Use the suggested substitution to write the expression as a
trigonometric expression. Simplify your answer as much as possible.
Assume 0≤θ≤π2.
√1−x^2, x=cos(θ)

The vertex form of the equation for function g is g(x) = a(x + 1)² - 38, where a is the same as in function f.

A quadratic function is one of the forms f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero.

The vertex form of the equation of a quadratic function is given by:

y = a(x - h)² + k

where (h, k) are the coordinates of the vertex. To find the vertex form of the equation for function g, we need to determine the vertex of function f and then adjust it to 2 units below and 3 units to the left.

To find the vertex of function f, we can use the formula:

h = -b / 2a

where a and b are the coefficients of the quadratic term and the linear term, respectively.

For function f(x) = x² - 4x - 32, we have a = 1 and b = -4, so the vertex has an x-coordinate of:

h = -b / 2a = -(-4) / 2(1) = 2

To find the y-coordinate of the vertex, we can substitute this value of h into the function:

f(2) = 2² - 4(2) - 32 = -36

So the vertex of function f is (2, -36).

To adjust the vertex of the function f 2 units below and 3 units to the left, we need to subtract 2 from the y-coordinate and add 3 to the x-coordinate. Therefore, the vertex of function g is:

(h, k) = (2 - 3, -36 - 2) = (-1, -38)

Therefore, the vertex form of the equation for function g is g(x) = a(x + 1)² - 38, where a is the same as in function f.

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Answer:

Yes, the triangle is right-angled since one of the angles is 90°

Step-by-step explanation:

The sum of the three angles of a triangle must add up to 180°

Here the three angles are 3x – 7, 4x – 1, and 5x + 20.

Adding them up gives

3x – 7 + 4x – 1 + 5x + 20 = 180

Grouping like terms:
3x + 4x + 5x - 7 - 1 + 20 = 180

12x + 12 = 180

12x = 180 - 12 = 168

x = 168/12 = 14

Substituting this value of x into each of the expressions:
3x – 7  = 3(14) - 7 = 42 - 7 = 35°

4x – 1 = 4(14) - 1 = 56 - 1 = 55°

5x + 20 = 5(14) + 20 = 70 + 20 = 90°

Since one of the angles is 90° it is indeed a right angled triangle

Answer:

8183.27

Step-by-step explanation:

Answer:

8183.3

Step-by-step explanation:

500(1+0.15)20=8183.27 round it to become 8183.3

The area of the circle is  490. 625 cm²

The formula for calculate the circumference of a circle is expressed as;

C  = 2πr

Given that;

From the information given, we have that;

Substitute the value of the circumference

25π = 2πr

Divide by the coefficient of r

r = 12. 5cm

Area of a circle = πr²

Substitute the values

Area = 3.14 × (12.5)²

Find the square

Area = 490. 625 cm²

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Answer:
(i) Rs. 3,300, Rs. 4,125, Rs. 1,650, and Rs. 2,475

(ii) 4,950

Step-by-step explanation:

i) To find Mr. Jha's monthly expenditure on each item, we can use the given percentages to calculate the amount he spends on each category:

Food expenditure = 20% of Rs. 16,500 = 0.20 x 16,500 = Rs. 3,300

House expenditure = 25% of Rs. 16,500 = 0.25 x 16,500 = Rs. 4,125

Fuel expenditure = 10% of Rs. 16,500 = 0.10 x 16,500 = Rs. 1,650

Miscellaneous expenditure = 15% of Rs. 16,500 = 0.15 x 16,500 = Rs. 2,475

Therefore, Mr. Jha's monthly expenditure on food, house, fuel, and miscellaneous items is Rs. 3,300, Rs. 4,125, Rs. 1,650, and Rs. 2,475 respectively.

(ii) To calculate how much Mr. Jha saves every month, we can subtract his total expenditure from his monthly income:

Total expenditure = Rs. 3,300 + Rs. 4,125 + Rs. 1,650 + Rs. 2,475 = Rs. 11,550

Savings = Monthly income - Total expenditure = Rs. 16,500 - Rs. 11,550 = Rs. 4,950

Therefore, Mr. Jha saves Rs. 4,950 every month.

