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Write a system of equations to describe the situation below, solve using substitution, and fill
in the blanks.
Peter works in the shipping department of a toy factory that makes radio-controlled
helicopters. Small helicopters weigh 3 pounds each, and are shipped in a container that
weighs 14 pounds. Large ones, on the other hand, weigh 4 pounds apiece, and are shipped in
a container that weighs 13 pounds. If these boxes can hold a certain number of helicopters
each, all of the packed containers will have the same shipping weight. What would the total
weight be? How many helicopters would fit in either container?

After solving, total 360 different executive boards can there be.

In the specified office have one member of the executive board is the president, another is the vice president, a third is the secretary, and the last one the treasurer.

So there are total 6 members.

The total number of members in executive board = 4

So, the ways of selecting different executive boards = [tex]^{6}P_{4}[/tex]

The ways of selecting different executive boards = [tex]\frac{6!}{(6-4)!}[/tex]

The ways of selecting different executive boards = [tex]\frac{6!}{2!}[/tex]

The ways of selecting different executive boards = [tex]\frac{6\times5\times4\times3\times2!}{2!}[/tex]

The ways of selecting different executive boards = 6 × 5 × 4 × 3

The ways of selecting different executive boards = 360

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The complete question is:

From a committee of 6 members, suppose that each member of the executive board also has a specified office: one member of the executive board is the president, another is the vice president, a third is the secretary, and the last one the treasurer. How many different executive boards can there be?

The smallest positive integer k such that 540k is a perfect cube is 8.

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Answer: The length of a side of a regular pentagon can be found using the formula, side = 2r * sin(π/5), where r is the radius of the circle that the pentagon is inscribed in. Using this formula, the length of a side of the pentagon is approximately 4.08 cm or 40.8 mm (rounded to the nearest mm).

Answer: -48x-32

Step-by-step explanation:

[tex](\stackrel{x_1}{-2}~,~\stackrel{y_1}{8})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{8}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{8}-\stackrel{y1}{8}}}{\underset{\textit{\large run}} {\underset{x_2}{4}-\underset{x_1}{(-2)}}} \implies \cfrac{0}{4 +2} \implies \cfrac{ 0 }{ 6 } \implies \text{\LARGE 0}[/tex]

A line that is perpendicular to the line y = -1/3x + 7 has a slope that is the negative reciprocal of -1/3 which is 3.

We know that the line intersects the point (4,2), so we can use this point and the slope to write the equation of the line in point-slope form:

y - y1 = m(x - x1)

where (x1, y1) is the point the line passes through, m is the slope, and y and x are the coordinates of any point on the line.

So the equation of the line that is perpendicular to y = −1/3x + 7 and intersects the point (4,2) is:

y - 2 = 3(x - 4)

Simplifying this, we get

y = 3x - 2

To convert this to the slope-intercept form we can rewrite it as

y = 3x + b

Therefore, the equation of the line that is perpendicular to y = −1/3x + 7 and intersects the point (4,2) is y = 3x - 2

The line that is parallel to y = 1/5x + 4, is A. 5 y - x = 6.

Parallel lines are any two or more lines that all lie in the same plane and never cross one another. They are equally spaced apart and have the same incline. No matter how far we extend a parallel line, it will never meet another parallel line.

As mentioned parallel lines will have the same slope so the line that is parallel to y = 1/5x + 4 would have the same slope of 1 / 5.

Converting the first option to slope intercept form gives:

5 y - x = 6

5 y = 6 + x

y = 6 / 5 + x / 5

y = 1.2 + 1 / 5 x

This slope is the same as y = 1/5x + 4 so they are parallel.

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The scale factor of figure A to fig B is 1 : 4

A scale factor is defined as the ratio between the scale of a given original object and a new object, which is its representation but of a different size (bigger or smaller).

Figure A is the small triangle and figure B is the small triangle.

