Select the correct answer. Suppose x varies indirectly as y, and x = 5 when y = 24. What is the value of x when y = 8? A. 15 B. 1. 67 C. 960 D. 38. 40 Re

    ΔFIT ≅  ΔRIS triangle  are congruent.

How do triangles work?

The three vertices of a triangle make it a three-sided polygon. The angles of the triangle are formed by the three sides' end-to-end connections at a point.

                       The triangle's three angles add up to a total of 180 degrees. A triangle is a closed, two-dimensional geometry with three sides, three angles, and three vertices. Polygons include triangles as well.

 In figure ,

              FI = IS

              RI = IT

            ∠FIT = ∠RIS

           ΔFIT ≅  ΔRIS

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Answer: The inverse of the logarithmic function f(x) = log2x can be found by interchanging the roles of x and y in the function and solving for y. The steps are as follows:

Step 1: Replace f(x) with y.

y = log2x

Step 2: Interchange x and y.

x = log2y

Step 3: Solve for y.

We need to isolate y on one side of the equation. To do this, we can rewrite the equation in exponential form. Recall that the logarithmic function with base b is defined as follows:

logb(x) = y if and only if b^y = x.

Using this definition, we can rewrite the equation x = log2y as follows:

2^x = y

This means that the inverse of f(x) = log2x is given by:

f –1(x) = 2^x

Therefore, the correct answer is:

f –1(x) = 2^x.

Step-by-step explanation:

Answer:

Step-by-step explanation:

The volume of the given prisms above is calculated as follows:

a.) = 432m³

b.) = 135m³

C.) = 1,332m³

For prism a:

Divide the prism to get two solid shapes and then add up the calculated volumes of each.

Vol of First shape = length× width× height

H = 12m

w = 4m

L = 7m

Vol = 12×4×7 = 336m³

Vol of second shape ;

=2( 4×4×3)

= 2 × 48

= 96m³

Therefore the volume of prism A = 336+96 =432

For prism b:

Volume= BH

B = area of the base

H = height of the prism

Volume = 1/2×5 × 6 × 9

= 5×3×9

= 135m³

For prism C:

Divide the prism to get two solid shapes and then add up the calculated volumes of each.

Vol of first shape = 5×15×12 = 900m³

Vol of second shape;

=2(6×12×3)

= 2(216)

= 432 m³

Volume of prism C = 900+432

= 1,332m³

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Answer:

Step-by-step explanation:

Center-radius equation for circle of radius r, centered at (h,k):(x−h)2+(y−k)2=r2

(h,k)=(0,0) and r=11−−√:

x2+y2=11

If she receives $2 off the regular price, p. by using a coupon. her total cost, including 20% for tax and tip, is $9. then the regular price of the lunch special is $11.

Let's figure out why that is so. Romita gets a sushi lunch special at Yoshi's, with a coupon. Romita received a discount of $2 off the regular price, P.

         Let's assume that the price of lunch before the discount is $P.

 Therefore, after the discount, the cost of the lunch is (P - 2).

 Now, let's determine the total cost of the lunch. The price of the lunch is inclusive of 20% tax and tip. This implies that Romita's total cost is 120% of the final cost of the lunch.

    We can represent this information in the equation below:

             Total cost = 120% of (P - 2)

             Total cost = 1.2(P - 2)

    From the given information, we know that the total cost is $9.

    This means we can solve the equation for P.P = (9/1.2) + 2P = 7.5 + 2P = $11

         The regular price of the lunch special is $11.

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Triangles are similar by SSS rule.

The triangle similarity rule states that if two triangles have corresponding angles that are congruent, then the triangles are similar. This means that their corresponding sides are proportional in length. More formally, if two triangles ABC and DEF are similar, then:

The corresponding angles are congruent: ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F.

The corresponding sides are proportional: AB/DE = BC/EF = AC/DF.

Similar triangles can also be used to find the height of tall objects or the distance between two objects of known height, using the angle of elevation and the properties of similar triangles.

In the Triangle ABC and CDA;

AB=CD(Given)

BC=AD(Given)

AC=AC(sides are common)

Hence,  Triangles are similar by SSS rule.