Answer: (i) Rs. 3,300, Rs. 4,125, Rs. 1,650, and Rs. 2,475 (ii) 4,950Step-by-step explanation:i) To find Mr. Jha's monthly expenditure on each item, we can use the given percentages to calculate the amount he spends on each category:Food expenditure = 20% of Rs. 16,500 = 0.20 x 16,500 = Rs. 3,300House expenditure = 25% of Rs. 16,500 = 0.25 x 16,500 = Rs. 4,125Fuel expenditure = 10% of Rs. 16,500 = 0.10 x 16,500 = Rs. 1,650 Miscellaneous expenditure = 15% of Rs. 16,500 = 0.15 x 16,500 = Rs. 2,475Therefore, Mr. Jha's monthly expenditure on food, house, fuel, and miscellaneous items is Rs. 3,300, Rs. 4,125, Rs. 1,650, and Rs. 2,475 respectively.(ii) To calculate how much Mr. Jha saves every month, we can subtract his total expenditure from his monthly income:Total expenditure = Rs. 3,300 + Rs. 4,125 + Rs. 1,650 + Rs. 2,475 = Rs. 11,550Savings = Monthly income - Total expenditure = Rs. 16,500 - Rs. 11,550 = Rs. 4,950 so Mr. Jha saves Rs. 4,950 every month. your welcome.

Answer: The surface area of a cylinder is given by the formula:

S = 2πr^2 + 2πrh

where r is the radius of the base, h is the height of the cylinder, and π is the mathematical constant pi.

In this case, the radius is 28 mm and the height is 30 mm.

Substituting these values into the formula, we get:

S = 2π(28)^2 + 2π(28)(30)

Simplifying the equation:

S = 2π(28)(28 + 30)

S = 2π(28)(58)

S = 32176π

Using a calculator, we can approximate the value of S to 3 significant figures:

S ≈ 101285.744 mm^2

Therefore, the surface area of the cylinder is approximately 101286 mm^2 (rounded to 3 significant figures).

Step-by-step explanation:

Cοmpletes the prοοf = (a + b + c) ± 2√a² + b² + c² - ab - bc - ac

An algebraic equatiοn is a mathematical statement that uses variables and mathematical οperatiοns such as additiοn, subtractiοn, multiplicatiοn, and divisiοn tο express the equality οf twο algebraic expressiοns.

The gοal οf sοlving an algebraic equatiοn is tο determine the values οf the variables that satisfy the equatiοn.

Tο prοve this statement, we will start by simplifying the given equatiοn:

√x-3a + √x-3b = √x-3c

Subtracting √x-3a frοm bοth sides, we get:

√x-3b = √x-3c - √x-3a

Squaring bοth sides, we get:

(x - 3b) = (x - 3c) + (x - 3a) - 2√(x - 3a)(x - 3c)

Simplifying the right-hand side, we get:

(x - 3b) = 2x - 3(a + b + c) - 2√(x - 3a)(x - 3c)

Mοving all the terms invοlving x tο οne side, we get:

x - 2x + 3(a + b + c) = 2√(x - 3a)(x - 3c) + 3b

Simplifying, we get:

x = (a + b + c) ± 2√(x - 3a)(x - 3c) + 2bc - 2ab - 2ac

Nοw, let's simplify the expressiοn under the square rοοt:

(x - 3a)(x - 3c) = x² - 3ax - 3cx + 9ac

= x² - 3(a + c)x + 9ac

Plugging this back intο the expressiοn fοr x, we get:

x = (a + b + c) ± 2√(x² - 3(a + c)x + 9ac) + 2bc - 2ab - 2ac

Expanding the square rοοt, we get:

x = (a + b + c) ± 2√[(x - (a + c))² - (a - c)²] + 2bc - 2ab - 2ac

= (a + b + c) ± 2√(x - a - c)² - (a - c)² + 2bc - 2ab - 2ac

= (a + b + c) ± 2√(x - a - b - c)² - 4ab - 4bc - 4ac + 2a² + 2b² + 2c²

= (a + b + c) ± 2√a² + b² + c² - ab - bc - ac

This cοmpletes the prοοf.

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Answer: 10.53 ft^2

Step-by-step explanation:

To get the area, multiply length by width

Therefore, the mark-up percentage for the model solar system is 200%. This means that the selling price is twice the cost price.

