform the function operation and then find the domain.
​g(x)​f(x)

Part 1
​g(x)​f(x)

enter your response here ​(Simplify your​ answer.)

The derivative of the function f(x) is:

f'(x) = (sinx +  2cosx)/(2√x)  

The derivative of the function f(x) is:

f'(x) = (sinx +  2cosx)/(2√x)  

Differentiation involves finding the derivative of a function. The derivative of a function represents the rate of change of the function concerning its input variable.

For any function of the form f(x) = u(x)·v(x). The derivative is given by:

f'(x) = u'(x)·v(x) + v'(x)·u(x)

f(x) = (√x)·sinx can be written as f(x) = x^1/2. sin x

Thus, if f(x) = (√x)·sinx, the derivative will be:

f'(x) = (1/2)x^1/2sinx + x^1/2cosx

f'(x) = sinx/(2√x) + cosx/(√x)  

f'(x) = (sinx +  2cosx)/(2√x)  

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The required dimensions of the kennel are 17 feet by 8.5 feet.

Let the length of the kennel be L and the width be W.

We know that the total length of fence available is 25 feet. Since one side of length 9 feet does not need fencing, the total length of the other three sides that need fencing is (L + 2W - 9).

Therefore, we have:

25 = L + 2W - 9

Simplifying the equation, we get:

L + 2W = 34

We also know that the area of the kennel is given by:

Area = Length x Width

Substituting L = 34 - 2W from the first equation into the above equation, we get:

Area = (34 - 2W) x W

Simplifying the equation, we get:

Area = 34W - 2W²

To maximize the area, we differentiate the above equation with respect to W, set it equal to zero, and solve for W:

d(Area)/dW = 34 - 4W = 0

Solving for W, we get:

W = 8.5

Substituting this value of W back into the equation L + 2W = 34, we get:

L + 2(8.5) = 34

L + 17 = 34

L = 17

Therefore, the dimensions of the kennel are 17 feet by 8.5 feet.

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Therefore, 34 student tickets and 126 adult tickets were sold.

Algebraic expressiοn can be defined as cοmbinatiοn οf variables and cοnstants.

Write twο equatiοns tο mοdel the prοblem:

Let x be the number οf student tickets sοld, and y be the number οf adult tickets sοld. Then, we can write the fοllοwing twο equatiοns:

5x + 10y = 1430 (the total amount earned from ticket sales is $1430)

x + y = 160 (the total number of tickets sold is 160)

Solve for one of the variables in terms of the other:

We can rearrange the second equation to solve for one of the variables in terms of the other:

x + y = 160

x = 160 - y (subtract y from both sides)

Substitute the expression found in step 2 into one of the equations from step 1:

We can substitute the expression x = 160 - y into the first equation:

5x + 10y = 1430

5(160 - y) + 10y = 1430 (substitute x = 160 - y)

800 - 5y + 10y = 1430 (distribute the 5)

5y = 630 (combine like terms)

y = 126 (divide both sides by 5)

Solve for the other variable:

Now that we know y = 126, we can use the expression x = 160 - y to find x:

x = 160 - y

x = 160 - 126

x = 34

Therefore, 34 student tickets and 126 adult tickets were sold.

Check the solution:

We can check our solution by plugging in x = 34 and y = 126 into the original equations:

5x + 10y = 1430

5(34) + 10(126) = 1430

x + y = 160

34 + 126 = 160

Therefore, Both equations check out, so our solution is correct.

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Therefore, the value of the car in the year 2003 will be approximately $8,962 when rounded to the nearest 50 dollars.

The annual rate of change is a measure that indicates the percentage increase or decrease in a value over a period of one year. It is commonly used to track changes in economic indicators such as Gross Domestic Product (GDP), inflation, and unemployment.

To calculate the annual rate of change, you need to first determine the starting value and ending value for the period in question. You then calculate the percentage change between the two values using the following formula:

[tex]Annual rate of change = ((Ending value - Starting value) / Starting value) * 100[/tex]

A) To find the annual rate of change between 1991 and 2000, we can use the formula:

[tex]r = (V1/V0)^{(1/n)} - 1[/tex]

where V0 is the initial value, V1 is the final value, and n is the number of years. Plugging in the given values, we get:

[tex]r = (12000/45000)^{(1/9)} - 1[/tex]

r ≈ -0.1049

Therefore, the annual rate of change between 1991 and 2000 is approximately -0.1049.