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The complete tables are mentioned below for Business Employees and For Killian employees.

In mathematics, "amount" typically refers to the quantity or total of something. It is a general term that can be used to describe any kind of numerical value or magnitude, and can be expressed in various units, such as dollars, meters, liters, etc.

In many mathematical contexts, the term "amount" is used interchangeably with other related terms, such as "quantity," "total," "sum," or "value." For example, in algebra, the amount of a variable can refer to the numerical value or quantity that it represents. In geometry, the amount of a particular measurement, such as area or volume, is the total quantity of that measurement in a given shape or object.

The concept of amount is fundamental in many areas of mathematics, from basic arithmetic to more advanced calculus and statistics. It plays a central role in various mathematical operations and concepts, such as addition, subtraction, multiplication, division, and integration.

Employee  C. North                N. Vanders              M. Clark

Position     Systems                 Web                         Data

                  Analyst                  Designer                  Entry

Level           2                           3                               3

Salary        $44,000                41,000                     31,500  

COLA           2.80%                   3.20%                     4.70%

Amount      $1,232                    $1,312                      $1,483

Merit            3.60%                    7.60%                     4.50%

Amount       $1,584                    $3,116                     $1,418  

New            $47,816                  $45,428                  $34,401

Salary

For Killian employees:

Employee  Annual   Merit         COLA       Total         New

                  Salary     Increase  Increase   Increase   Salary

D. Garcia   $18,000  6.00%      7.5%          12.50%     $20,250

J. Stites      21,750    3.50%      6.5%           10.00%    $24,090

J. Spar        31,860    5.10%       6.3%          11.40%      $35,472

T. Beggins  14,615     0.70%      7.5%          8.20%      $15,798

R. Stokes     27,916   6.250%    6.3%          12.525%   $31,373

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Step-by-step explanation:

Both of the expressions equal y....equate them

-6x + 8 = 5x-4

12 = 11x

x = 12/11 = 1  1/11

then y =  5x -4  = 5(12/11) - 4   = 1  5/11

( 1  1/11  ,  1  5/11 )

Answer:

Step-by-step explanation:

You want the missing segment lengths in the kite shown, where half-diagonals are 3 and 6, and one side length is 5.

The diagonals of a kite cross at right angles, so each of the triangles shown is a right triangle. The figure has left-right symmetry about the vertical axis, so x=z.

We recognize the top triangles as being 3-4-5 right triangles, so the missing side length is y = 4.

These lengths are the hypotenuse of a right triangle with legs 3 and 6. The Pythagorean theorem tells us they are ...

  x² = 3² +6²

  x² = 9 +36 = 45

  x = √(45) = √(9·5) = 3√5

The lengths x and z are ...

  x = z = 3√5

__

Additional comment

In case you have never heard of a 3-4-5 right triangle, you can find y from the Pythagorean relation:

  5² = 3² +y²

  y² = 25 -9 = 16

  y = √16 = 4

The {3, 4, 5} Pythagorean triple is one of several in common use in algebra, trig, and geometry problems. Others include {5, 12, 13}, {7, 24, 25}, {8, 15, 17}.

The given series converges only when x = 0

To determine the values of x for which the series n!x⁵ⁿ converges, we can use the ratio test, which states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges, and if the limit is greater than 1, the series diverges.

Applying the ratio test to the series, we obtain |aₙ ₊ ₁ /aₙ| = |(n+1)! x⁵(n+1)/(n! x⁵)| = |(n+1)x⁵|. We simplify this expression to obtain |(n+1)x⁵|.

We then take the limit as n approaches infinity of |(n+1)x⁵|. If x is not equal to zero, this limit will approach infinity, since n+1 grows without bound as n approaches infinity, and x⁵ is a constant factor that does not affect the growth rate. In this case, the limit will be greater than 1, indicating that the series diverges.

However, if x is equal to zero, then the limit of |(n+1)x⁵| is zero, indicating that the series converges.

Therefore, the series n!x⁵n converges only when x is equal to zero.

