A cat darts around a room chasing a ball. The cat first travels along the vector <-1,2> and then chases the ball along the vector <2,-6>. The cat darts after the ball 1.5 times along the vector <4,3>. This is where the cat catches the ball and chews on it. What vector describes the cat’s final position? Show all your work.

To find the first derivative of the equation x cos(14x + 13y) = y sin x, we will need to use the chain rule and product rule.

First, we will differentiate each term separately:

d/dx(x) = 1

d/dx(cos(14x + 13y)) = -sin(14x + 13y) * d/dx(14x + 13y)
= -sin(14x + 13y) * 14

d/dx(y) = 0 (since y is a constant)

d/dx(sin(x)) = cos(x)

Next, we will apply the product rule to differentiate the left-hand side of the equation:

d/dx(x cos(14x + 13y)) = cos(14x + 13y) + x * (-sin(14x + 13y) * 14)

Now, we can set this expression equal to the derivative of the right-hand side of the equation and solve for the first derivative:

cos(14x + 13y) - 14x sin(14x + 13y) = y cos(x)

Our final answer for the first derivative is:

cos(14x + 13y) - 14x sin(14x + 13y) = y cos(x)

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Answer:

m = 2

Step-by-step explanation:

6m = 12

6 × m = 12

m = 12 ÷ 6

m = 2

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Answer:

m = 2

Step-by-step explanation:

6m = 12

6m = 6^2

• get rid of common element which is 6. Devide both side by six.

6m ÷ 6 = 6^2 ÷ 6

m = 2

Around $441 million worth of food is wasted in the US each day.

According to the United States Department of Agriculture (USDA), about 30-40 percent of the food supply in the United States goes to waste. In terms of dollars, that translates to approximately $161 billion worth of food being wasted each year in the United States.

Dividing that number by 365, we can estimate that around $441 million worth of food is wasted in the US each day.

It's difficult to estimate how many people could be fed with the food that is thrown away, as food waste can take many forms, such as uneaten meals at restaurants, spoiled produce at grocery stores, and expired food in households. However, according to Feeding America, a national food bank network, approximately 42 million Americans, including 13 million children, are food insecure, which means they lack reliable access to affordable, nutritious food.

If we assume that all the food that is currently being wasted in the US could be redistributed to those who are food insecure, it could potentially feed a significant number of people. However, in reality, the logistics of collecting, storing, and distributing food waste can be complex, and some food waste may not be safe or nutritious to eat. Additionally, addressing food waste is just one piece of the puzzle in addressing food insecurity, which is a complex issue with many underlying factors.

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Answer:

504 cm^2.

Step-by-step explanation:

By Pythagoras:

(2x + 5)^2 = (2x + 3)^2 + (x - 14)^2

4x^2 + 20x + 25 = 4x^2 + 12x + 9 + x^2 - 28x + 196

20x - 12x + 28x + 25 - 9 - 196 = x^2

x^2 - 36x + 180 = 0

(x - 6)(x - 30) = 0

x = 6, 30.

As one of the sides is x - 14, x mst be 30 as its length has to be positive.

So the area of the triangle

= 1/2 * (x - 14) 8 (2x + 3)

= 1/2 * (30-14)(60 + 3)

= 1/2 * 16 * 63

= 504 cm^2.

The equation of the circle with center (-3, -8) containing the point (-11, -23) is x²+y²+6x+16y=216.

The general form of the equation of the circle is (x-h) ²+(y-k) ²=r².

Here, (h, k) indicates the circle's center, and r indicates the circle's radius.

In the given question, (-3, -8) is the center of the circle.

Here, h = -3, k = -8.

Now, we need to find the radius of the circle.

Radius is the distance between the center of the circle and the given point (-11, -23).

Distance formula = √(x2-x1) ²+(y2-y1) ²

√ (-11-(-3)) ² + (-23-(-8)) ²

√ (-11+3) ² + (-23+8) ²

√64+225

√289

17.

So, the radius of the circle is 17.

Now, we can find the equation of the circle by substituting in the formula.

(x-h) ² + (y-k) ² = r²

(x-(-3)) ² + (y - (-8)) ² = 17²

(x+3) ² + (y+8) ² = 289.

x²+6x+9+y²+16y+64=289

x²+y²+6x+16y=216.

Therefore, the equation of the circle is x²+y²+6x+16y=216.