We must divide the value by the entire value to find the percentage, then increase the resulting number by 100.

The mark-up percentage is the percentage by which the selling price exceeds the cost price.

In this case, the cost price of the model solar system is $24 and the selling price is $72. The mark-up amount is the difference between the selling price and the cost price, which is:

$72 - $24 = $48

To find the mark-up percentage, we divide the mark-up amount by the cost price and multiply by 100:

Mark-up percentage = (Mark-up amount / Cost price) x 100%

Mark-up percentage = ($48 / $24) x 100%

Mark-up percentage = 200%

Therefore, the mark-up percentage for the model solar system is 200%. This means that the selling price is twice the cost price.

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A population consists of five members 2,3,6,8,1). consider all possible samples of size two which can be doamn with replacement from this positioning population.  The mean is  4 and standard deviation of the population is 1.67. The mean and standard error of sample mean (x) from the sampling distribution of x is 4 and 1.18 respectively

i) The mean and standard deviation of the population:

Mean = (2+3+6+8+1)/5 = 20/5 = 4

Standard deviation = √[(2-4)²+(3-4)²+(6-4)²+(8-4)²+(1-4)²]/5 = √14/5 = 1.67


ii) The mean and standard error of sample mean (x) from the sampling distribution of x:

Mean of sample mean = Mean of population = 4

Standard error of sample mean = Standard deviation of population/√n = 1.67/√2 = 1.18


The mean of a population is calculated by adding up all the values and dividing by the number of values in the population. The standard deviation of a population is calculated by finding the difference between each value and the mean, squaring those differences, adding them up, and then dividing by the number of values in the population.

The mean of the sample mean is the same as the mean of the population, and the standard error of the sample mean is the standard deviation of the population divided by the square root of the sample size.

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Question 1.) The value of the angle C = 90° ( acute angle)

Question 2.) The area of the triangle = 240

Question 3.) The measure of the angle B = 65.4°

To calculate the area of a triangle the formula below is used:

Area of a triangle = 1/2 base × height.

where: base = 12

height = 40

area = 1/2 × 12× 40

= 480/2 =240

The measure of the angle B can be calculated using the SOHCAHTOA formula:

The sides of the triangle connected to angle B = Hypotenuse and opposite.

Sin B = opposite/hypotenuse

where,

opposite = 40

hypothenuse = 44

Sin B = 40/44 =0.909090909

B = sin-¹ (0.909090909)

B = 65.4°

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To find the inverse equation of f(x) = 3x + 5, we first replace f(x) with y:

y = 3x + 5

Next, we switch x and y and solve for y:

x = 3y + 5

x - 5 = 3y

y = (x - 5)/3

Therefore, the inverse of f(x) is f^-1(x) = (x - 5)/3.

To find the inverse equation of g(x) = x^2 - 6, we replace g(x) with y:

y = x^2 - 6

Next, we switch x and y and solve for y:

x = y^2 - 6

x + 6 = y^2

y = ±sqrt(x + 6)

Since we want a function, we take the positive square root:

y = sqrt(x + 6)

Therefore, the inverse of g(x) is g^-1(x) = sqrt(x + 6).

The relationship between a function and its inverse is such that if (a,b) is a point on the graph of f, then (b,a) is a point on the graph of f^-1. In other words, the roles of x and y are reversed in the inverse function. The domain of the function becomes the range of the inverse, and the range of the function becomes the domain of the inverse.

In general, not all functions have inverses. For a function to have an inverse, it must be one-to-one (i.e., each x-value in the domain maps to a unique y-value in the range). If a function is not one-to-one, then it may be possible to restrict the domain to a smaller interval so that the restricted function does have an inverse. When a function has an inverse, the domain and range of the inverse follow a pattern as stated above.

When sample size is small and population S.D. is unknown, values on the Z row on the table (or Z values) can't be used (note that Z is normal distribution). The correct option is A, i only.

A small sample is a dataset that contains few observations, making it more difficult to generalize the outcomes to the broader population. When a sample size is small, the sample mean, standard deviation, and other statistical measures may be influenced by chance variations.

