Triangle Similarity

To answer problem 1.) a. and b.

Refer to the attached image.

The unit tangent vector T and the curvature k is √2/3

Unit Tangent Vector and Curvature are two important concepts in vector calculus that are used to describe the behavior of curves in space.

To find the Unit Tangent Vector, we must first find the derivative of the curve. The curve is defined as (t) = <9 cos t, 9 sin t, √(3) t>, so taking the derivative with respect to t, we get:

d(t) / dt = < -9 sin t, 9 cos t, √(3) >

Next, we want to normalize this vector to get the Unit Tangent Vector, T. To do this, we divide the vector by its magnitude, which is given by:

=> √((-9 sin t)² + (9 cos t)² + (√(3))²)

=> √(81 + 81 + 3) = 9 √(3)

So, the Unit Tangent Vector is given by:

T = d(t) / dt / |d(t) / dt|

T = < -9 sin t / 9 √(3), 9 cos t / 9 √(3), √(3) / 9 √(3) >

T = < -sin t / √(3), cos t / √(3), 1 / 3 √(3) >

To find the Curvature, we use the formula:

k = |dT / dt|

where T is the Unit Tangent Vector and dT / dt is the derivative of the Unit Tangent Vector. To find dT / dt, we take the derivative of T with respect to t:

dT / dt = < cos t / √(3), -sin t / √(3), 0 >

Finally, the magnitude of dT / dt gives us the Curvature:

=> √((cos t / √(3))² + (-sin t / √(3))² + (0)²)

=> √(1 / 3 + 1 / 3) = √(2 / 3) / √(3)

So, the Curvature is given by:

k = √(2 / 3) / √(3)

k = √2 / 3

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The value of the exponent is 4¹⁴

Index forms are defined as mathematical models of writing numbers that are large or small in more convenient forms.

The other names are scientific notations and standard forms.

The rules of index forms;

From the information given, we have;

4^3/4^6 times 4^17

subtract the bases

4 ³⁻⁶ ⁺¹⁷

Add the values

4¹⁴

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The blood viscosity and vessel length remain relatively constant, the regulation of vessel radius becomes the primary mechanism for controlling resistance in the body.

Blood viscosity refers to the thickness or stickiness of blood, which is determined by the number and size of red blood cells, as well as the concentration of proteins and other substances in the plasma.

Blood viscosity affects resistance because the thicker the blood, the more difficult it is for it to flow through the blood vessels.

Blood vessel length refers to the distance that blood has to travel through the vessels, which affects resistance because the longer the vessel, the more difficult it is for blood to flow through it.

However, vessel length remains relatively constant in the body because it is determined by the body's anatomy and does not change significantly over time. Therefore, the only time when vessel length can change is during a surgical procedure or injury that alters the anatomy of blood vessels.

Mathematically, the resistance to blood flow (R) can be calculated using the formula R = (8 x viscosity x length) / (π x radius⁴).

This formula shows that blood viscosity and vessel length are directly proportional to resistance, while vessel radius is inversely proportional.

Therefore, any increase in blood viscosity or vessel length will increase resistance, while any decrease in vessel radius will increase resistance.

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Answer:

[tex](a+2)^2[/tex]

Step-by-step explanation:

Substitute [tex]x=a+2[/tex] into [tex]g(x)[/tex].

Answer: To solve this problem, we can use the Pythagorean theorem. Let's call the depth of the treasure "d".

We can create a right triangle with the following sides:

The sensor on the boat is at the top of the triangle and is 200 meters away from the treasure diagonally.

The line from the sensor to the treasure, which we can call "x", is at a 25-degree angle from the surface of the water.

The line from the treasure to the ocean floor, which we can call "d", is perpendicular to the surface of the water.

Since we know that the angle of depression is 25 degrees, we can use tangent to find the length of x.

tan(25) = x/d

x = d * tan(25)

Now, we have two sides of a right triangle and can use the Pythagorean theorem to find d:

x^2 + d^2 = 200^2

d^2 = 200^2 - x^2

d^2 = 200^2 - (d * tan(25))^2

Solving for d, we find that d = 174.59 meters. Rounding to the nearest tenth of a meter, we get d = 174.6 meters.