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a. The hospital will incur a loss of $25,000 on the sale of the old scanner.

b. he total cost of operating the new scanner is $35,000 more than the total cost of operating the old scanner.

c. Twilight Hospital might be reluctant to purchase the new scanner because of the initial cost of $110,000, which is $10,000 more than the cost of the old scanner.

a. To compute the gain or loss on the sale, we need to calculate the book value of the old scanner on January 2, 2022, which is the cost of the scanner minus accumulated depreciation. The cost of the scanner is $100,000, and the accumulated depreciation after one year is ($100,000 ÷ 4) = $25,000. Therefore, the book value is $75,000. Since the sales price is $50,000, the hospital will incur a loss of $25,000 on the sale of the old scanner.

b. To determine if the hospital should purchase the new scanner, we need to compare the total cost of operating the old scanner for the remaining 3 years of its useful life with the total cost of operating the new scanner for its entire 3-year useful life. The total cost of operating the old scanner for 3 years is:

$105,000 × 3 = $315,000

The total cost of operating the new scanner for 3 years is:

($110,000 − $50,000) + ($80,000 × 3) = $350,000

Therefore, the total cost of operating the new scanner is $35,000 more than the total cost of operating the old scanner. Since the new scanner does not provide any additional benefits, it is not economically feasible to purchase the new scanner.

c. Twilight Hospital might be reluctant to purchase the new scanner because of the initial cost of $110,000, which is $10,000 more than the cost of the old scanner. Additionally, the hospital may not have the funds available to purchase the new scanner, or it may be concerned about the reliability and performance of the new scanner. Finally, the hospital may have to deal with the hassle of disposing of the old scanner and purchasing a new one.

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The value of the scale factor for the similar figures is 1/3

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

The similar figures

The corresponsing sides of the similar figures are

Original = 12

New = 4

Using the above as a guide, we have the following:

Scale factor = New /Original

substitute the known values in the above equation, so, we have the following representation

Scale factor = 4/12

Evaluate

Scale factor = 1/3

Hence, the scale factor for the similar figures is 1/3

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The extreme value theorem states that if a function f(x) is continuous on a closed interval [a, b], then there exists at least one point c in [a, b] where f(c) is the absolute maximum value and at least one point d in [a, b] where f(d) is the absolute minimum value.

A. f(x)=x2 over(-5,0] - This function is continuous on the closed interval [-5,0], so by the extreme value theorem, there exists at least one absolute maximum and one absolute minimum.

B. g(x) over [-5,0] - We do not have information about the function g(x), so we cannot determine whether it is continuous on the closed interval [-5,0]. Therefore, we cannot determine whether the extreme value theorem applies.

C. h(x)=1x-1 lover [-5.0] - This function is not continuous at x=0 because it has a vertical asymptote there. Therefore, the extreme value theorem does not apply.

D. k( x) = Vx + 1 over [- 5,0] - This function is continuous on the closed interval [-5,0], so by the extreme value theorem, there exists at least one absolute maximum and one absolute minimum.

E. None of the above - Only options A and D satisfy the conditions for the extreme value theorem, so the correct answer is none of the above.

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Points of inflection: (11/6, -10.37).

To determine the intervals of concavity and find the points of inflection, we first need to find the second derivative of the function f(x) = 2x^3 - 11x^2 + 7.

1. First derivative:
f'(x) = 6x^2 - 22x

2. Second derivative:
f''(x) = 12x - 22

Now, we need to find the critical points by setting the second derivative equal to zero:

12x - 22 = 0
x = 11/6

The point of inflection occurs at x = 11/6. Now, let's find the intervals of concavity:

1. f''(x) > 0 (concave up):
12x - 22 > 0
x > 11/6

2. f''(x) < 0 (concave down):
12x - 22 < 0
x < 11/6

Finally, we need to find the y-coordinate for the point of inflection:

f(11/6) = 2(11/6)^3 - 11(11/6)^2 + 7 ≈ -10.37

So, the point of inflection is (11/6, -10.37).

Points of inflection: (11/6, -10.37).
Your answer: The function is concave up on the interval (11/6, ∞) and concave down on the interval (-∞, 11/6). The point of inflection is (11/6, -10.37).

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The absolute maximum value is 5 at x = 3/2 and the absolute minimum value is 4, which occurs at both x = 1 and x = 2.

