What is the length of the hypotenuse?
(Round your answer to the nearest tenth.)

The equation of the tangent line to the given path at the specified value of t is (x - sin(3), 3cos(3)(y - cos(3)), -3sin(3)(z - 1), 9/2).

To determine the equation of the tangent line to the given path at the specified value of t, we need to find the derivative of the given path with respect to t. The derivative of the given path is (3cos(3t), -3sin(3t), 9t^8/2). Now, we can plug in the specified value of t = 1 to find the slope of the tangent line at that point. The slope of the tangent line at t = 1 is (3cos(3), -3sin(3), 9/2).

Next, we can find the point on the given path at t = 1 by plugging in the specified value of t into the original equation. The point on the given path at t = 1 is (sin(3), cos(3), 1).

Finally, we can use the point-slope form of an equation to find the equation of the tangent line. The equation of the tangent line in (x, y, z) coordinates is (x - sin(3)) = 3cos(3)(y - cos(3)) = -3sin(3)(z - 1) = 9/2.

Therefore, the equation of the tangent line to the given path at the specified value of t is (x - sin(3), 3cos(3)(y - cos(3)), -3sin(3)(z - 1), 9/2).

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The equation of the given circle is: (x - 5)² + (y+ 1)² = 256

The center-radius form (most formally called the standard form) of a circle is usually expressed as;

(x - h)² + (y - k)² = r²

where (h, k) is the center and r is the radius.

We are given a circle with center T(5,-1) and a radius of 16 units

The equation then becomes:

(x - 5)² + (y - (-1))² = 16²

(x - 5)² + (y+ 1)² = 256

Thus, we can conclude that is the equation of the circle.

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Answer:

the vertex of∠5 is point M

Step-by-step explanation:

Answer: M

Step-by-step explanation: M represents the vertex. Think of the vertex almost like the starting point of an angle.

The financial product described is an annuity.

An annuity is a contract offered by insurance companies, where the buyer makes a lump-sum payment or a series of payments in exchange for regular payments made over time, usually until the end of the buyer's life. An annuity can be either fixed or variable, depending on the type of investment chosen by the buyer.

In a fixed annuity, the payments are predetermined and do not change over time, while in a variable annuity, the payments are based on the performance of the underlying investments. An annuity is often used as a retirement income stream.

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Answer:

[tex]x = -2[/tex]

Step-by-step explanation:

[tex] \frac{(x + 8)}{[(x + 3) (x + 4)]} = \frac{3}{(x + 3)}[/tex]

Multiplying both sides by (x + 3) (x + 4), we get:

[tex](x + 8) = 3 (x + 4)[/tex]

Expanding the right-hand side, we get:

[tex]x + 8 = 3x + 12[/tex]

Subtracting x and 8 from both sides, we get:

[tex]2x = -4[/tex]

Dividing both sides by 2, we get:

[tex]x = -2[/tex]

Therefore, the solution to the given equation is x = -2. We can check this solution by substituting x = -2 into the original equation and verifying that both sides are equal.

My answer:

There are 4 cards of number eight in a deck of 52 cards, namely: 8 Spades, 8 Diamond, 8 Clubs, and 8 Hearts. So the probable outcomes are 4 and the total cards are 52. So, x 100 = 1/13. So there is a 1/13 chance for getting an eight, i.e, for every 13 cards, one card will be an eight.

hope it helps :)

also please mark brainliest it means alot

The equation that shows a proportional relationship is y = 12x; where the constant of proportionality is 12

An equation is an expression that shows the relationship between two or more numbers and variables. Equations can either be linear, quadratic, cubic and so on depending on the degree.

A proportion relationship  is in the form:

y = kx; where x is the constant of proportionality

Let y represent the number of paper towel rolls and x represent the number of cases. Hence:

y = kx

From the table, using the point (1, 12):

12 = k(1)

k = 12

The equation is y = 12x

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The length of side 2 of the given triangle is 16.22.