B) To express the rate of change as a percentage, we can multiply it by 100:

[tex]r = -0.1049 * 100[/tex]

r ≈ -10.49%

Therefore, the correct answer to part A written in percentage form is approximately -10.49%.

C) Assuming the car value continues to drop by the same percentage, we can use the formula:

[tex]V = V_0 * (1 + r)^n[/tex]

where V0 is the initial value, r is the annual rate of change, and n is the number of years. Plugging in the given values, we get:

[tex]V = 12000 *(1 - 0.1049)^3[/tex]

V ≈ $8,961.75

Therefore, the value of the car in the year 2003 will be approximately $8,962 when rounded to the nearest 50 dollars.

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Answer:

3.1

Step-by-step explanation:

To express a percent as a mixed number, we need to divide the percent by 100 and convert it to a mixed number.

Let's do this for each option:

310% = 310/100 = 3.1

3.1 can be written as the mixed number 3 1/10.

49% = 49/100 = 0.49

0.49 cannot be expressed as a mixed number because it is less than 1.

7.4% = 7.4/100 = 0.074

0.074 cannot be expressed as a mixed number because it is less than 1.

0.001% = 0.001/100 = 0.00001

0.00001 cannot be expressed as a mixed number because it is less than 1.

Therefore, the percentage that can be expressed as a mixed number is 310%, which is equivalent to 3 1/10.

Step-by-step explanation:

pi ( x^2 - 36 )

pi ( x-6)(x+6)     Done.

Answer:

(D)

Step-by-step explanation:

The sum of the exterior angles of any polygon is [tex]360^{\circ}[/tex].

Wangari plants 12 trees every 3 hours, so her planting rate is 12 trees per 3 hours, or 4 trees per hour.

To find the equation that relates the number of trees Wangari plants (p) and the time she spends planting them (h) in hours, we can use the formula for direct variation:

p = k*h

where k is a constant of proportionality. Since Wangari plants at a rate of 4 trees per hour, k = 4:

p = 4h

Therefore, the equation that relates the number of trees Wangari plants (p) and the time she spends planting them (h) in hours is p = 4h.

The length of line EF is proved to be parallel to the length of line EF.

To prove that EF is parallel to BD, we need to show that the opposite interior angles are equal.

Let's denote the center of the circle as O, and the intersection point of BD and AE as X. Since ABCD is a square, we have:

From the above information, we can deduce that:

∠ABD = 45°, and

∠OBD = 90°.

Since line_AF = line_AE, we can also deduce that:

∠FAE = ∠FEA.

Now, let's consider the triangle AFE. We have:

∠AFE + ∠FAE + ∠FEA = 180° (sum of angles in a triangle)

∠AFE + 2∠FAE = 180° (substituting ∠FEA with ∠FAE)

∠AFE = 180° - 2∠FAE

Also, in triangle AXε, we have:

∠AXε + ∠AEX + ∠XAE = 180° (sum of angles in a triangle)

∠AXε + ∠AEX + ∠FAE = 180° (substituting ∠XAE with ∠FAE)

∠AXε + ∠FAE + ∠FAE = 180° (rearranging terms)

∠AXε + 2∠FAE = 180°

Now, let's consider the quadrilateral ABXE. We have:

∠ABX + ∠AXε + ∠AEB = 360° (sum of angles in a quadrilateral)

∠ABX + ∠AEX + ∠AEB = 360° (rearranging terms)

∠ABX + ∠FAE + ∠AEB = 360° (substituting ∠AEX with ∠FAE)

∠ABX + 2∠FAE = 360° (substituting ∠AEB with ∠FAE)

∠ABX = 360° - 2∠FAE

Finally, let's consider the triangle BDE. We have:

∠BDE + ∠BED + ∠EBD = 180° (sum of angles in a triangle)

