Geometry triangle congruence and proof problem

Answer:

1/4

Step-by-step explanation:

First, reduce the numbers with the greatest common factor of 2.

Then, you get your answer of 1/4.

Answer:

140J

Step-by-step explanation:

work done = Mass x acceleration x distance

mass = 7kg

acceleration = 10m/s^2

distance= 2m

work = 7 x 10 x 2

= 140j

Answer:

The answer is 140J or 140Nm

Step-by-step explanation:

Work done=mgh

W=7×10×2

W=140J or 140Nm

Answer:

67!

Step-by-step explanation:

A way to look at it is, since the number is 387, it can be broken into 6 parts. 50x6 is about 300. That leaves 87 to be left over. So it's safe to assume each number will be around 50-60, so I started to add up six consecutive numbers together until I reached my final set of numbers to reach 387.

My numbers were 62+63+64+65+66+67=387

I hope this makes sense!

The Intermediate Value Theorem concludes that for a continuous function on a closed interval, if there are two points in the interval such that the function takes on two different values, then there must be at least one point in the interval where the function takes on every value between those two values.

Assuming its assumptions are met, the Intermediate Value Theorem (IVT) concludes that:

If a continuous function, f(x), is defined on a closed interval [a, b], and k is any value between f(a) and f(b), then there exists at least one value c in the interval (a, b) such that f(c) = k.

In other words, if a function is continuous on a closed interval and k is a value between the function's values at the endpoints of the interval, the function must take on the value k at least once within that interval. This theorem is particularly useful in determining the existence of roots or zeroes for a function in a specified interval.

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The distance between successive fourth roots of the complex number √3/2 - 1/2i on the unit circle is 5π/24 units.

To find the distance between successive fourth roots of a complex number on the unit circle, we can use the concept of the angle between the roots. Let's proceed step by step:

The given complex number is √3/2 - 1/2i. This complex number lies on the unit circle because its magnitude is equal to 1.

1. Convert the given complex number to trigonometric form:

  √3/2 - 1/2i = cos(θ) + i*sin(θ)

  By comparing the real and imaginary parts, we can determine the angle θ:

  cos(θ) = √3/2

  sin(θ) = -1/2

  Using the unit circle, we can find that θ = 5π/6 (or 150 degrees). This angle represents the position of the given complex number on the unit circle.

2. Find the angle between successive fourth roots:

  Since we are interested in the fourth roots, we divide the angle θ by 4:

  θ/4 = (5π/6) / 4 = 5π/24

  This angle represents the angular distance between two successive fourth roots on the unit circle.

3. Calculate the distance between the two points:

  To find the distance, we multiply the angular distance by the radius of the unit circle (which is 1):

  Distance = (5π/24) * 1 = 5π/24

  Therefore, the distance between successive fourth roots of the complex number √3/2 - 1/2i on the unit circle is 5π/24 units.

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The value of the transformed function is h ( x ) = f ( x - 6 )² , where the value of c is -6

The transformation of a function may involve any change.

Usually, these can be shifted horizontally (by transforming inputs) or vertically (by transforming output), stretched (multiplying outputs or inputs), etc.

If the original function is y = f(x), assuming the horizontal axis is the input axis and the vertical is for outputs, then:

Horizontal shift (also called phase shift):

Left shift by c units: y=f(x+c) (same output, but c units earlier)

Right shift by c units: y=f(x-c)(same output, but c units late)

Vertical shift:

Up by d units: y = f(x) + d

Down by d units: y = f(x) - d

Stretching:

Vertical stretch by a factor k: y = k × f(x)

Horizontal stretch by a factor k: y = f(x/k)

Given data ,

Let the parent function be represented as f ( x )

Now , the value of f ( x ) = x²

And , when the function is shifted horizontally by a factor of -6 , we get

Horizontal shift (also called phase shift):

Left shift by c units: y=f(x+c)

Substituting the values in the equation , we get

h ( x ) = f ( x + c )

when c = -6

h ( x ) = f ( x - 6 )²

Hence , the transformed function is h ( x ) = f ( x - 6 )²

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Therefore, when the number of copies made in a month is above 4000, company A's charges are higher than company B's charges in the given equation.