The ratio of the corresponding lengths of the triangles must give thesame value.

scale factor = original length/ new length

scale factor = 14/56 = 1/4 = 1 : 4

also , 16/64 = 1/4 = 1:4

7/28 = 1/4 = 1:4

therefore since the ratio of all the sides are equal, the scale factor of figure A to fig B is 1:4

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The range of the values of x is 7<x<2.

In the given quadrilateral we can see that 2 of its sides are equal and the diagonal of the quadrilateral divides the quadrilateral into 2 triangles.

we can label the diagonal as "b" and the equal sides as "a".

when we apply the cosine law on the upper triangle, we get,

(2x-4)² = a² + b² - 2abcos45°

=(2x-4)² = a² + b² - 2ab×1/√2

=(2x-4)² = a² + b² - √2ab

Similarly, applying the cosine law on the lower triangle,

10² = a² + b² - 2abcos60°

= 10² = a² + b² - 2ab×1/2

= 10² = a² + b² - ab

we can clearly see from the above 2 equations formed by applying cosine law to the triangles that,

(2x-4)²<10²

2x-4<10

2x<14

x<7

and we know that 2x-4 has to be positive So,

2x-4>0

2x>4

x>2

combining these inequalities we get,

7<x<2

which gives the range, where x will belong

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Answer:

Step-by-step explanation:

The answer that thay already gave is backwards its actully 2<x<7 in acellus

Answer:

Step-by-step explanation:

Answer:
16.01 gallons of fuel.

Step-by-step explanation:

So we already know that there were initially 1.9 gallons. We have to add 14.11 gallons of fuel to the starting amount.

14.11 + 1.9 = 16.01 gallons

I hope this was able to help you :D

To find out how much fuel is in Leila's car now, we need to add the amount of fuel she just put in to the amount that was already in the car.

Leila put 14.11 gallons of fuel in the car and there were 1.9 gallons in the car to begin with.

So, 14.11 gallons + 1.9 gallons = 16.01 gallons

This is the correct answer because 14.11 gallons is the fuel Leila put into her car and 1.9 gallons is the fuel that was already in the car. Adding these two values together gives the total amount of fuel in the car now, which is 16.01 gallons.

The expression is equivalent to 2, and option C is correct.

To simplify simply means to make anything easier. In mathematics, simplifying an equation, fraction, or problem means taking it and making it simpler. Calculations and problem-solving techniques simplify the issue. By eliminating all common factors from the numerator and denominator and putting the fraction in its simplest/lowest form, we can simplify fractions.

Given expression

[tex](\frac{4^{5/4} 4^{1/4} }{4^{1/2} } )^{1/2}[/tex]

using properties,

aⁿ/aˣ = aⁿ⁻ˣ

aⁿaˣ = aⁿ⁺ˣ

(aⁿ)ˣ = aⁿˣ

so expression is

[tex]({4^{5/4} 4^{1/4} }{4^{-1/2} } )^{1/2}[/tex]

adding powers

5/4 + 1/4 - 1/2 = 6/4 -1/2

5/4 + 1/4 - 1/2 = 1

substitute the values,

expression is, √4 = 2

Hence option C is correct.

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Answer: -2

Step-by-step explanation:I'am in a rush right now sorry

Answer:2 and 3

Step-by-step explanation:

The population growth rate can be calculated using the equation x = x₀ (1+r)ⁿ. The population after 8 years will be 66 frogs.

The population growth is the increase in population after the specified number of years. Lets look into the data which is given.

The initial population is 6 tree frogs.

Population growth percent = 35%    

Rate of growth, r = 35/100 = .35

The population growth is calculated using the equation,  x = x₀ (1+r)ⁿ

     x₀ is  initial population

     r is the rate

    n is the number of years.

x = 6 (1+r)⁸ = 66.19

So the population of frog after 8 years will be 66.

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Answer:

14.87

Step-by-step explanation:

Pythagorean Thereom

c²= a²+b²

10²+11²=c²

100+121=c²

221=c²

c=14.866...

c=14.87

The  true statements about the unit prices :

spring water has a lower unit price of $0.11/ounce and fresh water has a lower unit price of $0.12/ounce

The correct answer: option (A) and option (C)

We use the unitary method in order to determine the unit price.