∠BDE + ∠AEB + ∠EBD = 180° (substituting ∠BED with ∠AEB)

∠BDE + ∠FAE + ∠EBD = 180° (substituting ∠AEB with ∠FAE)

∠BDE + 2∠FAE = 180° (since ∠FAE = ∠FEA)

∠BDE = 180° - 2∠FAE

From the above equations, we can see that:

∠ABX = ∠BDE (since both are equal to 360° - 2∠FAE)

∠ABD = ∠EBD (since both are equal to 45°)

Therefore, by the angle-angle (AA) criterion for similarity, we have:

triangle ABD is similar to triangle EXD.

This implies that:

BD/AD = XD/ED (by the property of similar triangles)

Since AD = BD (since ABCD is a square), we have:

BD/BD = XD/ED

1 = XD/ED

XD = ED

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Step-by-step explanation:

what is the answer to 100001/9

It is not safe to assume that candidate A will win the election

Statistical inference is the process of drawing conclusions or making decisions about a population based on sample data. It involves using statistical methods and techniques to analyze and interpret data, estimate population parameters, and assess the uncertainty of the results.

Here we have

A sample of 250 people was surveyed and a 95% Confidence interval was calculated. From this confidence interval, it can be concluded that between 48% and 60% of the population will vote for Candidate.

According to the given data, It is not safe to assume that Candidate A will win the election based solely on the confidence interval calculated from the sample.

A confidence interval is a range of plausible values for a population parameter, but it does not guarantee a particular outcome in the future.

Other factors such as the size and composition of the actual voting population, as well as the campaign strategies and performance of the candidates, should also be considered.

Hence,

It is not safe to assume that candidate A will win the election

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Step-by-step explanation:

10 = k * 5

1 = k * 1

10 / 1 = k * 5 / (k * 1)

10 = 5k

k = 2

y = kx

n = 2 * 1

n = 2

Based on the given statements, we can conclude:

If X, then not Y: This means that if X is true, then Y cannot be true.

If not Y, then Z: This means that if Y is not true, then Z must be true.

We are also given the information that Y is true. Therefore, we can conclude that:

Y is true, so not X: Since Y is true, X cannot be true, according to the first statement.

If not Y, then Z: Since Y is true, we cannot conclude anything about Z. However, we do know that Y cannot be false.

So the final conclusion is that X is false and Y is true, but we don't have enough information to determine whether Z is true or false.

Answer:

The system of equations has two real roots

Step-by-step explanation:            

y = x² + 3x + 4 ------------(I)      

y - x = 7 ------------------------(II)

y = 7 +x

Substitute y = 7 + x in equation (I),        

7 + x = x² + 3x + 4              

0 = x² + 3x + 4 - x - 7              

0 = x² + 3x - x + 4 - 7

Combine like terms,          

0 = x² + 2x - 3

x² + 2x - 3 = 0

a = 1; b = 2; c = -3

Discriminant = b² - 4ac                    

                     = 2² - 4*1*(-3)                    

                     = 4 + 12                    

                     = 16  

System of equations has two real roots as discriminant is greater than 0.

The probability of randomly selecting a student that didn't get an A is P = 0.61

We want to find the probability that the student did not get an A.

To get this, we need to take the quotient between the number of students that didn't get an A, and the total number of students.

In the table,  can see that there is a total of 69 students and we also can see that of these 69, 27 got an A.

Then the number that did not get an A is:

69 - 27 = 42

Then the probability is:

P = 42/69 = 0.61

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When expressed in scientific notation, extremely large or tiny numbers are easier to comprehend. The expression (6x10⁻³)(1.4x10¹) have the solution in scientific notation as 8.4 x 10⁻².

A number can be expressed using scientific notation if it cannot be conveniently expressed in decimal form due to its size or shape, or if doing so would require writing out an abnormally long string of digits. In the UK, it is also referred to as standard form, standard index form, and standard form.

Despite the fact that we are aware that whole numbers can never be exhausted, we are unable to record such vast amounts of data on paper. Moreover, a simpler method of representation is required for the numbers that appear at the millions place after the decimal. This may make it challenging to represent small numbers in their larger form. We employ a scientific notation as a result.