Let's start by setting up an equation to represent the cost of renting a copier from each company:

Cost for company A = 0.10x + 200

Cost for company B = 0.05x + 400

where x is the number of copies made in a month.

To find the number of copies above which company A's charges are higher than company B's charges, we need to set the two equations equal to each other and solve for x:

0.10x + 200 = 0.05x + 400

0.05x = 200

x = 4000

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The probability that x is greater than 77. 5 is 0.039828

Given,

Mean, μ = 80

Standard deviation, σ = 25

We have to find z score corresponding to 77.5

z = (x-μ) / σ

   \(=\frac{77.5-80}{25} \\= 0.1\)

Now we have to find P value from Z table:

P(x<77.5) = 0.46017

P(x>77.5) = 1 - P(x<77.5) = 0.53983

P(77.5<x<80) = 0.5 - P(x<77.5) = 0.039828

The probability of x greater than 77.5 is 0.039828

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The value of sinx is 1 - 3cosx

What is Trigonometry?

Trigonometry is a field of mathematics that explores the connections between triangle side lengths and angles. The topic arose in the Hellenistic civilization during the third century BC from geometric applications to astronomical research. Today, four varieties of trigonometry are used: core, plane, spherical, and analytic. The ratio of the sides and angles of a right triangle is the subject of core trigonometry.

Solution:

1/cosx - tanx = 3

we can write tanx as sinx/cosx

this is because we know that sinx = Perpendicular / Hypotenuse

and cosx = Base / Hypotenuse

so if we divide sinx by cosx we will find that it is equal to Perpendicular / Base which is nothing but tanx

1-sinx = 3cosx

sinx = 1 - 3cosx

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Answer:

Step-by-step explanation:

Area = length x width Area = 12 feet x 14 feet Area = 168 square feet So the area of Jacinda's bedroom is 168 square feet.

Answer:

168 ft²

Step-by-step explanation:

area of rectangle = length × width

A = LW

A = 12 ft × 14 ft

A = 168 ft²

Answer:

x=60

Step-by-step explanation:

The angles here are alternate interior angles, this means that they are congruent. Congruent means that they are equal to each other, so to solve just set them equal! (Your variable is x so you're solving for that)

\(2x=120\\x=60\)

Hope this helps! :)

The SSS Similarity Theorem states that if three sides of one triangle are equal to three sides of another triangle, then the triangles are similar. This theorem is also known as the Side-Side-Side Similarity Theorem or the SSS Similarity Postulate.

What is the similarity theorem of triangles?

Triangles with the same shape but different sizes are said to be similar triangles. Squares with any side length and all equilateral triangles are examples of related objects. In other words, if two triangles are similar, their corresponding sides are proportionately equal and their corresponding angles are congruent. Triangle similarity is indicated here by the symbol (~).

SSS or Side-Side-Side Similarity.
If all three sides of a triangle are in proportion to the three sides of another triangle, then the two triangles are similar.
Thus, if AB/XY = BC/YZ = AC/XZ then ΔABC ~ΔXYZ.
From this result, we can conclude that-
∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z
Hence, we have described the SSS similarity theorem.

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If the credit card did not have a promotional offer, Lori would have had to pay approximately $110.58 more in interest.

a. To determine how long it will take Lori to pay off the balance, we need to calculate the number of months it will take for the total balance to reach zero.

Let's assume it takes n months for Lori to pay off the balance. Each month, Lori pays $75 towards the balance. Since there is no interest during the promotional period, the balance decreases by $75 each month.