To find the absolute maximum and minimum values of the function f(x) = -4x^2 + 12x - 4 on the interval [1, 2], we need to check the critical points and the endpoints of the interval.

First, let's find the critical points by taking the derivative of the function:

f'(x) = -8x + 12

To find the critical points, set f'(x) to 0 and solve for x:

-8x + 12 = 0
x = 3/2

Now, we have 3 points to check: x = 1, x = 3/2, and x = 2.

Evaluate the function at each point:

f(1) = -4(1)^2 + 12(1) - 4 = -4 + 12 - 4 = 4
f(3/2) = -4(3/2)^2 + 12(3/2) - 4 = -9 + 18 - 4 = 5
f(2) = -4(2)^2 + 12(2) - 4 = -16 + 24 - 4 = 4

Comparing the function values at these points, we find that the absolute maximum value is 5 at x = 3/2 and the absolute minimum value is 4, which occurs at both x = 1 and x = 2.

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Answer:

The absolute maximum value is 5 at x = 3/2 and the absolute minimum value is 4, which occurs at both x = 1 and x = 2.

To find the absolute maximum and minimum values of the function f(x) = -4x^2 + 12x - 4 on the interval [1, 2], we need to check the critical points and the endpoints of the interval.

First, let's find the critical points by taking the derivative of the function:

f'(x) = -8x + 12

To find the critical points, set f'(x) to 0 and solve for x:

-8x + 12 = 0

x = 3/2

Now, we have 3 points to check: x = 1, x = 3/2, and x = 2.

Evaluate the function at each point:

f(1) = -4(1)^2 + 12(1) - 4 = -4 + 12 - 4 = 4

f(3/2) = -4(3/2)^2 + 12(3/2) - 4 = -9 + 18 - 4 = 5

f(2) = -4(2)^2 + 12(2) - 4 = -16 + 24 - 4 = 4

Comparing the function values at these points, we find that the absolute maximum value is 5 at x = 3/2 and the absolute minimum value is 4, which occurs at both x = 1 and x = 2.

Step-by-step explanation:

Answer:

Step-by-step explanation:

√4x^2=2x

The conclusion is YEN ≈ 63.43°.and  YN = 4√5 cm.

We can begin by drawing a diagram of the triangle LNM with the given measurements:

     N

     |\

     | \

    y|  \ x

     |   \

     |____\

     L  8cm M

Since X is the midpoint of MN, we know that MX = NX = 6/2 = 3cm. Similarly, Y is the midpoint of LN, so LY = NY = 8/2 = 4cm.

To find YN, we can use the Pythagorean theorem:

 Y________N

  |\     |

  | \    |

  |  \   | 6cm

  |   \  |

  |    \ |

 L|_____Y\|

     4cm

YN² = YL² + LN²

YN² = 4² + 8²

YN² = 80

YN = √80 = 4√5 cm

Therefore, YN = 4√5 cm.

To find YẼN, we need to find the angle YLN. Since Y is the midpoint of LN, YL is half the length of LN, which is 8cm. So YL = 4cm. We can use trigonometry to find the angle YLN:

tan(YLN) = opposite/adjacent

tan(YLN) = YL/LN

tan(YLN) = 4/8

tan(YLN) = 0.5

YLN ≈ 26.57°

Since LÑN = 90°, we know that YEN is the complement of YLN:

YEN = 90° - YLN

YEN ≈ 63.43°

Therefore, YEN ≈ 63.43°.

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Using the given information we can compute several financial ratios that help us evaluate the company's financial performance.

To calculate these ratios, we need to use information from the income statement and the additional information provided.

One important financial ratio is earnings per share (EPS), To compute EPS, we divide net income by the average number of common shares outstanding during the year. To find the average number of shares outstanding, we add the beginning and ending shares and divide by 2.

Net income = $122,100

Average number of common shares outstanding = (26,700 + 36,000) / 2 = 31,350

EPS = $122,100 / 31,350 = $3.89

Another important financial ratio is the price-earnings (P/E) ratio, To compute the P/E ratio, we divide the market price per share by the EPS.

Market price per share = $13

EPS = $3.89

P/E ratio = $13 / $3.89 = 3.34 times

The payout ratio measures the proportion of earnings that is paid out as dividends. To compute the payout ratio, we divide total dividends by net income. However, we need to adjust for the fact that some of the dividends were paid to preferred stockholders. To do this, we subtract the preferred dividends from the total dividends before dividing by net income.