The Law of Cosines is a mathematical formula that is used to calculate the length of sides and angles in a triangle. It is based on the Pythagorean theorem, which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides

The Law of Cosines helps to solve for the side length in triangles when we are given two sides and the angle between them. In this triangle we are given side 1 (15) and the angle between the two sides (33°). To find side 2, we will use the formula:

c² = a² + b²- 2ab*cosC

Where c is the length of the unknown side, a is the length of the given side, b is the length of the other given side, and C is the angle between the two given sides.

In this case, c is side 2, a is 15, b is unknown, and C is 33°. We will plug in the known values and solve for b:

b² = 15² + c² - 2*15*c*cos(33°)

b² = 225 + c² - 30c*cos(33°)

We can solve this equation for c:

c = (225 + b² - 30b*cos(33°))/2

Now, we can solve for b using the values for c and a given in the problem:

b²= 225 + 27² - 30*27*cos(33°)

b² = 225 + 729 - 690.8

b²= 264.2

b = 16.22

Therefore, the length of side 2 is 16.22.

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The volume generated by rotating the region in the first quadrant bounded by y = ex and the x-axis from x = 0 to x = ln(3) about the y-axis is (πln(3)3)/3.

To find the volume generated by rotating the region in the first quadrant bounded by y = ex and the x-axis from x = 0 to x = ln(3) about the y-axis, we can use the disk method. The disk method involves slicing the region into thin disks and adding up their volumes.

The volume of each disk is πr2h, where r is the radius of the disk and h is the thickness of the disk. In this case, the radius of each disk is x and the thickness is dx.

So, the volume of the region is:

V = ∫0ln(3)πx2dx

We can use the power rule for integration to solve this integral:

V = π∫0ln(3)x2dx = π[(x3)/3]0ln(3) = π[(ln(3)3)/3 - (03)/3] = (πln(3)3)/3

Therefore, the volume generated by rotating the region in the first quadrant bounded by y = ex and the x-axis from x = 0 to x = ln(3) about the y-axis is (πln(3)3)/3. This is the exact form of the volume.

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The lengths of the sides of the special 30°–60° right triangles are;

14. a = 8

b = 8·√3

c = 16

15. a = 4

b = 4·√3

c = 8

16. a = 7

b = 7·√3

c = 14

17. a = 12

b = 12·√3

c = 24

18. a = 5

b = 5·√3

c = 10

19. a = 9

b = 9·√3

c = 18

20. a = 10

b = 10·√3

c = 20

21. a = (√6)/2

b = 3·(√(2))/s

c = √6

Special right triangles are triangles that have 30°, 60°, and 45° interior angles.

The unknown lengths of the 30-60 special right triangle can be found as follows;

14. The length of the side a = 8

The length of the hypotenuse side, c can be found as follows;

sin(30°) = 1/2

sin(θ) = a/c

sin(30°) = 1/2 = 8/c

c = 2 × 8 = 16

c = 16

cos(θ) = b/c

cos(30°) = √3/2

Therefore; √3/2 = b/16

b = 16 × (√3/2) = 8·√3

b = 8·√3

15. a = 4, therefore;

c = 2 × 4 = 8

c = 8

b = (√3/2) × 8 = 4·√3

b = 4·√3

16. a = 7

b = 7·√3

c = 2 × 7 = 14

c = 14

17. a = b/√3, therefore;

a = 12·√3/√3 = 12

a = 12

b = 12·√3

c = 2 × a, therefore;

c = 2 × 12 = 24

c = 24

18. a = 5

b = a × √3, therefore;

b = 5 × √3 = 5·√3

b = 5·√3

c = 2 × a, therefore;

c = 2 × 5 = 10

c = 10

19. c = 18

a = c/2, therefore;

a = 18/2 = 9

a = 9

b = a × √3, therefore;

b = 9 × √3 = 9·√3

b = 9·√3

20. a = 10

b = a × √3, therefore;

b = 10 × √3 = 10·√3

b = 10·√3

c = 2 × a, therefore;

c = 2 × 10 = 20

c = 20

21. c = √6

a = c/2, therefore;

a = (√6 )/2

b = a × √3, therefore;

b = ((√6)/2) × √3 = (√(18))/2 = 3·√2/2

b = 3·√2/2

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The concept of the unit circle is presented throughout it's answer.