The initial balance is $562, so the remaining balance after n months can be represented as:

Remaining balance = Initial balance - (Monthly payment * Number of months)

0 = 562 - (75 * n)

Solving this equation for n:

75n = 562

n = 562 / 75

n ≈ 7.49

Therefore, it will take Lori approximately 7.49 months (or rounded up to 8 months) to pay off the balance.

b. To calculate the total interest paid, we need to subtract the initial balance from the total amount paid over the repayment period. The total amount paid is the monthly payment multiplied by the number of months.

Total interest paid = Total amount paid - Initial balance

Total amount paid = Monthly payment * Number of months

Total interest paid = (Monthly payment * Number of months) - Initial balance

Total interest paid = (75 * 8) - 562

Total interest paid = 600 - 562

Total interest paid = $38

Therefore, Lori will pay a total of $38 in interest.

c. If the credit card did not have a promotional offer, the interest would have been charged at a rate of 19.7% compounded daily after the promotional period.

To calculate the additional interest, we can use the formula for compound interest:

Additional interest = Initial balance * (1 + interest rate)^Number of months - Initial balance

Additional interest = 562 * (1 + 0.197/365)^(30 * 4) - 562

Calculating this value:

Additional interest ≈ $110.58

Therefore, if the credit card did not have a promotional offer, Lori would have had to pay approximately $110.58 more in interest.

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The slope of the tangent line will be 3/4. Then the correct option is C.

Given that:

The center of the circle is (4, 6).

Let r be the radius of the circle and the location of the center of the circle be (h, k). Then the equation of the circle is given as,

(x - h)² + (y - k)² = r²

Then the equation of the line is given as,

(x - 4)² + (y - 6)² = r²

The slope of the tangent line at (10, -2) is calculated as,

2(x - 4) + 2(y - 6)y' = 0

y' = -(x - 4) / (y - 6)

m = -(10 - 4) / (- 2 - 6)

m = 6 / 8

m = 3 / 4

The slope of the tangent line will be 3/4. Then the correct option is C.

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Answer:

3 units  

Step-by-step explanation:

The computation of the length of DC is shown below:

As we can see from the attached figure that the DE is parallel to BC with the help of  basic proportionality theorem

Now the equation is

\(\frac{AE}{EB} = \frac{AD}{DC}\)

\(\frac{12}{4} = \frac{9}{DC}\)

DC × 12 = 9 × 4

DC ×  12 = 36

So, DC

= 36 ÷ 12

DC = 3 units  

Hence, the correct option is b. which determines the length of DC

Answer:

The graph is attached below.

Step-by-step explanation:

The volume of the box containing the coffee mugs is,

\(V=x^{3}\)

Then the function representing the side length, in inches, for the box is:

\(g(x)=x\)

Now, it is provided that the company decides to double the volume of the box.

That is, the new volume will be:

\(V_{n}=2x^{3}\)

Then the side length, in inches, for the box will be:

\(g_{n}(x)=\sqrt[3]{2x^{3}} =\sqrt[3]{2}x\)

Then the graph representing the function, formed using the following points is:

\(x\ \ \ \ \ \ \ \ \ g_{n}(x)\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\0\ \ \ \ \ \ \ \ \ \ \ 0\\1\ \ \ \ \ \ \ \ \ \ \ 2^{1/3}\)

Answer:

c

Step-by-step explanation:

The study would use a simple random sampling because it would be easy to randomly select. The study would use cluster sampling because the of fall into naturally occurring subgroups. The study would use stratified sampling because it would be important to have members from each segment of the population.

The study is a sample survey, because the population is for it to be practical to record all of the responses.

a. The study would use a simple random sampling because it would be easy to randomly select members of the population. Simple random sampling is a type of probability sampling that is used when the population is homogenous and every member has an equal chance of being selected for the sample.

b. The study would use cluster sampling because the members of the population fall into naturally occurring subgroups. Cluster sampling is a type of probability sampling that is used when the population is heterogeneous and can be divided into naturally occurring subgroups.

c. The study would use stratified sampling because it would be important to have members from each segment of the population. Stratified sampling is a type of probability sampling that is used when the population is heterogeneous and can be divided into segments or strata based on certain characteristics.

d. The study is a sample survey, because the population is too large for it to be practical to record all of the responses. Sample survey is a type of survey that collects data from a sample of the population, rather than the entire population. This is often done when the population is too large or when it is not practical to survey the entire population.

