In ΔFGH, f= 67 inches, m∠G=66° and m∠H=30°. Find the length of g to the nearest inch

Answer:

To approximate the positive zero of the given polynomial function f(x), we can use numerical methods such as the Newton-Raphson method or the bisection method. Here, we will use the latter method.

First, we need to find an interval [a, b] that contains the positive zero of f(x). We can do this by observing the behavior of the function near the origin and near large positive values of x. We can see that f(0) = -1 is negative and that f(x) becomes more and more negative as x approaches infinity. This suggests that the positive zero of f(x) is somewhere in the interval (0, infinity).

We can evaluate f(x) at x = 1 and x = 2 to determine which half of the interval contains the positive zero. We have:

f(1) = 2(1)² - 1 - 16(1)³ - 3(1)² - 8(1) - 2 = -24

f(2) = 2(2)² - 1 - 16(2)³ - 3(2)² - 8(2) - 2 = -207

Since f(1) is negative and f(2) is very negative, we can conclude that the positive zero of f(x) is in the interval (1, 2).

Next, we can use the bisection method to refine the interval and approximate the positive zero to two decimal places. We start by evaluating f(c), where c is the midpoint of the interval (1, 2):

c = (1 + 2)/2 = 1.5

f(c) = 2(1.5)² - 1 - 16(1.5)³ - 3(1.5)² - 8(1.5) - 2 ≈ -97.875

Since f(c) is negative, the positive zero of f(x) must be in the interval (c, 2). We can repeat the process by finding the midpoint of this interval:

c = (1.5 + 2)/2 = 1.75

f(c) = 2(1.75)² - 1 - 16(1.75)³ - 3(1.75)² - 8(1.75) - 2 ≈ -38.104

Again, f(c) is negative, so the positive zero of f(x) must be in the interval (c, 2). We can repeat the process until the interval is small enough to obtain the desired level of accuracy.

Using a calculator or a computer program, we can continue the bisection method and find that the positive zero of f(x) is approximately 1.49, rounded to two decimal places.

Step-by-step explanation:

Answer:

The distance between points (4, -2) and (4,4) on a coordinate plane is 6 units. This is because the two points lie on the same vertical line, where the x-coordinate remains constant at 4. To find the distance between them, we simply need to subtract their y-coordinates: 4 - (-2) = 6. Therefore, the distance between the two points is 6 units.

The three numbers are 10, 20, and 40.'

Let's assign variables to each number:

According to the problem, the sum of these three numbers is 70:

x + y + z = 70

Now, we can substitute the expressions for y and z in terms of x:

x + 2x + 2(2x) = 70

Combine the terms:

x + 2x + 4x = 70

7x = 70

Now, divide by 7 to find the value of x:

x = 70 / 7

x = 10

Now that we have the value of x, we can find the values of y and z:

y = 2x = 2(10) = 20

z = 2y = 2(20) = 40

So, the three numbers are 10, 20, and 40.

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The question in English is

The sum of 3 numbers is 70, the second number is twice the first and the third is twice the second.

The diameter of the circle is approximately 947.15 meters.

What is a circle?

It is the center of an equidistant point drawn from the center. The radius of a circle is the distance between the center and the circumference.

The formula for the circumference of a circle is:

C = 2πr

where C is the circumference and r is the radius of the circle.

We can use this formula to find the radius of the circle given its circumference:

C = 2πr

942 = 2πr

Dividing both sides by 2π, we get:

r = 942/2π

Now, we can use the formula for the diameter of a circle in terms of its radius:

d = 2r

Substituting the value we found for r, we get:

d = 2(942/2π) = 942/π

Using a calculator to approximate π as 3.14, we can evaluate the expression to get:

d ≈ 947.15 meters

Therefore, the diameter of the circle is approximately 947.15 meters.

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Answer:

This is a right angle

Step-by-step explanation:

There is a square inside of the triangle.

Answer: 11

Step-by-step explanation: To find the value of (fog)(-1), we first need to find the value of g(-1), which means plugging -1 into the equation for g(x):

g(x) = x² – 5x - 5

g(-1) = (-1)² – 5(-1) - 5

g(-1) = 1 + 5 - 5

g(-1) = 1

Now we need to find (fog)(x) by plugging g(x) into f(x) and simplifying:

f(x) = -x - 12

fog(x) = f(g(x))

fog(x) = f(x² – 5x - 5)

fog(x) = -(x² – 5x - 5) - 12

fog(x) = -x² + 5x + 17

Finally, we can find (fog)(-1) by plugging -1 into the equation for fog(x):

fog(x) = -x² + 5x + 17

(fog)(-1) = -(-1)² + 5(-1) + 17

(fog)(-1) = -1 - 5 + 17

(fog)(-1) = 11

Therefore, the value of (fog)(-1) is 11.

The final amount with interest earned at a rate of 3.4% compounded semi-annually would be approximately $7,758.

Let's calculate the values based on the given information.

Given:

Initial deposit (P) = $7,500

Interest rate (r) = 3.4% or 0.034 (decimal)

Compounding frequency (n) = 2 (semi-annually)

We can use the compound interest formula to calculate the values:

A = P[tex](1 + r/n)^(nt)[/tex]

Where:

A = the final amount (including interest)

P = the principal amount (initial deposit)

r = the interest rate (in decimal)

n = the number of times interest is compounded per time period

t = the number of time periods

Let's calculate the final amount (A) after one year (2 semi-annual periods):

A =[tex]$7,500(1 + 0.034/2)^(2*1)[/tex]

A =[tex]$7,500(1.017)^2[/tex]

A = $7,500(1.034344) [rounded to 6 decimal places]

A ≈ $7,758 [rounded to the nearest dollar]

So, after one year, the final amount with interest earned at a rate of 3.4% compounded semi-annually would be approximately $7,758.