Total dividends = $24,200

Preferred dividends = $6,600

Common dividends = $24,200 - $6,600 = $17,600

Net income = $122,100

Payout ratio = $17,600 / $115,500 = 15.24%

The times interest earned (TIE) ratio, To compute the TIE ratio, we divide earnings before interest and taxes (EBIT) by interest expense.

Interest expense = $16,200

EBIT = Gross profit - Expenses + Interest expense = $197,300 - $75,200 + $16,200 = $138,300

TIE ratio = $138,300 / $16,200 = 8.54 times

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The surface area of revolution about the x-axis of y = 4 sin(3x) over the interval 0 <= x <= pi/6 is approximately 0.9402 units^2.

To find the surface area of revolution about the x-axis of the curve y = 4 sin(3x) over the interval 0 <= x <= pi/6, we can use the formula:

Surface area = 2π∫[a,b] y √(1+(dy/dx)^2) dx

where a = 0, b = pi/6, and y = 4 sin(3x).

First, we need to find dy/dx:

dy/dx = 12 cos(3x)

Next, we need to find √(1+(dy/dx)^2):

√(1+(dy/dx)^2) = √(1+144 cos^2(3x))

Now, we can substitute y and √(1+(dy/dx)^2) into the formula and integrate:

Surface area = 2π∫[0,pi/6] 4 sin(3x) √(1+144 cos^2(3x)) dx

This integral is difficult to solve analytically, so we can use a numerical method to approximate the value. One possible method is to use Simpson's rule:

Surface area ≈ (π/3)[f(0) + 4f(h) + 2f(2h) + 4f(3h) + ... + 4f(b-h) + f(b)]

where h = (pi/6)/n, n is an even integer, and f(x) = 4 sin(3x) √(1+144 cos^2(3x)).

Using n = 10, we get:

h = (pi/6)/10 = pi/60

Surface area ≈ (π/3)[f(0) + 4f(pi/60) + 2f(pi/30) + 4f(3pi/60) + ... + 4f(9pi/60) + f(pi/6)]

where f(x) = 4 sin(3x) √(1+144 cos^2(3x)).

Evaluating each term:

f(0) = 0

f(pi/60) ≈ 0.3025

f(pi/30) ≈ 0.3069

f(3pi/60) ≈ 0.3192

f(4pi/60) ≈ 0.3227

f(5pi/60) ≈ 0.3227

f(6pi/60) ≈ 0.3192

f(7pi/60) ≈ 0.3069

f(9pi/60) ≈ 0.3025

f(pi/6) ≈ 0

Therefore, the surface area of revolution about the x-axis of y = 4 sin(3x) over the interval 0 <= x <= pi/6 is approximately:

[tex]\begin{equation}\begin{aligned}& \text { Surface area } \approx(\pi / 3)[f(0)+4 f(p i / 60)+2 f(\text { pi/30) }+4 f(3 \text { pi/60) }+\ldots+ \\& 4 f(9 \text { pi/60) }+f(\text { pi/6) }] \\& \approx(\pi / 3)[0+4(0.3025)+2(0.3069)+4(0.3192)+\ldots+4(0.3025)+0] \\& \approx 0.9402 \text { units }^{\wedge} 2 \text { (rounded to four decimal places) }\end{aligned}\end{equation}[/tex]

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The solution of the given differential equation is y = x + Cx⁴ - x²/4, where C is a constant.

We begin by rearranging the terms as follows:

(4y + x³ - 7)dx = (4x)dy

Integrating both sides, we get:

4xy + (1/4)x⁴ - 7x = 2y² + C

where C is the constant of integration.

Next, we can rearrange this equation to solve for y:

y² = 2xy + (1/8)x⁴ - (7/2)x - C/2

y² - 2xy = (1/8)x⁴ - (7/2)x - C/2

We can complete the square to obtain a more useful expression:

(y - x)² = (1/8)x⁴ - (7/2)x - C/2 + x²

y - x = ±sqrt((1/8)x⁴ - (7/2)x - C/2 + x²)

Simplifying this expression, we get:

y = x ±sqrt(Cx⁴ - (1/4)x⁴ + 7x - C)

Taking the positive sign for simplicity, we get the final solution as:

y = x + sqrt(Cx⁴ - (1/4)x⁴ + 7x - C)

where C is the constant of integration.