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y = 0.6x + 3

To determine the equation of the line, we need to determine the slope and the y-intercept. Then, we will substitute the slope and the y-intercept in the slope intercept form equation to obtain the equation in slope intercept form.

Two points that lie on line: (0, 3) and (-5, 0)

\(\implies \text{Slope} = \dfrac{\text{Rise}}{\text{Run}} = \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}\)

\(\implies \text{Slope} = \dfrac{3 - 0 }{0- (-5)}\\\)

\(\implies \text{Slope} = \dfrac{3 }{0+5}\\\)

\(\implies \text{Slope} = \dfrac{3 }{5} = 0.6\)

\(\implies \text{Intersection of line on y-axis: 3}\)

\(\implies \text{y-intercept: 3}\)

Slope intercept form:

[Where "m" is the slope and "b" is the y-intercept]

Substitute the slope (m) and the y-intercept (b) in the formula;

\(\implies \text{Slope intercept form: y = (m)x + (b)}\)

\(\implies \text{Equation of line in slope intercept form: y = (0.6)x + (3)}\)

Open the parenthesis in the equation;

\(\implies\text{Equation of line in slope intercept form: \boxed{\text{y = 0.6x + 3}}}\)

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She used 29.8 feet of fencing for the fence -> The circumference of the circular garden is 29.8ft

Formula for circumference : 2πr

So, the radius is equal to : 29.8 : 2 : π = 4.74281730414... (round to 4.7)

Recheck : 4.7 x 2π = 29.5309709437... (close to 29.8)

We can use the position equation s = 1/2at^2 + v0t + s0 to find the position equation for the object.  This equation relates the object's position s at time t to its initial position s0, initial velocity v0, acceleration a, and time t.

To find the equation, we need to solve for a, v0, and s0 using the given information. We can start by using the equation with t=1, t=2, and t=3 to create a system of equations:

s1 = 1/2a(1^2) + v0(1) + s0

s2 = 1/2a(2^2) + v0(2) + s0

s3 = 1/2a(3^2) + v0(3) + s0

Plugging in the given values for s1, s2, and s3, we get:

152 = 1/2a + v0 + s0 (Equation 1)

120 = 2a + 2v0 + s0 (Equation 2)

56 = 9/2a + 3v0 + s0 (Equation 3)

Next, we can solve this system of equations for a, v0, and s0. One way to do this is to use elimination to solve for one variable at a time. Here, we'll solve for s0 first:

From Equation 1, we can solve for s0:

s0 = 152 - 1/2a - v0

We can then substitute this expression for s0 into Equations 2 and 3:

120 = 2a + 2v0 + (152 - 1/2a - v0)

56 = 9/2a + 3v0 + (152 - 1/2a - v0)

Simplifying these equations, we get:

-1/2a + v0 = -44 (Equation 4)

-5/2a + 2v0 = -96 (Equation 5)

Now we can solve for v0 by eliminating a from Equations 4 and 5:

-5(1/2a + v0) + 2(-1/2a + v0) = -5(-44) + 2(-96)

-5a + 14v0 = -332

Solving for v0, we get:

v0 = (-332 + 5a)/14

Substituting this expression for v0 into Equation 4, we get:

-1/2a + (-332 + 5a)/14 = -44

-7a/28 = -44 + 332/14

-7a/28 = -10

Solving for a, we get:

a = 40 ft/s^2

Finally, we can substitute the values of a and v0 into Equation 1 to solve for s0:

152 = 1/2(40)(1^2) + v0(1) + s0

152 = 20 + (-332 + 5(40))/14 + s0

152 = 20 - 18 + s0

s0 = 150 ft

Therefore, the position equation for the object is:

s = 1/2(40)t^2 + (-332 + 5(40))/14t + 150

= 20t^2/1 - 24t/7 + 150

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the number is 40.