Here, 16-ounce bottle of spring water is $1.76. A 20-ounce bottle of fresh water is $2.40

Let us assume that the unit price of spring water be 'm' and the unit price of freshwater be 'n'.

By unitary method the unit price of spring water would be,

m = 1.76/16

m = 0.11

This means,  1 ounce bottle of spring water is $0.11

And the unit price of freshwater be:

n = 2.40 / 20

n = 0.12

i.e., the unit price of freshwater is $0.12

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The complete question is:

A 16-ounce bottle of spring water is $1.76. A 20-ounce bottle of fresh water is $2.40. which statement about the unit prices is true?

A)spring water has a lower unit price of $0.11/ounce

B) spring water has a lower unit price of $0.12/ounce

C)fresh water has a lower unit price of $0.12/ounce

D)fresh water has a lower unit price of $0.11/ounce

The length of median A is 11.7.

The point of intersection of a triangle's medians is its centroid (the lines joining each vertex with the midpoint of the opposite side).

If the triangle has its vertices as   (x_1, y_1), (x_2, y_2) , \: (x_3, y_3), then the coordinates of the centroid of that triangle is given by:

[tex](x,y) = \left( \dfrac{x_1 + x_2 + x_3}{3} + \dfrac{y_1 + y_2 + y_3}{3} \right)[/tex]

Given;

The sides of triangle

a = 4, b = 5 and c = 6

The median of triangle =½√(2b2+2c2-a2).

=(2*5*5+2*6*6-4*4)

=(50+72-16)

=[tex]\sqrt{138}[/tex]

=11.7

Therefore, the median of triangle will be 11.7

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If the coin is tossed four times , then the probability of getting exactly two heads and two tails is 3/8   .

the  number of outcomes in tossing a coin is = 2 =  {H , T} ;

the number of outcomes in tossing the coin four times is = 2⁴ = 16 ;

we have to find the probability of getting exactly 2 heads and 2 tails :

So , the required outcomes will be = {HHTT, HTHT, HTTH, THHT, THTH , TTHH}  ;

the number of favorable outcomes is = 6 ;

the probability is = 6/16 = 3/8 .

Therefore , the required probability is 3/8 .

The given question is incomplete , the complete question is

If a coin is tossed 4 times, what is the probability of getting exactly two heads and two tails ?

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Answer: its 15

Step-by-step explanation:

Answer:

The expression that represents the given situation is 6P - 16.

Step-by-step explanation:

The teacher got six boxes of pencils, and each box contains P pencils. Since the teacher used 16 pencils in the first month of school, the total number of pencils left after the first month can be represented by the expression 6P - 16.

This expression means that the teacher started with 6 boxes of pencils, each containing P pencils, for a total of 6P pencils. But since the teacher used 16 pencils in the first month, the total number of pencils left after the first month is 6P - 16.

The area of the irregular figure is 65.12 cm².

We know that a shape is said to be irregular if the shape does not fit into any of the known patterns of shape that we have. In this case, we have a kind of shape that is made up of the rectangle and the semi circle.

We know that we can be able to obtain that area of the rectangle by the use of the formula;

A = l * w

A = area

l = length

w = width

We then have;

A = 8 cm * 5 cm

= 40 cm^2

For the semi circle;

A =  (π r²)/2

A = (3.14 (4²))/2 = (3.14*16)/2 = 50.24/2 = 25.12cm²

Total area of the figure = 40 cm² + 25.12 cm² = 65.12 cm²

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The length of the line segment is given by the distance equation

D = 7.2 units

The distance of a line between 2 points is always positive and given by the formula

Let the first point be A ( x₁ , y₁ ) and the second point be B ( x₂ , y₂ )

The distance between A and B is D , and the distance D is

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

Given data ,

Let the distance of the line segment between two points be D

Now , the equation will be

Let the first point be represented as P ( 1 , 6 )

Let the second point be represented as Q ( 7 , 2 )