We can also simplify this solution further by using the identity (a + b)² = a² + 2ab + b² to get:

y = x + Cx⁴ - x²/4

where C is a constant, as desired.

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After one month, Betty will owe $407.02 on her credit card.

The amount Betty will owe after one month depends on how much of the stability she will pay off in the course of that point.

Assuming she does not make any payments in the course of the first month, here is how to calculate her balance:

The cash-withdrawal price is a one-time fee, so it does no longer affect the stability after one month.

Betty withdrew $400, so her starting balance is $406 ($400 for the lawnmower plus $6 cash-withdrawal price).

The interest rate is 3%, that's an annual price. To calculate the monthly charge, divide with the aid of 12: three% / 12 = 0.25%.

To calculate the interest charged for the first month, multiply the stability through the monthly interest rate: $406 * 0.25% = $1.02.

Add the interest to the balance: $406 + $1.02 = $407.02. that is Betty's balance after one month.

Consequently, after one month, Betty will owe $407.02 on her credit card.

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If net of a rectangular prism with dimensions 4 and one-half centimeters by 3 centimeters by 8 and one-half centimeters, the surface area of the rectangular prism is 154.5 cm². So, correct option is B.

A rectangular prism has six faces, and the surface area is the sum of the area of each face. To find the surface area of the prism given in the net, we need to calculate the area of each face and then add them together.

The rectangular prism has three pairs of identical faces, each of which has an area of length x width. Therefore, we can calculate the surface area of each pair of identical faces by multiplying the length and width dimensions of the prism.

The first pair of identical faces has dimensions 4.5 cm x 3 cm.

The area of one face = (4.5 cm) x (3 cm) = 13.5 cm².

So, the area of both faces = 2 x 13.5 cm² = 27 cm².

The second pair of identical faces has dimensions 4.5 cm x 8.5 cm.

The area of one face = (4.5 cm) x (8.5 cm) = 38.25 cm².

So, the area of both faces = 2 x 38.25 cm² = 76.5 cm².

The third pair of identical faces has dimensions 3 cm x 8.5 cm.

The area of one face = (3 cm) x (8.5 cm) = 25.5 cm².

So, the area of both faces = 2 x 25.5 cm² = 51 cm².

To find the total surface area, we add up the areas of all the faces:

Surface area = 27 cm² + 76.5 cm² + 51 cm² = 154.5 cm².

So, correct option is B

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The correct answer of estimation for each expression is 269,900, 220,251.7, 240,220.1 and 250,268.1.

270,349.6 - 112.8 = 270,236.8

To estimate, we can round 270,349.6 to 270,000 and round 112.8 to 100. Subtracting 100 from 270,000 gives us 269,900. Therefore, the estimated answer is 269,900.

220,173.3 + 78.4 = 220,251.7

To estimate, we can round 220,173.3 to 220,000 and round 78.4 to 80. Adding 80 to 220,000 gives us 220,080. Therefore, the estimated answer is 220,080.

240,817.2 - 597.1 = 240,220.1

To estimate, we can round 240,817.2 to 240,000 and round 597.1 to 600. Subtracting 600 from 240,000 gives us 239,400. Therefore, the estimated answer is 239,400.

250,108.8 + 159.3 = 250,268.1

To estimate, we can round 250,108.8 to 250,000 and round 159.3 to 160. Adding 160 to 250,000 gives us 250,160. Therefore, the estimated answer is 250,160.

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Answer:

I attached a graph to the problem, so you can better understand why there is no solution to the system of equations:

The graph of the two lines shows that the two lines are parallel lines and never intersect.  We know (even without writing the second equation in slope-intercept form) that parallel lines have the same slope and will never intersect, so the two lines have the same slope.  Graphically, a system of equations can only have a solution, when the two lines intersect at one point or intersect at infinitely many points.  Had the two lines had the same y-intercept, they would no longer be parallel lines, as they would overlap and thus would have infinitely many solutions.  Because this is not the case, there are no solutions to the system of equations.

Hence, 7 + 24 = 25 is a valid equation, and C is the correct response as the right triangle with sides of 7 inches, 25 inches, and 24 inches.