In mathematics, the system of linear equations is the set of two or more linear equations involving the same variables. Here, linear equations can be defined as the equations of the first order, i.e., the highest power of the variable is 1. Linear equations can have one variable, two variables, or three variables. Thus, we can write linear equations with n number of variables. In this article, you will get the definition of the system of linear equations, different methods of solving these systems of linear equations and solved examples.

If the tens digit of the number is X and the ones digit is Y then the number is equal to 10*X + Y

If you reverse the digits then the number is equal to 10*Y + X

If reversing the digits decreases the number by 36 then

10*Y + X = 10*X + Y - 36

The sum of the digits is 4 so X + Y = 4  ==>  X = 4 - Y

Substitute this value of X into the 1st equation

10*Y + (4 - Y) = 10*(4 - Y) + Y - 36

9*Y + 4 = 40 - 9*Y - 36

18*Y = 0

Y = 0

So X = 4 - 0 = 4 and the number is 40

Check:  40 - 04 = 36

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The statement that shows a discrepancy between the check register and bank statement is: C. Gavin: The balance in his check register is $500 and the balance in his bank statement is $510.

The check register shows a balance of $500, while the bank statement shows a balance of $510.

In the case of Gavin, where the balance in his check register is $500 and the balance in his bank statement is $510, there is a $10 discrepancy between the two.

A possible explanation for this discrepancy could be outstanding checks or deposits that have not yet cleared or been recorded in either the check register or the bank statement.

For example, Gavin might have written a check for $20 that has not been cashed or processed by the bank yet. Therefore, the check register still reflects the $20 in his balance, while the bank statement does not show the deduction. Similarly, Gavin may have made a deposit of $10 that has not yet been credited to his account in the bank statement.

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Answer:

$30+m($5 x h)

he will be working for 21 h

Step-by-step explanation:

So, Steve has $30, so you want to add that amount to the amount he earn, which is $105.

$5 times 21 is $105

h = 21

but we still need for m but, we already have m

m = $105

Number of ice cream produced before = 45

Number of ice cream produced after = 54

Percent of speed the machine increase

Percent of speed the machine increase = [Number of ice cream produced after - Number of ice cream produced before] / Number of ice cream produced before

Percent of speed the machine increase = [(54 - 45) / 45]100

Percent of speed the machine increase = [9 / 45]100

Percent of speed the machine increase = 20%

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Percent of speed the machine increase is 20%

The area of the trapezoidal figure is 80 m²

An equation is an expression that shows how numbers and variables using mathematical operators.

The area of a figure is the amount of space it occupies in its two dimensional form

The area of trapezium = (1/2) * (sum of parallel sides) * height

Hence:

Area = (1/2) * (8 m + 12 m) * 8 m = 80 m²

The area of the figure is 80 m²

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The mean of all exam scores in the class is 72.67 ~ 73 . Mean is ratio of sum of score in all exam to the total numbers of all exams.

The exam scores are normally distributed , normally distributed is an arrangement in which most of values in middle and rest are symmetrically distributed at the ends 80th percentile of scores earned 85 it implies that 80% of students score is 85 or less and 30th percentile of scores earned 65 means 30% of students score is 65 or less .

