Show the optimal binary search tree for the following words, where the frequency
of occurrence is in parentheses: a (0.18), and (0.19), I (0.23), it (0.21), or (0.19).

Answer:

30

Step-by-step explanation:

get the x by itself on one side to solve

x-6/2=10

multiply both sides by 2 and divide by 6 which cancels out s x is by itself

x=10*6/2

simplify

x=60/2

x=30

Answer:

x = 13

Step-by-step explanation:

Given equation,

→ x - (6/2) = 10

Now the value of x will be,

→ x - (6/2) = 10

→ x = 10 + (6/2)

→ x = 10 + 3

→ [ x = 13 ]

Hence, the value of x is 13.

The number that twice the sum of it and negative seven is eight is 11.

Twice the sum of the number and negative seven is eight. The number can be found as follows:

let

the number  = x

According to the word expression, twice the value of  the addition of x and minus seven is equals to 8. Hence, we will find the value of x in the equation formed from the words expression.

The variable x is the required number.

Therefore,

2(x + (-7)) = 8

2(x - 7) = 8

open the brackets

2(x - 7) = 8

2x  - 14 = 8

add 14 to both sides of the equation

2x  - 14 = 8

2x  - 14 + 14 = 8 + 14

2x = 22

divide both sides by 2

x = 22 / 2

x = 11

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The required given function is a translation of the parent function by 8 units left and  3 units up.

Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.

Here,
Given function, g(x) = (x − 8)³+ 3

Parent function, f(x) = x³

From observation, it is seen that,
The given function is translated function of the parent function as,
As the parent function is translated left on the x-axis by 8 units, and 3 units up on the y-axis.

Thus, the required given function is a translation of the parent function by 8 units left and  3 units up.

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On solving the question, we that when focus is (0,4) and a directrix y is -4 coordinate is (0,0).

Parabola is the name of the graph of the quadratic function. Another way to think of this is the collection of all points whose distance from the focal point is equal to the distance from the line.

Focus (0,4) ,

the directrix y is (-4)

The vertex (h ,k) - halfway between the,

directrix and focus.

y = [y coordinate of focus + directrix] / 2.

The x coordinate will be the same as the x coordinate of the focus.

[tex]\frac{(0, 4-4)}{2} \\(0,0)[/tex]

Focus of (3,0),

a directrix of x = -3

The vertex (h , k) is halfway

between the directrix

and focus.

x = [x coordinate of focus + directrix] / 2.

The x coordinate will be the same as the y coordinate of the focus.

[tex]\frac{3-3,0}{3} \\(0,0)[/tex]

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Answer:

16.13

Step-by-step explanation:

multiply 10.75 and 1.5 which equals 16.125 so you would round up to 16.13 so your final answer would be $16.13 per hour

We can add 10 pounds of mixed nuts that contain 20% peanuts by solving equations.

Algebraically speaking, an equation is a statement that shows the equality of two mathematical expressions.

For instance, the two equations 3x + 5 and 14, which are separated by the 'equal' sign, make up the equation 3x + 5 = 14.

So, the quantity of mixed nuts we need is x pounds.

There are 0.70 pounds of peanuts in this mixture.

3 pounds of peanuts make up the 20% mix, or 0.20*15.

We got the equation:

0.70 x + 3 = 0.40(x + 15)

Now, solve this equation as follows:

0.70 x + 3 = 0.40(x + 15)

0.70 x + 3 = 0.40(x + 15)

0.70x + 3 = 0.40x + 6

0.70x - 0.40x = 6 - 3

0.30x = 3

x = 10 pounds

Therefore, we can add 10 pounds of mixed nuts that contain 20% peanuts by solving equations.

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The equation that represents the total cost you pay for cable television after x months is y = 35 + 29.95x.

What is an equation?

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

Given that a cable operator offers that buy a television at $29.95 per month plus a one-time installation fee.

The total cost that need to pay after 6 months is  $214.70.