Step-by-step explanation:

Answer:

Below    

Step-by-step explanation:

Since all of the lines are parallel and cut by a transversal .... the

 corresponding  angles are equal

8x - 11 = 7x +8

x = 19

Following are the given complete statements:

(a) Another name for plane X is plane EABC.

(b) F, D, and C are distinct points that are collinear.

(c) A, B, C and G are distinct points that are coplanar.

A plane is a flat, two-dimensional surface that extends infinitely in all directions. It is a mathematical concept used in geometry and is often represented as a flat sheet or a graph on a coordinate system. In the real world, a plane can be thought of as a flat surface such as a tabletop, a piece of paper or a wall.

(a) Another name for plane X is plane EABC.

(b) F, D, and C are distinct points that are collinear.

(c) A, B, C and G are distinct points that are coplanar.

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Errors in calculations are mistakes or inaccuracies that occur when performing mathematical calculations. They can be caused by a variety of factors, including human error, equipment malfunction, incorrect assumptions or data, and flawed methodology.

There are several types of errors that can occur in calculations, including:

Transcription errors: These occur when a number or a calculation is copied down incorrectly from one place to another, such as from a calculator to a piece of paper.

Calculation errors: These occur when the wrong calculation is used, or when a mistake is made in performing a calculation, such as forgetting to carry a digit when doing long division.

Rounding errors: These occur when numbers are rounded to a certain number of decimal places, and the resulting calculation is not as precise as it could be.

Measurement errors: These occur when the instruments used to measure quantities are not accurate, or when the measurements are not taken correctly.

Conceptual errors: These occur when the underlying concepts or principles of the problem are not fully understood, leading to incorrect calculations.

It is important to identify and correct errors in calculations to ensure the accuracy and validity of the results. Checking calculations multiple times and using multiple methods of verification can help to reduce the risk of errors.

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The surface area of the given pyramid is 336 square feet. the correct answer would be an option (F) 336 ft².

A square pyramid's total surface area is the total area covered by its four triangular sides and square base. The formula may be used to calculate the total surface area of a square pyramid with slant height.

The surface area of a square pyramid = a² + 2al

The figure of the square pyramid is given in the question, as shown

The surface area of the pyramid = a² + 2al

Here a = 12 ft and l = 8 ft

Substitute the values of a = 12 ft and l = 8 ft in the formula,

The surface area of the pyramid = 12² + 2 × 12 × 8

The surface area of the pyramid = 336 square feet

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The question seems incomplete, the correct question would be as:

The amount of the worker monthly tax will equals to K4,850.

An income tax refers to the tax imposed on individuals or entities in respect of the income earned by them. It is computed as the product of a tax rate times the taxable income. They may vary by type or characteristics of the taxpayer and the type of income

To calculate the worker's monthly tax, we need to determine the amount of salary that falls into each tax bracket and then calculate the tax owed for each bracket:

The worker's monthly tax is then computed as:

=  K1,500 + K3,000 + K350

= K4,850.

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To find the angle of elevation, we need to determine the height of the plane relative to the trees, and then use the tangent function to find the angle. Let's call the height of the plane "h".

We can set up a right triangle with the height of the plane, the distance from the runway to the trees, and the height of the trees.

We have:

h / (150 + 75) = 100 / h

Solving for h, we find that:

h = 125

So the height of the plane relative to the trees is 125 feet.

Now we can use the tangent function to find the angle:

tan(angle) = 100 / 125

angle = tan^-1(100 / 125)

angle = approximately 53.13 degrees

Rounding to the nearest hundredths, the angle of elevation that the pilot must take to avoid crashing into the trees is 53.13 degrees.

True. Because they have comparable angles, the offered figures are comparable figures.

Similar figures are described as two shapes that share the same shape but differ in size. For instance, similar objects are depicted but are not consistent in different-sized images of a person, such as passport size, stamp size, etc. In geometry, two comparable shapes with different side lengths or sizes are those whose dimensions are equal or have a common ratio. Examples of these shapes are similar triangles, similar rectangles, and similar squares. Scale factor is the name given to the common ratio. The matching angles are also of equal measure.