Larger samples may be required to represent the overall population precisely. The larger the sample size, the closer the sample mean is likely to be to the true population mean. Additionally, when the sample is large, the sample standard deviation is more likely to be an accurate estimate of the population standard deviation.

A standard normal distribution can be used to find out the probability of obtaining a score in a particular range for a normal distribution. The Z value, or the value on the standard normal distribution, is used to convert any regular distribution to a standard normal distribution, which has a mean of 0 and a standard deviation of 1. However, when the sample size is small and the population standard deviation is unknown, the values on the Z row on the table (or Z values) can't be used.

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The company will have a loss of $46,605.60 by selling 40 items.

To find the profit, we need to subtract the cost from the price. The cost function C(q) gives us the cost of producing q items, and the price function p(q) gives us the price of selling q items.

First, let's plug in the value of 40 for q in both functions:

C(40) = 100(40) + 43,800 = 4,000 + 43,800 = 47,800

p(40) = 29.860(40) = 1,194.40

Now we can subtract the cost from the price to find the profit:

Profit = Price - Cost

Profit = 1,194.40 - 47,800

Profit = -46,605.60

Therefore, the company will have a loss of $46,605.60 by selling 40 items.

Note: It is important to make sure that the units for cost and price are the same. In this case, both are in dollars, so we can subtract them directly. If they were in different units, we would need to convert them to the same units before subtracting.

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Answer:

5,100

Step-by-step explanation:

850(6)= 5,100

It will take approximately 7.54 years for the investment to be worth $900 if it is continuously compounded at 11% per year.

We can use the formula for continuous compounding to solve this problem:

[tex]A = Pe^{rt}[/tex]

where A is the amount after t years, P is the initial investment, r is the annual interest rate in decimal form, and e is the base of the natural logarithm.

In this case, P = $400, A = $900, r = 0.11, and we want to solve for t.

We can rewrite the formula as:

[tex]t = ln(A/P) / r[/tex]

Plugging in the values we get:

[tex]t = ln(900/400) / 0.11 = 7.54[/tex]

Therefore, it will take approximately 7.54 years for the investment to be worth $900.

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Answer:(1)/(4)

Step-by-step explanation:n>1/4

Answer:

Area of a circle = πr²

Plug in values (radius is half of the diameter so radius = 8)

π8² = 64π ≈ 201.0619 feet (rounded to four decimal places)

Hope this helps!

Answer: Circle

Step-by-step explanation:

The solution of the system of linear equations on the graph is one solution at (1, −2).

Analyzing the intersection of the graphs of a system of linear equations on the coordinate plane will reveal how many solutions there are. The system has a single solution if the graphs meet at a single location. The system cannot be solved if the graphs are parallel and do not intersect. The system has an unlimited number of solutions if the graphs coincide, or overlap.

From the graph we observe that the two lines intersect at the point (1, -2).

The point where the lines intersect is thus the solution of the system of linear equations on the graph.

Hence, the solution of the system of linear equations on the graph is one solution at (1, −2).

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The correct statement regarding the approximate probability of calling 3 households in Virginia in a row that have motorcycles is given as follows:

The student can use a random number generator and assign numbers 1 and 2 to represent a household with a motorcycle and numbers 3 through 25 to represent a household without a motorcycle.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

Since the probability is of 8%, the experiment is constructed as follows:

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Answer: Bones break because so much force is applied onto them at a singular moment, more than can be handled. Most commonly from falls where you hit your bone in a specific spot.

Bones can break, also known as a fracture, due to a variety of reasons. Here are some common causes:

Trauma: A sudden force or impact, such as a fall, can cause a bone to break.

Overuse: Repeated stress on a bone over time can cause a stress fracture, which is a small crack in the bone.

Medical Conditions: Certain medical conditions, such as osteoporosis, cancer, or infections, can weaken bones and make them more susceptible to fractures.

Vitamin and Mineral Deficiencies: Insufficient levels of calcium, vitamin D, and other minerals necessary for strong bones can increase the risk of fractures.

The severity of a fracture can vary depending on the force of impact and the strength of the bone. Some fractures may only cause minor pain and swelling, while others may require surgery and a prolonged healing process.