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Answer:

Step-by-step explanation:

Answer:

x = 5

Step-by-step explanation:

[tex]3x + 11 = 2(x + 8)[/tex]

[tex]3 x + 11 = 2x + 16[/tex]

[tex]x = 5[/tex]

The value of "n" is 6.

"n" is related to a circle with intersecting chords labeled 5, n+4, 7, and n+8 [1]. To find the value of "n", we can use the property that states that if two chords intersect inside a circle, the products of their segments are equal. Using this property, we can set up the following equation:

7(n+4) = 5(n+8)

Expanding the brackets, we get:

7n + 28 = 5n + 40

Simplifying, we get:

2n = 12

n=6

A circle is a two-dimensional geometric shape that is defined as a set of all points in a plane that are at a fixed distance (called the radius) from a given point (called the center). It is a closed shape, meaning that it has no beginning or end, and its boundary is a continuous curve.

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The function table for the function y=−2x+1 if given below so that Output of  y  could be –2, –1, 0, 1.

[tex]\left[\begin{array}{ccc}X&Y\\1/2&0\\1&1\\0&-1\\-1/2&-2\end{array}\right][/tex]

Here, Y and x are related in such a way that there is a distinct value of y for each value of x, and this relationship is frequently represented as y = f(x), or "f of x." In other words, the same x cannot have more than one value for f(x). The domain and range of the function are the set of values of f(x) that are produced by the values in the domain of the function, which is the set of values of x. In addition to f(x), other abbreviated symbols for functions of the independent variable x include g(x) and P(x), especially when the nature of the function is ambiguous or unspecified.

The function is an expression, rule, or law that establishes the relationship between two variables (the dependent variable). In mathematics, functions are everywhere, and they are crucial for constructing physical relationships in the sciences. German mathematician Peter Dirichlet first offered the modern definition of function in 1837.


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Using diagonals from a common vertex of a decagon (a polygon with ten sides), a total of seven triangles can be formed.

To see why, we can apply the formula for the number of triangles formed by drawing all the diagonals from a single vertex of a polygon, which is (number of sides - 2). For a decagon, the formula gives us:

number of triangles = (10 - 2) = 8

However, we need to subtract one from the total because the vertex where the diagonals are drawn from is not included in any of the triangles. Therefore, the number of triangles formed by drawing diagonals from a common vertex of a decagon is:

number of triangles = 8 - 1 = 7

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$1330.97 is the amount needed to repay the loan in full using the Rule of 78.

In finance and ecοnοmics, interest is payment frοm a bοrrοwer οr depοsit-taking financial institutiοn tο a lender οr depοsitοr οf an amοunt abοve repayment οf the principal sum, at a particular rate. It is distinct frοm a fee which the bοrrοwer may pay the lender οr sοme third party.

We wοuld subract interest  saved  frοm  the tοtal payment which Grant wοuld have tο pay if he cοntinued the lοan fοr 30 mοnths :

        7050-79.03= $6970.97

The Amοunt οf mοney Grant have already paid fοr 24 mοnths

= 24*235= $5640

The amοunt needed tο repay the lοan in full

= 6970.97 - 5640 = $ 1330.97

Thus, option a $ 1330.97 is correct.

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Answer:

58.5

Step-by-step explanation:

[tex]A=\frac{h_bb}{2}[/tex]

[tex]\frac{6.5*18}{2}[/tex]

[tex]58.5[/tex]

Two methods for proving that two triangles on a Cartesian coordinate plane are congruent are:

Two methods for proving that two triangles on a Cartesian coordinate plane are congruent are:

Side-Side-Side (SSS) congruence theorem: Two triangles are congruent if their corresponding sides are equal in length.

Side-Angle-Side (SAS) congruence theorem: Two triangles are congruent if two corresponding sides and the included angle are equal in both triangles.

To prove that two triangles on a Cartesian coordinate plane are congruent using coordinate geometry, we can follow these steps:

Identify the coordinates of the vertices of the two triangles.

Use the distance formula to find the lengths of the sides of each triangle.