Now , distance between P and Q is D , and the distance D is

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

D = √ ( 1 - 7 )² + ( 6 - 2 )²

On simplifying the equation , we get

D = √ ( -6 )² + ( 4 )²

D = √ ( 36 + 16 )

D = √ 52

D = 7.2 units

Hence , the distance is 7.2 units

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The equation of the parabola with a vertex (2,3) and contains the point (0,0) is:  y = -3/4(x - 2)² + 3

Parabolic equitation or Quadratic equation with a vertex (2,3) and contains the point (0,0)  is written using the vertex form of the equation which is:

(x - h)² =  4P (y - k)

OR

standard vertex form, y = a(x - h)² + k     where a = 1/4p

The vertex

v (h, k) = (2, 3)

h = 2

k = 3

substitution of the values into the equation gives

y = a(x - 2)² + 3

considering the parabola passed the point (0, 0)

y = a(x - 2)² + 3

0 = a(0 - 2)² + 3

-3 = a(4)

a = -3/4

substituting for a

y = -3/4(x - 2)² + 3

hence the required equation is y = -3/4(x - 2)² + 3

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New weight = 225 x (1 - (16 / 100)) = 225 x 0.84 = 189 pounds.

So, after dieting and exercising, Jim now weighs 189 pounds.

Answer: 189 pounds

Step-by-step explanation:

If he lost 16% of his weight, Jim is now 84% of his original weight.

84% = 0.84

225 x 0.84

189 pounds

The solution of the system of linear equations is :-

x = 6 hours .

y = 330 miles.

Now, According to the question:

Let x be the number of hours that have elapsed since Zac started traveling .

Let y be the number of miles traveled .

We know that,

Speed = Distance / Time.

so,

Speed of Zac = D / T = 123.75 / 2.25 = 55 miles / hour.

Speed of Lynn = D / T = 105 / 1.75 = 60 miles / hour.

then, Distance travelled by Zac in x hours = S * T

y = 55 × x

y = 55x miles

and, Distance travelled by Lynn in (x - 0.5) hours = S * T

y = 60 × (x - 0.5)

y = 60x - 30

Therefore, a system of linear equations that represents the distance each of them has traveled since Zac left on his trip are :-

y = 55x ----------- Eqn.(1)

y = 60x - 30 ------------- Eqn.(2)

Now, putting value of Eqn.(1) in Eqn.(2) , we get,

55x = 60x - 30

60x - 55x = 30

5x = 30

x = 6 hours.

Putting value of x in Eqn.(1),

y = 55 * 6

y = 330 miles.

Hence, the solution of the system of linear equations is :-

x = 6 hours .

y = 330 miles.

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The probability of the car encountering it's first red light on the third day, i.e., P(Y = 3) will be 0.1479

According to the question,

we have to calculate the probability of the car encountering it's first red light on the third day, i.e., P(Y = 3)

Here,

Probability of encountering a Red light = 0.35, since traffic light at the intersection is red 35% of the time.

And Y is the number of days that pass up to and including the first time the car encounters a red light, i.e.,

The distribution here is a geometric one, where

P(Y = 3)

= P(Not red light in initial 2 lights) x P(red light in third light)

= [(1 - 0.35)² X 0.35]

= 0.1479

so, the probability of the car encountering it's first red light on the third day, i.e., P(Y = 3) will be 0.1479.

Note:

mean and variance of this question is not possible to find out.

the complete question ends till the probability to find the first red light on the third day.

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The sum of the given expressions -3x +  9  and 7x^2 - 2x + 1  is

= 7x^2 - 5x +10

What is Algebraic expression ?

Algebraic expression can be defined as the combination of variables and constants.

Given expressions are ,

-3x +  9  and 7x^2 - 2x + 1

So,

The sum of the expressions could be

= -3x+9 + 7x^2 - 2x + 1

= 7x^2 -3x-2x + 9+1

=7x^2 - 5x +9+ 1

= 7x^2 - 5x +10

Hence, The sum of the given expressions -3x +  9  and 7x^2 - 2x + 1  is

= 7x^2 - 5x +10

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