Given:

= (6x1.4)(10⁻³x10¹) = (8.4x10¹)(10⁻²)

= 8.4x (10¹ x 10⁻²)

= 8.4x (10¹ x 10⁻²)

= 8.4 x 10⁻²

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Thus, the polynomial at x=i needs to be evaluated: As a result, when x-1 is divided by [tex]-3x^4 - 2x^3 + 5x^2 - 7x,[/tex] the remaining is -8i - 5.

Using just the activities of addition, removal, multiplication, and non-negative decimal exponents, a polynomial is a mathematical equation made up of variables and coefficients. Polynomials can contain one or perhaps more variables, and they can be categorised based on their degree, which is the polynomial's highest exponent. The most familiar example of polynomial is the exponential, which has a rank of 2 and may be expressed in the form ax2 + bx + c. The shortest polynomials be monomials, which have only one term. Algebra, algebra, and number theory are just a few of the mathematical areas where polynomials are used.

given

The remainder theorem can be used to get the remaining when[tex]-3x^4, 2x^3, 5x^2[/tex], and 7x are divided by x-i.

The remainder is p when a polynomial p(x) is divided by (x-a), according to the theorem (a).

In this instance, we must determine the remaining after dividing [tex]3x^4[/tex] by x-i and adding [tex]2x^3 , 5x^2 ,7^x.[/tex]

Thus, the polynomial at x=i needs to be evaluated: As a result, when x-1 is divided by [tex]-3x^4 - 2x^3 + 5x^2 - 7x,[/tex] the remaining is -8i - 5.

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The percentage of compaction of the tested soil is 92.1%.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can include numbers, variables, and mathematical operations, such as addition, subtraction, multiplication, and division.

a) To compute the water content, we need to find the weight of water in the soil sample.

Wet unit weight of soil = (weight of wet soil)/(volume of soil)

The volume of soil can be found using the weight of sand (sand core) to fill the test hole:

Volume of soil = Volume of sand cone = (weight of sand)/(bulk density of sand)

Volume of soil = 1636 g / 15.73  = 0.104

Wet unit weight of soil = (2100 g - 1636 g) / 0.104  = 5490

The dry unit weight of soil can be found by dividing the dry weight of the soil sample by its volume:

Dry unit weight of soil = (weight of dried soil)/(volume of soil)

Dry unit weight of soil = 1827 g / 0.104 = 17558.5

b) To compute the in-place dry unit weight of tested soil, we need to know the water content of the soil.

Water content = [(weight of wet soil - weight of dry soil) / weight of dry soil] x 100%

Water content = [(2100 g - 1827 g) / 1827 g] x 100% = 14.4%

Dry unit weight of tested soil = (dry unit weight of soil) / (1 + water content)

Dry unit weight of tested soil = 17558.5 / (1 + 0.144) = 15294.3

c) To compute the percentage of compaction, we need to compare the in-place dry unit weight to the maximum dry unit weight.

Maximum dry unit weight = 18.09

Optimum moisture content = 13%

Maximum wet unit weight = maximum dry unit weight / (1 - optimum moisture content/100)

Maximum wet unit weight = 18.09  / (1 - 0.13) = 20.805

Maximum weight of soil = maximum wet unit weight x volume of soil

Maximum weight of soil = 20.805  x 0.104 = 2.161 kN

Actual weight of soil = (dry unit weight of tested soil) x (1 + water content) x volume of soil

Actual weight of soil = 15.294  x (1 + 0.144) x 0.104  = 1.990 kN

Percentage of compaction = (actual weight of soil / maximum weight of soil) x 100%

Percentage of compaction = (1.990 kN / 2.161 kN) x 100% = 92.1%

Therefore, the percentage of compaction of the tested soil is 92.1%.

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Answer: 106 stickers

Step-by-step explanation:

38 + 75 + 18 = 131 (total amount of stickers she got)

131 - 25 = 106

To determine the actual height of the climbing frame, we need to know the scale factor of the model. If the scale factor is, for example, 1:50, it means that every 1 cm on the model represents 50 cm in real life.

