The game of clue involves 6 suspects, 6 weapons, and 9 rooms. one of each is randomly chosen and the object of the game is to guess the chosen three. (a) how many solutions are possible? in one version of the game, the selection is made and then each of the players is randomly given three of the remaining cards. let s, w, and r be, respectively, the numbers of suspects, weapons, and rooms in the set of three cards given to a specified player. also, let x denote the number of solutions that are possible after that player observes his or her three cards. (b) express x in terms of s, w, and r. (c) find e[x].

We are given a function H(t) that represents the height of a cannon ball after t seconds as:

                  [tex]H(t)=-16t^2+48t+12[/tex]

and a second cannon ball is represented by the function g(t) as:

                       [tex]g(t)=10+15.2t[/tex]

PART A:

  t           0            1             2            3

H(t)        12          44          44           12

g(t)         15.2       25.2      40.4       55.6

Hence, between t= 2 seconds and t=3 seconds the ball will meet

     such that H(t)=g(t)

Since, we know that the Height H(t) decreases from 44 feet to 12 feet between t=2 to t=3 seconds.

and height g(t) increases from 40.4 feet to 55.6 feet between t=2 to t=3 seconds.

Hence, the two cannon balls will definitely meet between t=2 to t=3 seconds.

and the time at which they meet is calculated by solving:

[tex]H(t)=g(t)\\\\i.e.\\\\\\-16t^2+48t+12=10+15.2t\\\\\\i.e.\\\\\\16t^2-32.8t-2=0\\\\\\i.e.\\\\\\t=-0.059\ and\ t=2.109[/tex]

As t can't be negative.

Hence, we get:

t=2.109 seconds

PART B:

The solution from PART A means that the one of the ball first reach the highest point at 44 feet and then returns back to the initial position and hence follows a parabolic path while the second cannon ball reach a greater height with the increase in time and hence in this phenomena the two balls will definitely meet.

The length and width of the rectangle are 11 m and 7 m respectively.

A rectangle is a part of a quadrilateral, whose sides are parallel to each other and equal.

Given that,

Length of the rectangle is 10 m less  than three times the width,

And area of rectangle = 77 m²

Let the width of rectangle is x m,

Then length of rectangle = 3x -10 m.

The area of rectangle = 77

length × width = 77

x × (3x - 10) = 77

The value of x = 7 satisfy the equation,

Therefore, the width of rectangle is x = 7 m                  

And length of the rectangle is 3x - 10 = 21 - 10 = 11 m

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Answer:

B. {-4,3,5,8}

Step-by-step explanation:

We are given,

The set representing the function is {(-1,5),(2,8),(5,3),(13,-4)}.

It is required to find the range set of the function.

Now, as we know,

In the ordered pair [tex](x,y)[/tex], the y co-ordinate 'y' represents the range values of a function.

So, we see that,

The range values of the given function are 5, 8, 3 and -4.

Thus, the set representing the range of the function is {-4,3,5,8}.

Answer:

(A)[tex]\frac{x^{\frac{1}{4}}}{y^{\frac{5}{6}}}[/tex]

Step-by-step explanation:

The given expression is:

[tex]\frac{x^{\frac{1}{2}}y^{\frac{-1}{3}}}{x^{\frac{1}{4}}y^{\frac{1}{2}}}[/tex]

Upon solving the given expression, we get

=[tex]{x^{\frac{1}{2}-\frac{1}{4}}{\cdot}}{y^{\frac{-1}{3}-\frac{1}{2}}}[/tex] (using the property of exponents and powers that if base is same then the powers gets added.)

=[tex]x^{\frac{1}{4}}{\cdot}y^{\frac{-5}{6}}[/tex]

=[tex]\frac{x^{\frac{1}{4}}}{y^{\frac{5}{6}}}[/tex]

which is the required simplified form of the given equation.

Hence, option A is correct.

Answer:

[tex]10x^4+65x^3-25x^2[/tex]

Step-by-step explanation:

Multiply 5x^2(2x^2 + 13x − 5)

[tex]5x^2(2x^2 + 13x - 5)[/tex]

Multiply 5x^2 inside the parenthesis

[tex]5x^2 * 2x^2 = 10x^4[/tex]

[tex]5x^2 * 13x = 65x^3[/tex]

[tex]5x^2 * (-5) = -25x^2[/tex]

Collect all the terms together

[tex]10x^4+65x^3-25x^2[/tex]

Answer:

She ran 5 miles

Step-by-step explanation:

To solve this, we can easily use proportion.

The question state that there are 5,280 feet in a mile, We are ask to find the number of miles Rily run, if she ran 26,400 feet.

Using proportion;

Let x = the number of miles Rily  can run

5,280 feet     = 1 mile

26,400 feet    = x

Cross-multiply

5280x = 26,400

To get the value of x, we will divide both-side of the equation by 5280

5280x/5280 = 26400/5280

(On the left-hand side of the equation, 5280 will cancel-out 5280 leaving us with x and on the right-hand side of the equation 26400 will divide 5280)

x = 26400 / 5280

x=5 mile

Therefore, If Rily ran 26400 feet, then she has ran 5 miles.