If two figures are similar, then they are represented by the symbol ‘∼’. Let us say there are two triangles, ABC and PQR, which are similar, then they are represented as;

∆ABC ~ ∆ PQR

Now if the two triangles are similar to each then their corresponding sides should be in proportion. Hence,

AB/PQ = BC/QR = AC/PR

∠A = ∠P, ∠B = ∠Q, ∠C = ∠R

Since, we can see that the angles in both the given figures are exactly the same. Thus, the given figures are similar figures.

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Jimmy’s total income is $91838.61

You should understand that total income' refers to the sum of certain incomes (in cash and, in some circumstances, in kind) of the statistical unit during a specified reference period

The given parameters that will help us to determine the income are

wages from accounting job,             $75,937.84;

interest,                                                   $839.40

wages from museum job,                  $14,937.92;

dividends,                                                $123.45.

Therefore Jimmy's total income is the sum of all the incomes

$(75,937.84 + 839.40 + 14,937.92 + 123.45)

Total income = $91838.61

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Answer:

42/11!

Step-by-step explanation:

So first, you need to make sure you have the formula which is V= Base X height. The base is the area of 1 circle, so you multiply the radius X radius X PI.

1/3 x 1/3 is 1/9. 22/7 is used as pi so multiply 1/9 by 22/7 and you get 22/63. Now you divide that by 4/3. When dividing fraction, you keep the first decimal (4/3) the same, turn the division symbol into a multiplication one, and flip the last fraction (so 22/63 is now 63/22!).

Now you just solve 4/3 x 63/22, which will give you 252/66. Now simplify that by dividing both numbers by 3 until you get the smallest answer, which is 42/11!

The height of the cylinder is approximately 42/11 inches. The answer is option C.

Here, we have,

we know that,

The formula for the volume of a cylinder is V = πr²h,

where V is the volume, r is the radius, and h is the height.

We are given that the volume of the cylinder is 1 and one-third in^3 and the radius is one-third in. Substituting these values into the formula, we get:

1 and one-third = 4/3

V = π(1/3)²h = 4/3

Simplifying the equation, we get:

h = (4/3) / (π(1/3)²)

  = (4/3) / (π/9)

  = (4/3) * (9/π)

  = 12/π

Approximating π as 22/7, we get:

h ≈ (12/π)

  ≈ (12/(22/7))

  = 42/11 inches

Therefore, the height of the cylinder is approximately 42/11 inches. The answer is option C.

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After considering all the combinations of three pieces of wood, we can conclude that Ennis can make 2 different triangular borders for his vegetable garden.

A triangle is a shape with three sides, and each side can be considered as a side length. In order to form a triangle, the sum of the lengths of the two shorter sides must be greater than the length of the longest side.

To find the number of different triangular borders that Ennis can make, we need to consider all the combinations of three pieces of wood as the sides of the triangle. There are four pieces of wood, so the number of combinations of three pieces is 4 choose 3, which is 4.

Now, we need to determine if each combination of three pieces of wood forms a triangle. If the sum of the lengths of the two shorter sides is greater than the length of the longest side, it forms a triangle. If not, it does not form a triangle.

If Ennis uses the 8-inch and 10-inch pieces as two sides, he can use the 12-inch or 18-inch piece as the third side, and it forms a triangle.

If he uses the 8-inch and 12-inch pieces as two sides, he can only use the 10-inch piece as the third side, and it forms a triangle. If he uses the 8-inch and 18-inch pieces as two sides, it does not form a triangle.

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Answer: 16*(1/8)

Step-by-step explanation:

So, based on the theoretical likelihood, we anticipate that 35 times out of 140 repeats, both spins will be odd numbers.

Probability is a branch of mathematics that deals with the study of random events and the likelihood of their occurrence. Probability is expressed as a number between 0 and 1, with 0 indicating that an event is impossible to occur and 1 indicating that an event is certain to occur. The probability of an event A, denoted by P(A), is calculated as the number of favorable outcomes for the event divided by the total number of possible outcomes. For example, if a fair six-sided die is rolled, the probability of rolling a 3 is 1/6 because there is only one favorable outcome (rolling a 3) out of the total 6 possible outcomes. Probabilities can be used to make predictions about the likelihood of future events and to make decisions under uncertainty. Probabilities can also be used to describe the distribution of random variables and to quantify the relationship between different events. Probability theory is widely used in many fields, such as statistics, engineering, finance, physics, and biology, among others.