Assuming that we have the scale factor, we can use the following proportion:

model height / actual height = scale factor

We know that the model height is 10 cm, and we want to find the actual height. Let's say the scale factor is 1:100. Then we have:

10 cm / actual height = 1/100

Multiplying both sides by the actual height, we get:

actual height = (10 cm) x (100/1) = 1000 cm

Therefore, the actual height of the climbing frame in this example is 1000 cm, or 10 meters.

The three options that represent the correct steps to solve quadratic equation are:

What is equation?

In mathematics, an equation is a statement that two expressions are equal. It typically consists of two sides, called the left-hand side (LHS) and the right-hand side (RHS), connected by an equal sign.

To solve the quadratic equation [tex]8x^2 + 16x + 3 = 0[/tex], Patel could use the following steps:

Use the quadratic formula: x = (-b ± √([tex]b^2[/tex] - 4ac)) / 2a, where a = 8, b = 16, and c = 3. This formula gives the solutions to any quadratic equation of the form [tex]ax^2[/tex] + bx + c = 0.

Factor the quadratic equation by finding two numbers that multiply to give ac (8 * 3 = 24) and add to give b (16).

This can be a bit tricky, but in this case, the factors are (4, 6). So we can write [tex]8x^2[/tex] + 16x + 3 as [tex]8x^2[/tex] + 4x + 2x + 3, and then group the terms as ([tex]8x^2[/tex] + 4x) + (2x + 3) = 4x(2x + 1) + 1(2x + 3).

Use the factored form of the equation to set each factor equal to zero and solve for x.

So we have 4x(2x + 1) + 1(2x + 3) = 0, which gives us two possible solutions: 2x + 1 = 0, which gives x = -1/2, and 2x + 3 = 0, which gives x = -3/2.

Therefore, the three possible steps Patel could use to solve the quadratic equation are:

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[tex]Therefore, (gof)(x) = 0.9(x - 220).[/tex]

This function models the price of the computer after first taking a $220 discount and then taking a 10% discount on the discounted price.

Discount refers to a reduction in the price of a product or service. It is often used as a marketing strategy to attract customers and increase sales. The discount can be in the form of a percentage reduction from the original price, a fixed amount off the price, or other incentives such as free gifts or coupons. Discounts can be offered for various reasons, such as to clear out inventory, reward loyal customers, or to promote a new product or service. Businesses use discounts as a way to create a sense of urgency and encourage customers to make a purchase.

Sure, here's a step-by-step explanation:

Given:

The regular price of a computer is x dollars.

[tex]f(x) = x - 220[/tex] is a function that gives the price of the computer after a $220 discount.

[tex]g(x) = 0.9x[/tex] is a function that gives the price of the computer after a 10% discount.

To understand what these functions model in terms of the price of the computer, we can break down each function:

Function f(x):

[tex]f(x) = x - 220[/tex]

This function takes the regular price of the computer (x) and subtracts $220 from it.

Therefore, the result of this function gives us the price of the computer after a $220 discount.

Function g(x):

[tex]g(x) = 0.9x[/tex]

This function takes the regular price of the computer (x) and multiplies it by 0.9 (which is the same as taking 10% off).

Therefore, the result of this function gives us the price of the computer after a 10% discount.

In summary, function f models the price of the computer after a $220 discount, while function g models the price of the computer after a 10% discount.

To find (gof)(x), we need to first find g(f(x)):

[tex]g(f(x)) = g(x - 220)[/tex] [Substituting f(x) into g(x)]

[tex]= 0.9(x - 220)[/tex] [Substituting g(x) into the above expression]

[tex]Therefore, (gof)(x) = 0.9(x - 220).[/tex]

This function models the price of the computer after first taking a $220 discount and then taking a 10% discount on the discounted price.

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In the word problem , the length of wood Mr. DeWitt carved off is 3.6cm.

What is word problem?

Word problems are often described verbally as instances where a problem exists and one or more questions are posed, the solutions to which can be found by applying mathematical operations to the numerical information provided in the problem statement. Determining whether two provided statements are equal with respect to a collection of rewritings is known as a word problem in computational mathematics.