Assume that a be the cost one-time installation fee.

Then total cost that need to pay after 6 months is  (6 × 29.95) + a = 179.7 + a

Therefore,

179.7 + a = $214.70

a = 35.

After x month, he needs to pay 35 + 29.95x.

Assume that y represents total cost after x months.

y = 35 + 29.95x

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Answer:

B

Step-by-step explanation:

x -5 ≥ 0

x ≥ 5

the function exists only in the interval [5, +oo) and it keeps growing

785.4 cubic feet

Step 1: Start by calculating the area of the base of the storage tank. The area of a circle is calculated by: A = π*r², where r is the radius of the circle.

Step 2: For Emma’s cylindrical storage tank, the radius of the base is 5 feet. Substitute this value in the formula and calculate the area of the base, which is equal to: A = π*5² = 78.54 square feet.

Step 3: Now we need to calculate the volume of the cylindrical storage tank. The volume of a cylinder is calculated by: V = π*r²*h, where r is the radius of the circle and h is the height of the cylinder. For Emma’s tank, the height is 10 feet. Substitute these values in the formula and calculate the volume, which is equal to: V = π*5²*10 = 785.4 cubic feet.

Step 4: Therefore, Emma’s cylindrical storage tank has a capacity of 785.4 cubic feet.

If you have any additional questions or need further assistance, please let me know.

The measures are  NE = 7.5 units, GO = 9 units, AG = 12 units, and the perimeter of the ΔOAG is 36 units.

What is the perimeter of the triangle?

A triangle's perimeter is measured in the same unit as its side lengths. If the lengths of its sides are measured in a different unit, first convert them to that unit.

We have,

OA = (4x + 3)  .............(1)

NE = (2.5x)     .............(2)

RN = 4.5         .............(3)

ER = (2x)         .............(4)

Mid-point segment of the triangle

EN = 1/2(OA)         .............(5)

from equations (1), (2), and (5)

2.5x = 1/2(4x + 3)

5x = 4x + 3

x = 3           .............(6)

OA = 4x + 3 = 15

NE = (2.5x) = 7.5

ER = (2x) = 6

RN = 1/2(GO)

GO = 2 * RN = 2 * (4.5)

GO = 9

ER = 2x = 2*3 = 6

AG = 2(ER) = 2 * 6 = 12

The perimeter of the OAG is OA + AG + GO = 15 + 12 + 9

= 36 units.

Hence, the asked measures are  NE = 7.5 units, GO = 9 units, AG = 12 units, and the perimeter of the ΔOAG is 36 units.

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Answer: centimeters 3

The length of the major arc AB is 15.44 units

Given:  line x + 2y = −5 and circle of radius 5 centered at the origin

The equation of a circle of radius r whose center is the origin O (0, 0) is given by x²+y² = r². Thus:

x²+y² = 5²

x²+y² = 25

To find the points where the line intersects the circle, we can substitute the equation of the circle into the equation of the line.

Substituting this into the equation of the line, we have;

x + 2y = -5

x²+y² = 25

Solving for x, we have

x = (-5 ±√(33))/2

Since the circle is centered at the origin, both of these points are on the circle. Thus, the coordinates of the points A and B are

A: (-5 +√(33))/2, (-5 - √(33))/4

B: (-5 - √(33))/2, (5 - √(33))/4

To find the length of the major arc AB, we can use the formula

length = θ/360 × 2πr

where r is the radius of the circle, and θ is the central angle formed by points A and B. The central angle can be found using the formula

θ= 180 - 2cos⁻¹((AB² + r² - AC²)/(2 × AB × r))

where AB is the distance between the points A and B, and AC is the distance between point A and the center of the circle (which is the origin)

We can use the Pythagorean Theorem to find AB and AC:

AB = √(((-5 + √(33))/2 - (-5 - √(33))/2)² + ((-5 - √(33))/4 - (5 - √(33))/4)²)

     = √(66)