Here,

The theoretical probability of both spins being odd numbers is 9 over 36, which means that for every 36 times the experiment is repeated, we expect 9 of those times to result in both spins being odd numbers.

If the experiment is repeated 140 times, we can use the theoretical probability to estimate the number of times both spins will be odd numbers as follows:

140 * (9/36) = 35

So, based on the theoretical probability, we predict that both spins will be odd numbers 35 times out of 140 repetitions.

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When a negative number is divided by a positive number, the result is always negative.

Let's use the example (-6) / 2 again. We can write this expression as:

(-6) / 2 = (-1) * 6 / 2

We can then simplify the expression on the right-hand side by multiplying -1 and 6, which gives us:

(-1) * 6 / 2 = -6 / 2

Now, we can divide 6 by 2, which gives us:

-6 / 2 = -3

So we see that the quotient is negative.

We can generalize this to any negative number divided by a positive number. Let's say we have a negative number -a and a positive number b. We can write the expression as:

(-a) / b = (-1) * a / b

Multiplying -1 and a gives us:

(-1) * a / b = -a / b

So we see that the quotient is negative, since -1 times a is negative, and dividing a negative number by a positive number results in a negative quotient.

Therefore, we can conclude that when a negative number is divided by a positive number, the result is always negative.

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The area of the left over wood is

Area of the kite

perimeter of the kite

The area of the left over wood is given by

= area of the circle - area of the pentagon

area of the circle

= pi * r^2

= pi * 5^2

= 78.54 square cm

area of the pentagon

= 5/2 * a * 5

solving for a

360 / 5 = 72

72 / 2 = 36

cos 36 = a / 5

a = 5 cos 36 = 4

area of the pentagon = 5/2 * 4 * 5 = 50 sq cm

left over wood = 78.54 - 50 = 28.54 square cm

Area and perimeter of kite

area of kite = product of the two diagonals

length of one of the diagonals = 2 * KT = 2 * 12 = 24 cm.

The second diagonal

tan 30 = KT / x

x = 12 / tan30 = 20.78

the length = 12 + 20.78 = 32.78

area of the kite = 32.78 * 24 = 786.72 square cm

perimeter of the kite

= addition of the sides = 2(shorter side + longer side)

Using Pythagoras rule

shorter side = √(12² + 12²) = 16.97

longer side =  √(20.78² + 12²) = 24

perimeter = 2 (16.97 + 24) = 81.94 cm

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The X, Y, and Z axes define a three-dimensional coordinate system, which is commonly used to represent physical objects or mathematical functions in space.

The X, Y, and Z axes are terms commonly used in mathematics, physics, engineering, and computer graphics to refer to the three dimensions in a three-dimensional coordinate system.

The X-axis is the horizontal axis and is used to represent values along the horizontal plane. It is often called the "abscissa" and is typically drawn as a horizontal line that intersects with the Y-axis at the point where they meet, which is called the origin.

The Y-axis is the vertical axis and is used to represent values along the vertical plane. It is often called the "ordinate" and is typically drawn as a vertical line that intersects with the X-axis at the origin

.The Z-axis is the third axis and is used to represent values along the depth or height plane. It is often called the "applicate" and is typically drawn as a line that is perpendicular to both the X and Y axes, intersecting with both at the origin.

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Step-by-step explanation:

Perimeter of a Semicircle

[tex](\pi )(r) + 2r[/tex]

R = 0.6 since that's half the diameter

[tex](3.14)(0.6) + (2)(0.6)[/tex]

[tex]1.884 + 1.2 = 3.084[/tex]

Which rounds to 3.1

In a linear equation, 9.14 does he save by paying off the loan when the next payment is due .

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.

                               Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.