Z-score value for 80th percentile is 0.841

Z-score value for 30th percentile is – 0.524

Formula for normally distributed,

Z =( X – μ) / σ

where μ is mean of scores , σ is standard deviations of scores ,X is sample mean

For 80th percentile , X= 85 , Z= 0.841

0.841=( 85- μ) /σ ----(1)

-0.524= (65-μ) / σ ----(2)

Solving equations (1) and (2)

- (841/524)= (85-μ )/( 65-μ) => 841μ – 841(65) = -524μ +85(524)

⇒ 1365μ= 99,205 => μ= 72.67

So, the value of mean of scores obtained by students in a class is 72.67 ~ 73

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Answer:

Horizontal line ⇒ y = -4

Vertical line ⇒ x = -3

Step-by-step explanation:

∵ A horizontal line passes through the point (-3, -4)

→ That means the y-coordinates of all points on the line have

   the same value

∴ The equation of the line is y = -4

∵ A vertical line passes through the point (-3, -4)

→ That means the x-coordinates of all points on the line have

   the same value

∴ The equation of the line is x = -3

Answer:

not sure what you are asking.

Step-by-step explanation:

Answer:

see below

Step-by-step explanation:

y=6x^2 +2x - 20

a=6  b= 2  c = -20

Quadratic Formula

-b ± sqrt( b^2 - 4ac)

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

   2a

-2 ± sqrt( 2^2 - 4*6(-20))

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

   2*6

-2 ± sqrt( 2^2 - 4*6(-20))

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

   2*6

-2 ± sqrt(484)

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

   2*6

-2 ± 22

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

   12

20/12 = 5/3        -24/12 = -2

solutions are 5/3 and -2

Factoring

0=6x^2 +2x - 20

0 =2( 3x^2 +x - 10)

   =2 ( 3x-5) ( x+2)

Using the zero product property

3x-5 = 0   x+2 = 0

3x=5         x =-2

x = 5/3      x = -2

Completing the square

y=6x^2 +2x - 20

Add 20

20=6x^2 +2x

Divide by 6

20/6 = x^2 +2/6x

10/3 = x^2 +1/3x

Take the coefficient of x and divide by 2

1/3 /2 = 1/6

Square it

(1/6) ^2 = 1/36

Add it to each side

10/3 + 1/36 = x^2 +1/3x+ 1/36

120/36 + 1/36  = ( x+ 1/6) ^2

121/ 36= ( x+ 1/6) ^2

Take the square root of each side

sqrt(121/ 36) = sqrt( ( x+ 1/6) ^2)

±11/6 = x+1/6

Subtract 1/6 from each side

-1/6 ±11/6 = x

-1/6 - 11/6 = x      -1/6 + 11/6 = x

-12/6 =x                10/6 =x

-2 =x                         5/3 =x

Graphing

Answer: and Step-by-step explanation:

y=6x² +2x - 20

let a=6  

let b= 2

let c = -20

Using Quadratic Formula by hand Using Factoring and the Zero Property Product Using Completing the Square Using Graphing on the Calculator

using Quadratic Formula

-b ± √( b² - 4ac)

             2a

-2 ± √( 2² - 4*6(-20))

             2*6

-2 ± √( 2² - 4*6(-20))

              2*6

-2 ± √(484)

       2*6

20/12 = 5/3      

-24/12 = -2

the answers are 5/3 ;  -2

use by Factoring

0=6x² +2x - 20

0 =2( 3x² +x - 10)

   =2 ( 3x-5) ( x+2)

by using the zero product property

3x-5 = 0

x+2 = 0

3x=5        

x =-2

x = 5/3    

x = -2

by using completing the square

y=6x² +2x - 20   Add 20

20=6x² +2x        Divide by 6

20/6 = x² +2/6x

10/3 = x^2 +1/3x    Take the coefficient of x and divide by 2

1/3 /2 = 1/6             Square it

(1/6) ² = 1/36        Add it to both side s

10/3 + 1/36 = x² +1/3x+ 1/36

120/36 + 1/36  = ( x+ 1/6)²

121/ 36= ( x+ 1/6)²  Take the square root of both side

√(121/ 36) = √( ( x+ 1/6)²)

±11/6 = x+1/6          Subtract 1/6 from both side

-1/6 ±11/6 = x

-1/6 - 11/6 = x      

-1/6 + 11/6 = x

-12/6 =x                

10/6 =x

x = -2  

x = 5/3

by using a graphing computer