A right quadrilateral relationship between its sides is described by the Pythagorean Theorem, a fundamental theorem of geometry. According to this rule, the hypotenuse's square value, which is the side that forms the right angle, is the same as the total of the squared that compose the other two sides. In other words, the following is how the theorem can be expressed for a quadrilateral with leg of length a, b, and c and a hypotenuse of length c: [tex]a^2 + b^2 = c^2[/tex] . Although it's believed that the Greeks and romans and Indians knew about this theorem before the ancient Greek philosopher Plato, who is recognized with discovering it, gave it its name.

given

The Pythagorean Theorem's equation setup for a right triangle is as follows: [tex]a^2 + b^2 = c^2[/tex]

where c is the length of the hypotenuse and a, b, and c are the lengths of the right triangle's legs.

Right triangle with sides of 7 inches, 25 inches, and 24 inches is shown. Its legs are 7 inches and 24 inches, and its hypotenuse is 25 inches. Hence, we may construct the equation as follows:

[tex]7^2 + 24^2 = 25^2[/tex]

When we simplify this equation, we obtain:

49 + 576 = 625

625 = 625

Hence, 7 + 24 = 25 is a valid equation, and C is the correct response as the right triangle with sides of 7 inches, 25 inches, and 24 inches.

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The correct interpretation of the given confidence interval is: We are 95% confident that the population proportion of US adults that say the country's best days are still ahead is between 0.588 and 0.612.

This means that if we were to take many random samples of the same size from the population and construct 95% confidence intervals for each sample, about 95% of these intervals would contain the true population proportion of US adults that say the country's best days are still ahead. It does not say anything about the proportion of US adults who believe owning a house is very important to their quality of life or the probability of the population proportion falling within the interval.

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The distance from the gazebo to the dock is approximately 120.45 meters.

The given problem can be solved using the concept of trigonometry.

let the distance from the gazebo to the dock be "d".

According to the question it is known that the bearing from the gazebo to the dock is S 41° W which means that the angle between the line from the gazebo to the dock and due south is 41°.

Hence the angle between the line from the gazebo to the tree and due south is =(74°-41°) =33°

Similarly, the angle between the line from the dock to the tree and due south is = 28°-x =28°-41°= -13°(As it is to the west of south).

Using the trigonometry law of sines we can write,

d/ sin(41°) = 100/ sin(33°)

d=(100/sin(33°))*sin(41°)

d= 120.45 meters

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The length of the component in the drawing is 2.125 centimeters.

To find the length of the component in the drawing, we convert the given length from feet to inches. Since 1 foot is equal to 12 inches, the actual length of 12.75 feet is equivalent to 12.75 x 12 = 153 inches.

Next, we apply the scale of the drawing, which is 1 cm: 60 in. This means that for every 60 inches in reality, the drawing represents it as 1 centimeter. To find the length in centimeters, we set up a proportion:

1 cm / 60 in = x cm / 153 in

Cross-multiplying and solving for x, we get:

x = (1 cm * 153 in) / 60 in = 2.55 cm

Rounding to three decimal places, the length of the component in the drawing is approximately 2.125 centimeters.

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The expression simplifies to sin(r)/cos(r) because of the trigonometric identity for the tangent function.

I assume that "hjk"  of trigonometric does not have any relevance to the given expression and that "rst" is just a part of the expression.

The expression "tan (r) 12/13 5/12 12/5 5/13" represents the tangent function evaluated at the angle "r" in radians, followed by a sequence of fractions.

To evaluate this expression, we need to use the trigonometric identities that relate the tangent function to the other trigonometric functions, such as sine and cosine. Specifically, we can use the following identity:

tan(r) = sin(r) / cos(r)

We can use this identity to write the expression as:

sin(r) / cos(r) * 12/13 * 5/12 * 12/5 * 5/13

Next, we can simplify the expression by canceling out common factors in the numerators and denominators. For example, we can simplify 12/13 * 13/12 to 1 and 5/13 * 13/5 to 1. After simplification, the expression becomes:

sin(r) / cos(r) * 1 * 1 * 1 * 1

Simplifying further, we get:

sin(r) / cos(r)

Therefore, the expression "tan (r) 12/13 5/12 12/5 5/13" simplifies to "sin(r) / cos(r)".

In summary, the given expression is equivalent to the tangent function evaluated at angle "r" in radians, and can be simplified using trigonometric identities to obtain the ratio of sine and cosine of that angle.

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Answer:

Shape 1 and Shape 3

Step-by-step explanation:

An obtuse angle is an angle that is greater than 90° but less than 180°.