According to the given ,

Length of wood = 20.3 cm

The length of boat = 16.7 cm.

Now Length of wood he carved off is,

=> Length of wood - length of boat

=> 20.3 - 16.7

=> 3.6 cm.

Hence the length of wood Mr. DeWitt carved off is 3.6cm.

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x²+(9/2)x-3/2

=2x²+9x-3

The statements which are true are that the radius of the circle is 3 units, the center of the circle lies on the x-axis and the radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

What is a circle?

A circle is formed by all points in a plane that are at a particular distance from the center. To put it another way, it is the curve that a moving point in a plane draws to maintain a constant distance from another point.

We are given a circle whose equation is [tex]x^{2}[/tex] + [tex]y^{2}[/tex] - 2x - 8 = 0.

We know that the general form of a circle is  [tex]x^{2}[/tex] + [tex]y^{2}[/tex] + 2gx + 2fy + C = 0, where (-g, -f) is the center and √[tex]g^{2}[/tex] + [tex]f^{2}[/tex] - C is the radius.

So, in the equation, g is -1 and f is 0.

So, the center is (1, 0) which represents that the center lies on the x - axis.

Now,

⇒ Radius = √[tex]g^{2}[/tex] + [tex]f^{2}[/tex] - C

⇒ Radius = √1 + 0 - (-8)

⇒ Radius = √1 + 8

⇒ Radius = √9

⇒ Radius = 3 units

So, the radius of the circle is 3 units.

Now, radius of circle x² + y² = 9 is

[tex]r^{2}[/tex] = 9

r = 3

So, the  radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

Hence, the required solution has been obtained.

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The complete question has been attached below.

Answer: Mark as brainliest

Option A shows the correct method to calculate the volume of the cylinder-shaped mold.

Step-by-step explanation:

The correct method to calculate the number of cubic units of cake batter needed to fill the mold is:

V = πr^2h

Where:

V = volume of the cake batter needed

π = 355/113 (approximate value of pi)

r = radius of the cylinder (diameter/2 = 10/2 = 5)

h = height of the cylinder (7)

Substituting the values in the formula, we get:

V = (355/113) x 5^2 x 7

V = 616.07 cubic inches (rounded to two decimal places)

Therefore, approximately 616.07 cubic inches of cake batter are needed to fill the mold.

After answering the presented question, we can conclude that As a quadrilateral result, the value of x is 112 degrees.

In geometry, a quadrilateral is a four-sided polygon with four edges and four corners. The term is derived from the Roman terms quadri and latus (meaning "side"). A rectangle is a two-dimensional form with four sides, four vertices, and four corners. Concave and convex surfaces are basically of two types. In addition, trapezoids, parallelograms, rectangles, rhombuses, and squares are subclasses of convex quadrilaterals. A rectangle is a two-dimensional structure with four straight sides. Quadrilaterals come in a variety of shapes, including parallelograms, trapezoids, rectangles, kites, squares, and rhombuses.

Because PQRS is inscribed in circle O, the quadrilateral's opposite angles sum up to 180 degrees. Therefore,

QOS angle + QOR angle = 180 degrees

Angle QOS x (Angle R x Angle OQR) = 180°

Angle QOS = 180 degrees + (180 degrees - Angle P + 38 degrees)

Angle P = 180 degrees - Angle QOS + 218 degrees

Angle P - 38 degrees - 218 degrees = Angle QOS

QOS angle = Angle P - 256 degrees

Lastly, we may get Angle P by using the knowledge that the total of the angles of a triangle is 180 degrees.

QOS angle = 8 degrees, 38 degrees, and 218 degrees

QOS angle = -248 degrees

QOS angle = -248 degrees plus 360 degrees = 112 degrees

As a result, the value of x is 112 degrees.

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If the height of the triangular roof is half its base, then the height of the roof is:

h = (1/2) * 32 = 16 feet

The area of a triangle is given by the formula:

A = (1/2) * base * height

Plugging in the values we have:

A = (1/2) * 32 * 16

A = 256 square feet

Therefore, the area of the triangular roof is 256 square feet.