The unit circle is a circle with a radius of 1 unit that is centered at the origin (0,0) of a coordinate plane. It is a fundamental concept in trigonometry and geometry, and it is used to define the trigonometric functions (sine, cosine, and tangent) of angles in the Cartesian plane.

The format of each coordinate on the unit circle is given as follows:

[tex](\cos{\theta}, \sin{\theta})[/tex].

Hence we can calculate the trigonometric measures for any angle, as the tangent, the secant, the cossecant and the cotangent of an angle are all functions of the sine and the cosine obtained by the unit circle.

The problem is incomplete, hence the unit circle concept is presented in this answer.

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Answer:

a. An absolute value inequality that represents the possible actual temperature t is:

|t - 72| < 2

b. To find the range of possible actual temperatures, we can solve the inequality:

|t - 72| < 2

We can split this into two separate inequalities, one for when t is greater than 72 and one for when t is less than 72:

t - 72 < 2 and -(t - 72) < 2

Simplifying these inequalities, we get:

t < 74 and t > 70

Therefore, the range of possible actual temperatures is 70°F < t < 74°F.

Step-by-step explanation:

Answer:

To find all the real zeros of a polynomial function algebraically, use the Rational Root Theorem to generate a list of possible rational zeros and use synthetic division to test each possible zero. The zeros that remain after testing all the possible zeros are the real zeros of the polynomial function.

To find all the real zeros of a polynomial function algebraically, you can use the Rational Root Theorem and synthetic division. The Rational Root Theorem states that if a polynomial function has integer coefficients, then any rational zero (i.e., a zero that can be expressed as a fraction) must have a numerator that is a factor of the constant term and a denominator that is a factor of the leading coefficient.

Here are the steps you can follow:

1. Write the polynomial function in descending order of degree, with all like terms combined.

2. Use the Rational Root Theorem to generate a list of possible rational zeros. This list will be a set of all the possible fractions that can be formed by dividing a factor of the constant term by a factor of the leading coefficient.

3. Use synthetic division to test each possible zero. Start with the smallest possible denominator and work your way up. If a possible zero is not a zero of the polynomial function, cross it off the list.

4. Repeat step 3 until all the possible zeros have been tested. The zeros that remain are the real zeros of the polynomial function.

For example, let's say we have the polynomial function f(x) = x^3 - 6x^2 + 11x - 6.

Write the polynomial function in descending order of degree: f(x) = x^3 - 6x^2 + 11x - 6.

Use the Rational Root Theorem to generate a list of possible rational zeros: ±1, ±2, ±3, ±6.

Use synthetic division to test each possible zero. We start with x = 1:

1 │ 1 -6 11 -6

│ 1 -5 6

└─────────────

1 -5 6 0

Since the remainder is zero, we have found a zero of the polynomial function at x = 1. We can write the factorization f(x) = (x - 1)(x^2 - 5x + 6).

Now we use synthetic division to test the other possible zeros:

2 │ 1 -6 11 -6

│ 2 12 46

└─────────────

1 -4 23 40

x = 2 is not a zero of the polynomial function.

3 │ 1 -6 11 -6

│ 3 15 78

└─────────────

1 -3 26 72

x = 3 is not a zero of the polynomial function.

-1 │ 1 -6 11 -6

│ -1 7 -4

└────────────

1 -7 18 -10

x = -1 is not a zero of the polynomial function.