AC = √(((-5 + √(33))/2)² + ((-5 - √(33))/4)²)

    = √(25 + 33)/2

    = √(58)/2

Substituting these values into the formula for θ, we have

θ = 180 - 2cos⁻¹((√(66)² + 5² - (√(58)/2)²)/(2 × √(66) ×5))

  = 180 - 2cos⁻¹((66 + 25 - 29)/(20))

  = 180 -2cos⁻¹(62/20)

  = 180 - 2cos⁻¹(3.1)

  = 180 - 2 × 1.43

  = 180 - 2.86

  = 177.14

Finally, substituting this value into the formula for the length of the major arc, we have

length = 2 × π × 5 × (177.14/360)

           = 2  × π  5 ×  (0.4925)

           = 2 ×  3.14 ×  5 ×  (0.4925)

           = 6.28 × 5 ×  (0.4925)

           = 31.4 ×  (0.4925)

           = 15.44 units

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Answer: Sonya's heart rate increased by 75% after running a mile on the treadmill.

Step-by-step explanation:

To find the percent increase in Sonya's heart rate, we can use the following formula:

Percent increase = (New value - Old value) / Old value * 100

Substituting the given values into the formula, we get:

Percent increase = (133 beats per minute - 76 beats per minute) / 76 beats per minute * 100

Calculating the result, we get:

Percent increase = 57 beats per minute / 76 beats per minute * 100

= 0.75 * 100

= 75%

Therefore, Sonya's heart rate increased by 75% after running a mile on the treadmill.

The mid point of the line segment is (1,9)

The general formula to calculate the midpoint is written as

(x, y) = (x1+x2/2) , (y1 + y2)/2

Where (x1, y1) and (x2, y2) refers the end points.

Here we have given that line segment with endpoints a (-3, 7) and b (1, 11).

And we need to find the midpoint.

While we looking into the given question we have the end point of the line segment

Then the values of (x1, y1) is (-3, 7) and the value of (x2, y2) is (1, 11),

Then the midpoint is calculated by apply the values on the formula, then we get,

=> (1 - 3)/2 , (11 + 7)/2

When we do the appropriate mathematical operation, then we get

=> 2/2 , 18/2

Therefore, the midpoint is (1, 9).

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The probability of drawing purple and green marbles will be 1/80.

The probability of an event occurring is defined by probability.

Probability is also called chance because if you flip a coin then the probability of coming head and tail is nothing but chances that either head will appear or not.

As per the given,

Number of purple marble = 5

Number of green marble = 1

Total marble = 5 + 6 + 8 + 1 = 20

Probability drawing a purple marble = 5/20

Probability of drawing a green marble = 1/20 (because the number of marble remain the same)

Probability of drawing a purple and green = 5/20 x 1/20 = 1/80

Hence "There is a 1/80 chance of drawing purple and green marbles.".

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Answer:

200 b/c she will use 50cm² by Wall :)

7*2*4 = 56.

7 is the length, 2 is the height, 4 is the widght

Equation for volume is l*h*w

When the number of machines increased by factor 2 the order would have been finished in 11 hours,

Given that,
All the fudge machines at a chocolate factory work at the same rate. Six machines working simultaneously can complete a big order in 22 hours.

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

here,
Number of hours as of the current number of working machines = 22
The number of machinery now doubled so working hours = 22 / 2 = 11 hours.

Thus, When the number of machines increased by factor 2 the order would have been finished in 11 hours,

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Step-by-step explanation:

[tex]2x + 3y = 5[/tex]

[tex]2x = 5 - 3y[/tex]

[tex]x = \frac{5 - 3y}{2} \\ [/tex]

Answer: Add 3y to both sides of the equation. 2x=5+3y Divide each term in 2x=5+3y by 2 and simplify.

By using compound interest, it can be calculated that-

A) Deposit is $45544.69

B) Interest = $84455.31

What is compound interest?