Jan  293  18  311  1,507 16.25%

Feb  296  15  311  1,212 32.67%

Mar 298           12  311   913 49.25%

Apr  301            9  311   612 66.00%

May  304    6  311  308 82.92%

Jun  308    3  311    0        100.00%

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a. Two out of every 3 seniors at Lewis High apply for college, so the number of seniors who apply for college is 2/3 (or approximately 0.67) times the total number of seniors at Lewis High.

b. The number of seniors who apply for college is one-third (or approximately 0.33) less than the number of seniors who don't apply.

c. One-third (or approximately 0.33) of the graduating seniors at Lewis High do not apply to college.

d. Three out of every 5 seniors who apply for college are female students, so two out of every 5 seniors who apply for college are male students. Therefore, the fraction of seniors who are males who apply for college is 2/5 (or approximately 0.4).

The number of males who apply for college is 2/3 of the total number of seniors who apply for college, which is (2/3) * (2/5) = 4/15 (or approximately 0.27) times the total number of graduating seniors at Lewis High.

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The length of the right triangle x is 2√5 units.

A right triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.

Therefore, the sides of the right triangle can be found using Pythagoras's theorem,

Hence,

c² = a² + b²

where

Therefore,

x² = 6² - 4²

x² = 36 - 16

x² = 20

square root both sides

x = √20

x = 2√5

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Answer:

Alaskan huskies take 10h 34min.

Siberian huskies take 8h 45min.

Step-by-step explanation:

Alaskan huskies : 15km/h + 18min rest after 3h

18/3= 6min rest/h

1h= 60min

60/15= 4min/km

4*145= 580min

         = 9h 40min

There is three 3h in 9h 40min.

3*18= 54min

9h 40min+54min= 10h 34min

Siberian huskies : 20km/h + 30min rest after 2h

30/2= 15min rest/h

60/20= 3min/km

3*145=435min

         = 7h 15min

There is three 2h in 7h 15min.

3*30= 90min

       = 1h 30min

7h 15min+1h 30min= 8h 45min

Answer:

Step-by-step explanation:

sin angle S = [tex]\frac{opposite}{hypotenuse}[/tex]

sin S = [tex]\frac{30}{hypotenuse}[/tex]

To complete the ratio we need to find the hypotenuse:

a² + b² = c²

16² + 30² = c²

256 +  900 = c²

1156 = c²

√1156 = √c²

34 = c

Now substitute 34 in the sin ratio:

[tex]\frac{30}{34}[/tex] = .8824

nearest hundredth = .88

The length of the rectangular tray is 1 9/10

What is a Rectangle?

Rectangle is a four-sided flat shape where every angle is a right angle (90°).

Area of a Rectangle = Length * Width

where,

Area = 4 3/4

Length = ?

Width = 2 1/2

To find the length of the tray,

Length = Area/Width

Length = 4 (3/4) / (2 1/2)

Length = (19/4) / (5/2)

Length = 19/4 * 2/5

Length = 19/10

Length = 1 9/10

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The correct statement regarding the solution to the system of equations is given as follows:

b. (1,2) is a solution to both the lines A and B.

We want to obtain the graphic solution for the system of equations, hence we obtain the point of intersection of the two equations.

This point of intersection is a solution to both of the equations, hence it is the solution to the system of equations.

The coordinates of the point of intersection in this problem are given as follows:

(1,2).

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Answer:

Щоб визначити, яка авіакомпанія має кращий досвід прибуття вчасно, ми можемо переглянути їхню статистику ефективності вчасності. Згідно з даними Міністерства транспорту США, у 2020 році показник своєчасних прильотів у SkyWest Airlines становив 85,8%, а у Southwest Airlines – 83,5%. Це означає, що у 2020 році SkyWest Airlines мала кращий досвід прибуття вчасно, ніж Southwest Airlines.

Однак варто зазначити, що своєчасна продуктивність може змінюватися залежно від низки факторів, включаючи погоду, управління повітряним рухом і проблеми з обслуговуванням. Також важливо врахувати інші фактори, які можуть бути для вас важливими, наприклад вартість авіаквитків, наявність рейсів у бажані пункти призначення та якість обслуговування під час польоту. Таким чином, незважаючи на те, що оперативність є важливим фактором, який слід враховувати при виборі авіакомпанії, це може бути не єдиним фактором, який вам слід враховувати.

Step-by-step explanation:

Answer:

The solution to this problem is x = 26 over 11 , y =  6 over 11.

Step-by-step explanation:

Hope this helped!