-2 │ 1 -6 11 -6

│ -2 16 -50

└────────────

1 -8 27 -56

x = -2 is not a zero of the polynomial function.

-3 │ 1 -6 11 -6

│ -3 27 -102

└────────────

1 -9 38 -108

x = -3 is not a zero of the polynomial function.

6 │ 1 -6 11 -6

Hope this helps, I'm sorry if it doesn't! :]

Answer:

a

Step-by-step explanation:

Answer:45%

Step-by-step explanation:

:))))

Answer:

45%

Step-by-step explanation:

99/180 = 55%

100% - 55% = 45% decrease

Answer: 45%

Answer:

240

Step-by-step explanation:

15% of 54=8.1

15% of 240=36

15% of 360=54

15% of 540=81

so the answer is 240

36 is 15 percent of 240.

We have,

To find out what number 36 is 15% of, we can set up the equation:

36 = 0.15x

Here, x represents the unknown number we are trying to find.

To solve for x, we divide both sides of the equation by 0.15:

36 / 0.15 = x

Simplifying the right side gives:

240 = x

Therefore,

36 is 15 percent of 240.

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Answer:

look below

13,800

Around 62.47% of the data population is situated between a z-score of -1.5 and 0.5.

Assuming a standard normal distribution, the percentage of the data population that falls between a z-score of -1.5 and 0.5 can be calculated using a standard normal table or a calculator.

Using a standard normal table, we can find the area under the curve between z = -1.5 and z = 0.5.

The area to the left of z = 0.5 is 0.6915, and the area to the left of z = -1.5 is 0.0668. Therefore, the area between z = -1.5 and z = 0.5 is:

0.6915 - 0.0668 = 0.6247

So, approximately 62.47% of the data population falls between a z-score of -1.5 and 0.5.

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Answer:

150 Lanterns

Step-by-step explanation:

Total lanterns : 300

[tex]\frac{2}{6} * 300 = 100[/tex]
[tex]\frac{1}{6} * 300 = 50[/tex]

so He sold 150 lanterns in total , leaving 300-150 = 150 Lanterns left.

Answer:

The formula for compound interest is:

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

Where:

A = the final amount

P = the principal (initial amount)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the time the money is invested (in years)

In this case, P = $500, r = 0.12, n = 2 (compounded twice a year), and t = 2.

So, A = 500(1 + 0.12/2)^(2*2) = $673.01

Therefore, Sam's total investment will be worth $673.01 after 2 years.

Step-by-step explanation:

A table that represents a linear function include the following: A.

x −4 −2 0 2 4

y   5 1 −3 −7 −11

In Mathematics, a linear function can be defined as a type of function whose equation is graphically represented by a straight line on the cartesian coordinate.

This ultimately implies that, a linear function has the same (constant) slope and it is typically used for uniquely mapping an input variable to an output variable, which both increases simultaneously.

Next, we would determine the slope by using the points contained in the table as follows;

Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (1 - 5)/(-2 + 4) = (-3 - 1)/(0 + 2) = (-7 + 3)/(2 - 0)

Slope (m) = -2.

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[tex]\textit{area of a circular ring}\\\\ A=\pi (R^2 - r^2) ~~ \begin{cases} R=\stackrel{outer}{radius}\\ r=\stackrel{inner}{radius}\\[-0.5em] \hrulefill\\ R=4\\ r=2 \end{cases}\implies A=\pi (4^2 - 2^2) \\\\\\ A=12\pi \implies A\approx 37.70~in^2[/tex]

Answer:

12πin² = 37.7in² (3sf)

Step-by-step explanation:

area of whole circle = π(4)² = 16π

area of small circle = π(2)² = 4π

area of shaded = 16π - 4π = 12π

12π = 37.6991... ≈ 37.7in² (3sf)

(a) When [tex]x=-2, 2^x = 2^(-2) = 1/2^2 = 1/4[/tex]. So, the expression 2^x when x=-2 is 1/4.