The scale ratio is the ratio of the length of one side of a polygon with the length of a corresponding side of a similar polygon.

The ratio you're referring to is called the scale factor. The scale factor is the ratio of the length of one side of a polygon to the length of the corresponding side of a similar polygon. Let's denote this scale factor as ( k ).

To calculate the scale factor, you need to choose corresponding sides from the two similar polygons. Once you've chosen a pair of corresponding sides, divide the length of one side by the length of the corresponding side from the other polygon.

Let's say we have two similar polygons, Polygon A and Polygon B, and we want to find the scale factor between them. Let's denote the length of a side of Polygon A as [tex]\( L_A \)[/tex] and the length of the corresponding side of Polygon B as [tex]\( L_B \)[/tex]. Then, the scale factor, ( k ), is given by:

[tex]\[ k = \frac{L_A}{L_B} \][/tex]

This ratio will give you the scale factor between the two polygons.

Assembly Line A will produce 24975 units during the same time that Line B produces 555 units.

Let's assume that the time it takes for Assembly Line A to produce 45 units is equal to the time it takes for Assembly Line B to produce 37 units. This means that the time it takes for Line A to produce 1 unit is equal to the time it takes for Line B to produce 1 unit.

Now, we can set up a proportion to find how many units Line A will produce during the same time that Line B produces 555 units:

45 units / 1 unit = x units / 555 units

Cross-multiplying, we get:

45 * 555 = x units * 1

x units = 24975 units

Therefore, Line A will produce 24975 units during the same time that Line B produces 555 units.

Answer:

Step-by-step explanation:

The answer is actually B. $160,000; $320,000

The reason for this is because -

$10,000 is the constant, so let a= 10,000.

The investment doubles every 13 years, so let b=2.

Let x= the number of 13-year periods.

The value of a $10,000 investment doubling every 13 years will be $160,000 after 52 years and $320,000 after 65 years.

To calculate the value of an investment that doubles every 13 years, you can determine the number of times the investment will double over a given period. In this case, after 52 years, the investment will have doubled 4 times (since 52 divided by 13 is 4), and after 65 years, it will have doubled 5 times (since 65 divided by 13 is 5).

The formula for the future value of an investment that doubles is: Future Value = Present Value × (2^number of doublings).

After 52 years, the investment's worth would be calculated as follows:

$10,000 × (2^4) = $10,000 × 16 = $160,000

After 65 years, the investment's worth would be:

$10,000 × (2^5) = $10,000 × 32 = $320,000

Therefore, the correct answers are $160,000 after 52 years and $320,000 after 65 years, which corresponds to choice B.

Answer:

Close to 240 times but probably not exactly 240 times

Step-by-step explanation:

The simplification form of the expression 90/ [10+(3² - 4)] is 6 the answer is 6.

It is defined as the operation in which we do the addition of numbers, subtraction, multiplication, and division. It has a basic four operators that is +, -, ×, and ÷.

As we know, the expression can be defined as the combination of constants and variables with mathematical operators.

To begin, either multiply or divide the value in parentheses (32-4), which equals 5.

Add the number in the parentheses once more: 90/[10+5 (always do the parenthesis first) [10+5]= 15.

Finally, divide 90/15 by 6, always keeping that in mind (PEMDAS)

Thus, the simplification form of the expression 90/ [10+(3² - 4)] is 6 the answer is 6.

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The Cartesian form of equation is,

3x + 4y = 12

We have to given that,

The parametric equations are,

x = 4 sin² θ

y = 3 cos² θ

Now, Change the parametric equations into cartesian form as,

x = 4 sin² θ

sin² θ = x/4  .. (i)

y = 3 cos² θ

cos² θ = y/3 (ii)

Add equation (i) and (ii),

sin² θ + cos² θ = x/4 + y/3

1 = x/4 + y/3

1 = (3x + 4y)/12

12 = 3x + 4y

3x + 4y = 12

Therefore, The Cartesian form of equation is,

3x + 4y = 12

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The image of point (5, -1) under the same translation that maps point P to P' is (1, -8). This translation vector is obtained by subtracting the coordinates of P from P'.

To find the image of point (5, -1) under the same translation T that maps point P to P', we can calculate the vector that represents the translation from P to P' and then apply the same translation to point (5, -1).

The translation vector is given by:

Translation vector = P' - P = (-6, -4) - (-2, 3) = (-4, -7)

Now, to find the image of point (5, -1) under the same translation, we simply add this translation vector to (5, -1):

Image of (5, -1) = (5, -1) + (-4, -7) = (5 - 4, -1 - 7) = (1, -8)

So, the image of point (5, -1) under the translation T is (1, -8). Therefore, the correct answer is B. (1, -8).

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2x +4 =14

2x =10

x =5

 the number is 5