Use the angle formula to find the measure of one angle in each triangle.

Check if the corresponding sides and angles of the two triangles are equal.

If the corresponding sides and angles are equal, the two triangles are congruent by SAS or SSS congruence theorem.

Here are the coordinates of the Triangles:

ΔABC = A(-3, 4) B (-4, 1) C (0,3)

ΔDEF = D(0, 2) E(3, 1) F(2, -2)


To prove whether the two triangles ΔABC and ΔDEF are congruent or not, we need to check if their corresponding sides and angles are equal.

Using the distance formula, we can find the lengths of the sides of each triangle:

AB = √[(4 - 1)² + (-4 - (-3))²] = 3.16

BC = √[(3 - 1)² + (0 - 4)²] = 4.47

AC = √[(3 - 4)² + (0 - (-3))²] = 3.16

DE = √[(2 - 1)² + (3 - 0)²] = 3.16

EF = √[(2-3)² + (-2- 1)²] = 3.16

DF = √[(0-2)² + (2 - - 2)²] = 4.47


Thus, on the basis of the side-side-side postulate, both triangles are congruent because the three sides have an equivalent on each triangle.

AB = DE
AC = EF
BC = DF

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The calculated probability of the coin landing face-up on "tails" on all six flips is 1/64.

The probability of getting tails on a single flip of a fair coin is 1/2. Since each flip is independent of the others, the probability of getting tails on all six flips is:

P = (1/2) x (1/2) x (1/2) x (1/2) x (1/2) x (1/2)

When evaluated, we get

P = 1/64

Therefore, the probability of the coin landing face-up on "tails" on all six flips is 1/64.

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The minimum value is 16 , the maximum value is 50, the first quartile is 20, the third quartile is 38 and the median is 24.

The box plot that represents the dataset is the second box plot.

The numbers would first be rearranged in ascending order. The numbers arranged in ascending order is:

16, 18, 22, 23, 25, 32, 43, 50

The five number summary are the median, minimum, first quartile, third quartile and maximum.

Median = (n + 1) / 2 = (8 + 1) / 2 = 4.5th term = (23 + 25) / 2 = 24

Minimum = 16

Maximum = 50

First quartile = (n + 1) / 4 = (8 + 1) / 4 = 2.25 term = 20

Third quartile = 3/4 x (n + 1) = 3/4 x (9/4) = 6.75th term = 38

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The steps that you should take to determine if these figures are similar include the following:

2. Multiply the vertices of polygon ABCD by 1/20.

5. Reflect the intermediate image.

In Mathematics and Geometry, two geometric shapes are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.

In Mathematics and Geometry, the scale factor of a geometric figure can be calculated by dividing the length of the image by the length of the pre-image or by multiplying the vertices by a specific scale factor.

This ultimately implies that, we would multiply the vertices of polygon ABCD by 1/20 and then apply a reflection of the intermediate image;

Vertices A' = (1 × 1/20, -6 × 1/20) = (1/20, -6/20)

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The bacteria pοpulatiοn after 3 hοurs is 36864.

Expοnential grοwth:  

The cοncept used here is expοnential grοwth, which οccurs when the rate οf increase οf a quantity is prοpοrtiοnal tο its current value.

In this case, the bacteria pοpulatiοn dοubles every 15 minutes, which means that its rate οf increase is prοpοrtiοnal tο its current value.

This leads tο an expοnential grοwth curve, where the pοpulatiοn increases rapidly οver time.    

Here we have

An experiment begins with 9 bacteria that grοw at such a rate their pοpulatiοn dοubles every 15 minutes.

Since there are 4 sets οf 15 minutes in an hοur,

The pοpulatiοn will dοuble 4 times in οne hοur.

Therefοre, the bacteria pοpulatiοn after οne hοur will be:

=> 9 × 2⁴ = 9 × 16 = 144

After twο hοurs, the pοpulatiοn will dοuble 4 times again: Hence, the pοpulatiοn after 2 hοurs

=> 144 × 2⁴ = 144 × 16 = 2,304

Similarly, After three hοurs,

2,304 × 2⁴ = 2,304 × 16 = 36864

Therefοre,

The bacteria pοpulatiοn after 3 hοurs is 36864.