(b) When [tex]x=3, 2^x = 2^3 = 8[/tex]. So, the expression 2^x when x=3 is 8.

The expression 2^x represents exponential growth, where the base 2 is multiplied by itself x number of times. When x is a negative number, the expression becomes a fraction because we are dividing 1 by the base raised to the absolute value of x. In this case, 2^(-2) is equivalent to 1/2^2, which is equal to 1/4. On the other hand, when x is a positive number, the expression grows exponentially. When x=3, the expression becomes 2^3, which is equal to 2x2x2 = 8.

In summary, the evaluation of 2^x when x is a negative number produces a fraction, whereas when x is a positive number, it produces a positive integer.

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The quotient of [tex]40y^(4)-5y^(3)-30y^(2)-10y[/tex] divided by [tex]8y^(3) - y^(2) - 6y - 2[/tex]

You can use the polynomial long division method to find the product of two polynomials. Following are the steps:

To divide [tex]40y^(4)-5y^(3)-30y^(2)-10y[/tex] by 5y, we utilize long division.  Divide dividend by divisor to obtain first term of the quotient:

[tex]40y^(4) / (5y) = 8y^(3)[/tex]

Multiply divisor (5y) by first term of quotient (8y^(3)) to obtain first term of partial product:

[tex](8y^(3))(5y) = 40y^(4)[/tex]

Subtract the partial product from the dividend to obtain remainder:

[tex]40y^(4) - 40y^(4) = 0[/tex]

[tex]-5y^(3) / (5y) = -y^(2)(-y^(2))(5y) = -5y^(3)-5y^(3) - (-5y^(3)) = 0[/tex]

[tex]-30y^(2) / (5y) = -6y(-6y)(5y) = -30y^(2)-30y^(2) - (-30y^(2)) = 0[/tex]

Repeat process:

[tex]-10y / (5y) = -2(-2)(5y) = -10y-10y - (-10y) = 0[/tex]

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Answer:

6%

Step-by-step explanation:

0.06 is 6% as a decimal

the approximate volume of the tissue box to the nearest whole number is 300 cubic centimeters.

The volume of a rectangular prism is given by the formula V = lwh, where l, w, and h are the length, width, and height of the prism, respectively.

Substituting the given values, we get:

V = (5.2 cm)(9.6 cm)(6 cm)

Simplifying, we get:

V = 299.52 cubic centimeters

Therefore, the approximate volume of the tissue box is 299.52 cubic centimeters.

Since the dimensions of the tissue box are given to only two decimal places, we can reasonably assume that the answer accurate to two decimal places is a good approximation. If we want to express the answer to the nearest whole number, we would round it to the nearest integer.

Rounding 299.52 to the nearest integer, we get:

V ≈ 300 cubic centimeters

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The angle between the line of sight from the lighthouse to the tugboat is 27° and the angle between the line of sight from the lighthouse to the cargo ship is 54°.

Let's call the angle between the line of sight from the lighthouse to the tugboat "x". Then, we know that the angle between the line of sight from the lighthouse to the cargo ship is x+27, since the given angle between the lines of sight is 27°.

We also know that the angle between the lines of sight at the cargo ship is twice the angle between the lines of sight at the tugboat. Using this information, we can set up the equation:

2x = x+27

Solving for x, we get:

x = 27

So the angle between the line of sight from the lighthouse to the tugboat is 27°. Then, we can use this to find the angle between the line of sight from the lighthouse to the cargo ship:

x+27 = 27+27 = 54

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Madison will have $322.65 in her bank account in 10 years. Here is the calculation to find this amount:

Starting balance: $200

Year 1: $200 x 1.05 = $210

Year 2: $210 x 1.05 = $220.50

Year 3: $220.50 x 1.05 = $231.53

...

Year 10: $279.10 x 1.05 = $322.65

Therefore, after 10 years, Madison will have $322.65 in her bank account due to the annual interest earned.

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