If the interest on a certain principal at a certain rate over a certain period of time increases exponentially rather than linearly, the interest earned is called compound interest.

If p is the principal, r is the rate and n is the time in years, then amount is calculated by

[tex]A = p(1 + \frac{r}{100})^n[/tex]

The difference between the amount and the principal is the compound interest

A)Here,

Let the principal be $p

Rate = 6%

Time = 18 years

Amount = $130000

[tex]130000 = p(1 + \frac{6}{100})^{18}[/tex]

[tex]130000 = p(1 + \frac{3}{50})^{18}[/tex]

[tex]130000 = p(\frac{53}{50})^{18}[/tex]

[tex]130000 = p(1.06)^{18}[/tex]

[tex]p = \frac{130000}{1.06^{18}}[/tex]

p = $45544.69

B) Interest = $(130000 - 45544.69)

                  = $84455.31

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Answer:

  C.  4² +8² ≠ 10²

Step-by-step explanation:

You want to know why it is not possible for the lengths 4, 8, and 10 to form a right triangle.

The side lengths of a right triangle must satisfy the Pythagorean relation:

  a² +b² = c²

These numbers do not:

  4² +8² = 16 +64 = 80 ≠ 10² = 100

A right triangle is not formed because ...

  4² +8² ≠ 10²

__

Additional comment

The smallest Pythagorean Triple is {3, 4, 5}. Doubling these numbers gives {6, 8, 10}. If the two longest sides are 8 and 10, the shortest must be 6 for a right triangle to be formed.

When the side lengths are reduced to a mutually prime set of numbers, their sum must be even.

   4:8:10 reduces to 2:4:5 which has an odd sum.

The Cramer rule can be applied to solve the system of equations and the value of x₂ is x₂ = 12.

For a given system of equations ax + by = c ; px + qy =r, if the determinant of the coefficient matrix [tex]A = \begin{bmatrix}a&b\\p&q\end{bmatrix}[/tex] is non-zero, then as per the Cramer rule, the solution to the system of equations is:

[tex]x=\frac{ \begin{vmatrix}c&b\\r&q\end{vmatrix}}{\begin{vmatrix}a&b\\p&q\end{vmatrix}}\quad\text{and}\quad y=\frac{ \begin{vmatrix}a&c\\p&r\end{vmatrix}}{\begin{vmatrix}a&b\\p&q\end{vmatrix}}[/tex]

The given system of equations is:

-2x₁ - x₂ = 2

-3x₁ - 2x₂ = -3

The coefficient matrix of the system of equations is:

[tex]A = \begin{bmatrix}-2&-1\\-3&-2\end{bmatrix}[/tex]

Now, find the determinant of the coefficient matrix:
[tex]A = \begin{vmatrix}-2&-1\\-3&-2\end{vmatrix}\\A = 4-3\\A =1[/tex]

Since the determinant of the coefficient matrix is non-zero, the Cramer rule can be applied.

According to the Cramer rule, it follows:

[tex]x_2=\frac{\begin{vmatrix}-2&2\\-3&-3\end{vmatrix}}{\begin{vmatrix}-2&-1\\-3&-2\end{vmatrix}}\\\\x_2=\frac{6-(-6)}{1}\\\\x_{2}=12[/tex]

Hence, the Cramer rule can be applied to solve the system of equations and the value of x₂ is x₂ = 12.

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The density of the drink is 1.04 grams per cm³

Given:

density of grape juice = 1.07 grams per cm³, density of fruit syrup = 1.6 grams per cm³ and density of carbonated water = 0.99 grams per cm³

volume of grape juice = 35 cm³, volume of fruit syrup = 25 cm³ and volume of carbonated water = 310 cm³

First, find the mass of grape juice, fruit syrup and carbonated water using their volume and density:

density = mass / volume

=> mass = density × volume

mass of grape juice = 1.07 × 35 = 37.45 grams

mass of fruit syrup = 1.6 × 25 = 40 grams

mass of carbonated water = 0.99 × 310 = 306.9 grams

Total mass = 37.45 + 40 + 306.9  = 384.35 grams

Total volume = 35 + 25 + 310 = 370 cm³

density of the drink = Total mass / Total volume

density of the drink = 384.35 / 370 = 1.04 grams per cm³

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The monomial that completes the expression is x²

From the question, we have the following parameters that can be used in our computation:

36 - 12x + ...