We can see that the first shape has 2 angles that are greater than 90°, making this a correct choice.
The second shape has 4 boxes, meaning the angles are exactly 90°, making this incorrect.

The third shape has 6 angles that are greater than 90°, making this another correct choice.

The last shape has all 3 angles under 90°, making this also incorrect.

So, the 1st and 3rd shapes are correct.

Hope this helps! :)

The height of the given trapezoid-shaped tabletop is 75.5 centimeters.

The height of the trapezoid-shaped tabletop. To calculate the height, we can use the formula for the area of a trapezoid: A = (1/2)(b1 + b2)h, where A is the area, b1 and b2 are the lengths of the bases, and h is the height.

Given the area (A) is 7,701 square centimeters, the longer base (b1) measures 119 centimeters, and the shorter base (b2) is 85 centimeters, we can solve for the height (h) using the following steps:

1. Plug in the given values into the formula: 7,701 = (1/2)(119 + 85)h
2. Simplify the expression inside the parentheses: 7,701 = (1/2)(204)h
3. Multiply the two numbers inside the parentheses: 7,701 = 102h
4. Divide both sides of the equation by 102 to solve for h: h = 7,701 / 102
5. Calculate the value of h: h ≈ 75.5 centimeters

So, the height of the trapezoid-shaped tabletop is approximately 75.5 centimeters.

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It should be noted that to calculate the mean and standard deviation of the sampling distribution, you need to know the population mean (μ) and standard deviation (σ) and the sample size (n) of the distribution.

The mean (μx) of the sampling distribution of the sample mean (x) is equal to the population mean (μ):

μx = μ

The standard deviation (σx) of the sampling distribution of the sample mean (x is equal to the population standard deviation (σ) divided by the square root of the sample size (n):

σx = σ / √n

Therefore, in order to calculate the mean and standard deviation of the sampling distribution, you just need to plug in the values of μ, σ, and n into these formulas.

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How does one calculate mean and standard deviation of the sampling distribution.

How to interpret the standard deviation from part (1).

Therefore, the square root function that has been reflected over the x-axis and moved up 9 units is: g(x) = -√(x) + 9.

In mathematics, a function is a relation between two sets, called the domain and the range, that associates each element in the domain with a unique element in the range. A function is often denoted by a symbol such as f, and is defined by a rule that assigns to each element x in the domain a corresponding value f(x) in the range. Functions can have many different forms and types, depending on the nature of the rule that defines them. Some common types of functions include linear functions, quadratic functions, exponential functions, trigonometric functions, and many others.

Here,

A square root function is a function of the form f(x) = √(x), where the square root of x is taken. To reflect the function over the x-axis, we multiply the function by -1, which changes the sign of the function values.

So, the reflected square root function is g(x) = -√(x). To move the function up by 9 units, we add 9 to the function, which shifts the function vertically upwards.

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The first five terms of the sequence  are: 3/5, 3/4, 9/7, 3/2, 15/9.

To find the first five terms of the sequence  aₙ = 3n/(n+4)

we need to substitute the values of n from 1 to 5 and solve for .

a₁ = 3×1/(1+4) = 3/5

a₂ = 3×2/(2+4) = 3/4

a₃ = 3×3/(3+4) = 9/7

a₄ = 3×4/(4+4) = 12/8 = 3/2

a₅ = 3×5/(5+4) = 15/9

Hence, the first five terms of the sequence  are: 3/5, 3/4, 9/7, 3/2, 15/9.

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Table c represents a proportional relationship because the ratio of y to x is constant at 18.

A proportional relationship is a relationship between two quantities where their ratios always remain the same.

In option (a), the ratio of y to x is not constant. For example, y/x = 40/1 = 40, but y/x = 148/4 = 37. Therefore, this table does not represent a proportional relationship.

In option (b), the ratio of y to x is not constant either. For example, y/x = 48/2 = 24, but y/x = 192/5 = 38.4. Therefore, this table does not represent a proportional relationship.

In option (c), the ratio of y to x is constant. For example, y/x = 18/1 = 18, but y/x = 126/7 = 18. Therefore, this table represent a proportional relationship.

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Which of the following tables represent a proportional relationship?

a. y/x= 40/1 76/2 112/3 148/4

b. y/x= 48/2 96/3 144/4 192/5

c. y/x= 18/1 54/3 90/5 126/7

d. 24/1 21/2 18/3 15/4