For the right triangles below, find the values of the side lengths c and h.

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:

Step-by-step explanation:

=====================================

Work Shown:

[tex](\text{x}-7)^2 = 36\\\\\sqrt{(\text{x}-7)^2} = \sqrt{36}\\\\\text{x}-7 = \pm\sqrt{36}\\\\\text{x}-7 = \pm6\\\\\text{x}-7 = 6 \ \text{ or } \ \text{x}-7 = -6\\\\\text{x} = 6+7 \ \text{ or } \ \text{x} = -6+7\\\\\text{x} = 13 \ \text{ or } \ \text{x} = 1\\\\[/tex]

-------------

Check:

Plug in x = 13

[tex](\text{x}-7)^2 = 36\\\\(13-7)^2 = 36\\\\(6)^2 = 36\\\\36 = 36 \ \ \checkmark\\\\[/tex]

This confirms x = 13

Now check x = 1

[tex](\text{x}-7)^2 = 36\\\\(1-7)^2 = 36\\\\(-6)^2 = 36\\\\36 = 36 \ \ \checkmark\\\\[/tex]

The value x = 1 is confirmed as well.

Both solutions are confirmed.

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

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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The story: John requires 2 liters of oil for a trip. they have  3/4 liters of oil at home, how many liters of oil will he get from his father to make the trip?

x in the equation is the liters of oil required from the father. It was solved to be 1 1/4

In the problem, the equation made available is 3/4 + x = 2

The solution of the equation is as follows

3/4 + x = 2

collecting like terms

x = 2 - 3/4

x = 5/4

x = 1 1/4

The value of x = 1 1/4

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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

Answer: its 15

Step-by-step explanation:

Answer:

Simple interest is calculated by multiplying the principal (the initial amount of money deposited), the interest rate, and the number of years the money is on deposit.

To find out how much interest Diane will be paid in the first 2 years, we need to calculate the total interest earned over that time period.

First, we find the interest rate by multiplying the annual interest rate of 5% by .01 (to convert the percentage to decimal form)

0.05 * .01 = 0.05

Next, we find the total interest earned by multiplying the principal (400), the interest rate (0.05), and the number of years (2)

400 x 0.05 x 2 = 40

So, in the first 2 years, Diane will be paid $40 in interest on her deposit of $400.

Answer:

in total, Diane will be paid $20 + $40 = $60 in interest over the first 2 years.

Step-by-step explanation:

Simple interest is calculated as:

I = Prt

where:

I = Interest

P = Principal (initial amount deposited)

r = Interest rate (expressed as a decimal)

t = Time (in years)

So in this case, with a principal of $400 and an interest rate of 5% (or 0.05 as a decimal), the interest paid in the first year would be:

I = $400 * 0.05 * 1 = $20

And in the second year, it would be:

I = $400 * 0.05 * 2 = $40

So in total, Diane will be paid $20 + $40 = $60 in interest over the first 2 years.

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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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  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

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 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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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.

[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]

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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To learn more about

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?

Answer:2 and 3

Step-by-step explanation:

Cam can run 9.3205 miles when she finishes.

mile, any of several distance measurements, such as the statute mile of 5,280 feet (1.609 km). It was derived from the Roman mille passus, or "thousand paces," which was equivalent to 5,000 Roman feet.

The term "mile" is derived from the Latin "mille passus," which means "one thousand paces," and a mile was 1,000 Roman strides, with a stride equaling two steps. In 1592, the English Parliament standardised the Mile measurement at eight furlongs (660 feet).

Cam signed up to run a 15-kilometre running race.

So, here, we will convert kilometers to miles-

I kilometer is =0.621371 miles

And, I mile =1.609344 kilometres.

Thus, converting kilometres to miles.

Simply multiply the Number of kilometres by 0.62137.

Therefore, 1 kilometre =0.621371 miles

1.5 kilometer=15*0.621371 miles =9.32057 miles

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The area of the rectangle is 20 square unit.

Let the length and width of the rectangle are respectively l and w.

Let another side of square is a.

Given that length of the rectangle is 5 more than the side of square.

l = a + 5

The width of the rectangle is 5 less than the side of the square.

w = a – 5

Given that the area of square is 45 square unit.

So, a² = 45

Now calculate the area of rectangle:

Area of rectangle = l * w

= (a + 5)(a - 5)

= a² - 25

= 45 – 25

= 20

Therefore, the area of rectangle is 20 square unit.

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

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Correct option is B, the given triangles are congruent by LL.

According to the leg-leg theorem (LL theorem), if the legs of two right triangles are the same length, then the complete triangle must be congruent.

Keep in mind that in this instance, the shorter (non-hypotenuse) sides of the triangle are referred to as the "legs." Right triangle sides all adhere to a particular pattern known as the Pythagorean theorem, which is why this theorem holds true.

According to the Pythagorean theorem, the length of the hypotenuse, squared, equals the length of each of the two legs, squared. Congruence states that two triangles must have equal hypotenuses and equal legs in order to be congruent.

Thus, the side-side-side (SSS) theorem, which was previously mentioned, is simply condensed into the leg-leg theorem.

In the given triangles,

the base of the triangles is same

the height of the triangles is common,

So, by Leg - Leg theorem, triangles are congruent.

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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: 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).

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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Amy will earn $26 in interest after 4 years.

Simple interest is a type of interest calculation where interest is only earned on the original principal (initial deposit) amount. It does not take into account any interest that has already been earned and is not compounded.

To calculate the simple interest earned on a deposit, you can use the formula: I = P * r * t, where I is the interest earned, P is the principal (initial deposit), r is the annual interest rate as a decimal, and t is the time in years.

In this case:

I = (100)(0.065)(4)

I = 26

The interest is: $26.

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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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