Complete the expression with a variable

So, we have

36 - 12x + y

Rewrite as

y - 12x + 36

The given terms of the expression are factors of 6

By trial and error, we make use the following representation

y - 12x + 36 = (x - 6)²

Expand the expression

y - 12x + 36 = x² - 12x + 16

Evaluate the like terms

y = x²

Hence, the monomial is x²

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The equation in the standard form of a parabola is y = -3x² - 7x - 2

What is the standard form of a parabola?

Another equation that, when graphed, produces a parabola is one written in standard form. Like the letters in the vertex form equation, each letter in the standard form equation provides a nugget of knowledge about the parabola.

Here, we have

The standard equation of a parabola is

y = ax² + bx + c

Substitute the given values in this equation

When x = - 3 and y = -8

-8 = a (-3)² + b (-3) + c

-8 = 9a - 3b + c …. (1)

When x = 0 and y = - 2

-2 = a (0)² + b (0) + c

c = -2....(2)

When x = 2 and y = -28

-28 = a (2)² + b (2) + c

-28 = 4a + 2b + c …. (3)

Equations (1) and (3) can be written as

9a - 3b + c = -8 …. (4)

4a + 2b + c = -28 …. (5)

Multiply equation (4) by 2 and equation (5) by 3 to eliminate 'b'

18a - 6b + 2c = -16

12a + 6b + 3c = -84

Subtracting these equations

30a + 5c = -100

Substitute the value of 'c' here

30a + 5(-2) = -100

30a = -100 + 10

30a = -90

a = -3

Substitute the value of a and c in equation (1)

-8 = 9a - 3b + c

-8 = -27 - 3b -2

-6 = -27 - 3b

3b = -21

b = -7

Here the value of a = -3, b = -7, and c = -2.

Hence, the equation in the standard form of a parabola is y = -3x² - 7x - 2

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Using simple mathematical operations, we know that a-b can be -1, -2, and 0.

A mathematical "operation" is the process of calculating a value utilizing operands and a math operator.

The supplied operands or integers must adhere to a set of predefined rules that are connected to the symbol of the math operator.

Parentheses, Exponents, Multiplication, Division, and Addition Subtraction are also known as PEMDAS (from left to right).

So, we have: 2 < a < 5, 3 < b < 7

Now, a - b will be:

the values of a can be 3 and 4

Similarly, the value of b can be 4 and 5.

So there are 4 possibilities as shown:

3 - 4 = -1

3 - 5 = -2

4 - 4 = 0

4 - 5 = -1

Therefore, using simple mathematical operations, we know that a-b can be -1, -2, and 0.

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The given expression in simplified radical form is  2x⁴ y³  [tex]\sqrt[4]{2y^3}[/tex]

Define Radicals.

The term "radical" is used to represent the square root or nth roots.

Expression with a square root is referred to as a radical expression.

Radicand: The value or phrase contained within the radical symbol.

Given expression is,

[tex]\sqrt[4]{32x^1^6 y^1^5}[/tex]

Rewrite this expression,

we know, 2⁵ = 32

[tex]\sqrt[4]{(2x^4.y^3)^4 .y^3 .2}[/tex]

You see this expression is equal to given expression i.e, [tex]\sqrt[4]{32x^1^6 y^1^5}[/tex]

Take out the term from under the radical,

2x⁴ y³  [tex]\sqrt[4]{2y^3}[/tex]

Hence, the given expression in simplified radical form is  2x⁴ y³  [tex]\sqrt[4]{2y^3}[/tex]

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