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Answer:

18ft cubed

Step-by-step explanation:

To solve for the volume of a triangular prism

It's Base x Height x 0.5

V = bxhx0.5

V = 4x9x0.5

V = 36x0.5

V = 18ft cubed

Answer:

400 students altogether

Step-by-step explanation:

(100% -5%) - (45% + 25%) + 25% = 95% - 70% + 25% = 50%

200 ÷ 50% = 400

The function is defined by this set of rules and we got the cost of mailing a letter weighing 2.5 ounces from the United States to Mexico in 2018 using function P is $2.89.

Based on the rules, the function P is defined as follows:

P(w) = 1.15, if 0 < w ≤ 1

P(w) = 1.99, if 1 < w ≤ 2

P(w) = 2.89, if 2 < w ≤ 3

P(w) = 3.30 + 0.45(w-3), if 3 < w

This function gives the cost, in dollars, of mailing a letter from the United States to Mexico in 2018 based on the weight of the letter in ounces, w.

To find the cost of mailing a letter weighing 2.5 ounces using function P, we need to evaluate P(2.5). Based on the rules defined above, we can see that 2 < 2.5 ≤ 3. Therefore, the cost of mailing a letter weighing 2.5 ounces would be:

P(2.5) = 2.89

So, it would cost $2.89 to mail a letter weighing 2.5 ounces from the United States to Mexico in 2018 using function P.

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A rectangular pyramid has a rectangular base and four triangular faces that meet at a common vertex.

To match each net with its name, we need to understand the basic properties of both rectangular prism and rectangular pyramid.

A rectangular prism is a 3D shape that has six faces, and each face is a rectangle. A rectangular pyramid is also a 3D shape, but it has a rectangular base and four triangular faces that meet at a single point.
It consists of six rectangles connected along their edges to form a box-like structure.

On the other hand, the net for a rectangular pyramid is also a flat pattern, but it consists of a rectangle and four triangles attached to the edges of the rectangle.
To match each net with its name, we need to compare the features of both nets with the basic properties of both shapes. If we observe the net for a rectangular prism, we can see that it has six rectangles, which represents the six faces of the rectangular prism.

Also,

The edges of each rectangle connect with other rectangles in such a way that it forms a box-like structure. Hence, the net with six rectangles is the net for the rectangular prism.
The net for a rectangular pyramid consists of a rectangular base and four triangles attached to the edges of the rectangle.

The rectangular base is one of the features of a rectangular pyramid, and it matches with the rectangle in the net. The other four triangles are attached to the edges of the rectangle, forming a pyramid-like structure.

Therefore,

The net with a rectangle and four triangles is the net for the rectangular pyramid.
The net with six rectangles is the net for the rectangular prism, and the net with a rectangle and four triangles is the net for the rectangular pyramid.

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The probability that no more than 3 of them weigh 65 lb or more is 0.0165 (approximately).

The Australian Sheep Dog is a breed renowned for its intelligence and work ethic. It is estimated that 45% of adult Australian Sheep Dogs weigh 65 pounds or more. A sample of 16 adult dogs is studied. We need to determine the probability that no more than 3 of them weigh 65 pounds or more. Solution:Step 1The number of trials (n) is 16. The probability of success in one trial is p = 0.45. The probability of failure is q = 1 - p = 1 - 0.45 = 0.55.Step 2We need to find the probability that no more than 3 of them weigh 65 lb or more.P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X distribution with parameters n = 16 and p = 0.45.Using the binomial probability formula,P(X = k) = nCk * pk * q(n - k)P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)= 16C0 * 0.45^0 * 0.55^16 + 16C1 * 0.45^1 * 0.55^15 + 16C2 * 0.45^2 * 0.55^14 + 16C3 * 0.45^3 * 0.55^13≈ 0.0165 (rounded to 4 decimal places)Therefore, the probability that no more than 3 of them weigh 65 lb or more is 0.0165 (approximately).

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