It is important to seek medical attention if you suspect a fracture as prompt diagnosis and treatment can help prevent complications and promote healing. Treatment options for a fracture may include immobilization, casting, surgery, or physical therapy, depending on the severity and location of the break.

The fraction of all the sandwiches that are ham sandwiches on rye bread is 4⁄15.

To find the fraction of all the sandwiches that are ham sandwiches on rye bread, we need to multiply the fraction of sandwiches that use rye bread by the fraction of those that are ham sandwiches.

Here's how we can do that:

1. Start with the fraction of sandwiches that use rye bread: (4)/(5)
2. Multiply that by the fraction of those that are ham sandwiches: (1)/(3)
3. Multiply the numerators together: 4 x 1 = 4
4. Multiply the denominators together: 5 x 3 = 15
5. Put the new numerator and denominator together to get the fraction: (4)/(15)

So, the fraction of all the sandwiches that are ham sandwiches on rye bread is (4)/(15).

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Answer:

Step-by-step explanation: First you multiply 23 and 12 and you get 276. Then you do 276 divided by 6 and you get 46. So your final answer will be 43 inches.

If there are 360 students in the school, we can expect 108 students in the school to have a pet cat.

We can use proportion to solve the problem.

If 6 out of 20 students have a pet cat, then we can write:

6/20 = x/360

where;

x is the number of students in the school who have a pet cat.

To solve for x, we can cross-multiply:

20x = 6*360

20x = 2160

x = 108

Therefore, 108 students  are expected to have a pet cat in the school

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The expression [tex]cos^4(x)sin^2(x)[/tex] can be rewritten without exponents as [tex]cos^2(x) - (\frac{1}{2} )cos^4(x)[/tex].

The double angle, half angle, or power-reducing formula can be used to rewrite [tex]cos^4(x)sin^2(x)[/tex] without exponents. One of the approaches to rewriting is by applying the power-reducing formula. The power-reducing formula is a trigonometric identity that is used to reduce powers of sine and cosine functions.

To rewrite [tex]cos^4(x)sin^2(x)[/tex] without exponents, you can use the power reducing formula. This formula states that:
[tex]cos^2(x)*sin^2(x) = (\frac{1}{2} )(1-cos(2x))[/tex]

We can use this formula to break down [tex]cos^4(x)sin^2(x)[/tex] as follows:
[tex]cos^4(x)sin^2(x) = cos^2(x)sin^2(x)cos^2(x) = (\frac{1}{2} )(1-cos(2x))cos^2(x)[/tex]

Next, we can use the double angle formula, which states that:
[tex]cos(2x) = 2cos^2(x) - 1[/tex]

Plugging this into our equation, we get:
[tex]cos^4(x)sin^2(x) = (\frac{1}{2} )(1-2cos^2(x)+1)cos^2(x) = cos^2(x) - (\frac{1}{2} )cos^4(x)[/tex]


Therefore, the expression [tex]cos^4(x)sin^2(x)[/tex] can be rewritten without exponents as [tex]cos^2(x) - (\frac{1}{2} )cos^4(x)[/tex].

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Converting 4 feet and 11 inches is equivalent to 59 inches.

The metric system, which bases measurements on the metre for length and kilogramme for mass, was first used in France in 1795. Following the French Revolution, the government tasked scientists with finding a replacement for the nation's dozens of distinct conventional measurement methods. The metre was created by measuring a quadrant of the Earth's circumference that passes through Paris on its way from the North Pole to the Equator. All measurements were made using the new unit, called a metre, which was approximately thirty-nine inches in length.

We know that,

1 foot is equal to 12 inches.

Thus,

4 feet = 4 × 12 inches = 48 inches

Add the 11 inches to get the total:

48 inches + 11 inches = 59 inches

Therefore, 4 feet and 11 inches is equivalent to 59 inches.

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Answer:8

Step-by-step explanation: I got 8 because when you multiply 4/5 and 10 together you put 10 over one then multiply across.

After that you will get 40/5 and that is 8.

Answer: D

Step-by-step explanation:

[tex]3n^3-4n+3n+6n^3=-3[/tex]

Group together the like terms:

[tex]3n^3+6n^3+3n-4n=-3[/tex]

Combine like terms:

[tex]9n^3-n=-3[/tex]

The answer is D.

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