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Answer:

75=x

Step-by-step explanation:

The only function that represents a range as less than zero only is: Option C: 1/(x - 2)²

The range of a function is defined as all the possible values y could be. The formula to find the range of a function is y = f(x). In a relation, it is only a function if every x value corresponds to only one y value.

Now, we want to find the function for which the range is a set of all real numbers less than zero. This means that the range is made up of only negative numbers.

The only option that denotes the range to be less than zero is option C because no matter the domain value, the range will remain a negative number.

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Thus, the length of walk way around the filed is the perimeter of the right triangle which is found as: 206.023 m .

The quickest approach to determine a triangle's perimeter is to sum up all of its sides' lengths, but you must first determine each side's length if you don't already know it.

The walk way around the filed is the perimeter of the right triangle.

walk way = base + height + hypotenuse

walk way = B + P + H

base B = 80 - 10 = 70 m

Altitude P = 70 - 20 = 50 m

For the hypotenuse H , use the Pythagorean theorem:

H² = B² + P²

H² = 70² + 50²

H² = 4900 + 2500

H² = 7400

H = 86.023

So,

walk way = B + P + H

walk way = 70 + 50 + 86.023

walk way = 206.023 m

Thus, the  walk way around the filed is the perimeter of the right triangle which is found as: 206.023 m .

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In terms of part A, the solution set is a triangular region with vertices at (0,4), (2,3), and (4,0).

Part B: Yes, the point (2, 3) included in the solution area for the system.  The solution of the inequalities is (4, 0).

Part C. I choose the point (1,3) in the solution set. This implies that Michael can buy 1 cupcake and 3 pieces of fudge with $16 as well as feed at least 4 siblings.

Part A: The inequalities limit Michael's purchase of cupcakes and fudge with $16 to feed at least 4 siblings. The first one, 4x + 2y ≤ 16, restricts the total cost to be $16 or less. To graph y ≤ -2x + 8, draw the line with slope -2 and y-intercept 8, and shade below it. The inequality x + y ≥ 4 means Michael must feed at least 4 siblings. It can be graphed as y ≥ -x + 4, a line with slope -1 and y-intercept 4. To graph the inequality, draw a dashed line with slope -1 and y-intercept 4. Shade the region above the line for y ≥ -x + 4. The solution set is the intersection of the shaded regions. Triangle with vertices (0,4), (2,3), and (4,0).

Part B:

To check if (2,3) is in solution area, we test if it satisfies both inequalities. We substitute x=2, y=3 into 4x + 2y ≤ 16 and get 14. When x=2 and y=3 in x+y≥4, we get 5, which is true. However, (2, 3) is not included in the solution area because it does not satisfy 2x+y≤8.

Part C:

Let select (2,2) in the solution set. Michael can buy 2 cupcakes and 2 pieces of fudge while feeding 4 siblings within his budget. Michael can spend $8 on 2 cupcakes and $4 on 2 fudge pieces to feed his siblings. Michael can buy various combinations of cupcakes and fudge within his budget to feed at least 4 siblings.

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See full question below

Michael has $16 and wants to buy a mixture of cupcakes and fudge to feed at least 4 siblings. Each cupcake costs $4, and each piece of fudge costs $2.

This system of inequalities models the scenario:

4x + 2y ≤ 16

x + y ≥ 4

Part A: Describe the graph of the system, including shading and the types of lines graphed. Provide a description of the solution set. (4 points)

Part B: Is the point (2, 3) included in the solution area for the system? Justify your answer mathematically. (3 points)

Part C: Choose a different point in the solution set and interpret what it means in terms of the real-world context. (3 points)

Answer:

Step-by-step explanation:

64 - 16= 48

[tex]\sqrt{48}[/tex]

[tex]\sqrt{3*16}[/tex]

4[tex]\sqrt{3}[/tex]

The values are given below:

In mathematics, a function is a rule that assigns to each input value (or argument) from a set called the domain, a unique output value from a set called the range.

The term "maths function" is typically used to describe a specific type of function that is studied in mathematics, which can be represented using algebraic or other mathematical expressions.

Functions can be used to describe various phenomena in mathematics, science, and other fields, and they play a fundamental role in many areas of mathematics, including calculus, linear algebra, and number theory.

Some common examples of mathematical functions include polynomial functions, trigonometric functions, exponential functions, and logarithmic functions.

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The pattern represents a straight line that passes through the origin with a slope of 1.

The form Ax + By + C = 0 is a two-variable linear equation, where A and B are the coefficients, C is a constant term, and x and y are the two variables, with each having a degree of 1.

The numerical pattern that represents the rule y = x + 0, starting with x = 0, is:

x y

0 0

1 1

2 2

3 3

4 4

... ...

This example shows that when x increments by 1, y additionally increments by 1. Since the rule for y is y = x + 0, we can see that adding 0 to x doesn't change its value, so y is just the same as x. As a result, x and y have a linear relationship with a slope of 1 and a y-intercept of 0.

To put it another way, the pattern is a straight line with a slope of one that runs through the origin.

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part a:

Henry should take a route that bypasses the park on the way to school and on the way home from school in order to ride the least possible distance.

He should take Route C on his way to school and Route B on his back.

part b: no values were given to calculate the distance.

Distance is described as a numerical or occasionally qualitative measurement of how far apart objects or points are.

Since the park near Henry's house is closed before school, so he cannot ride through the park on his way to school, he should take Route C on his way to school and Route B on his back.

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The mean absolute deviation of the entire data set is 4.

To find the mean absolute deviation (MAD) of the given data set, we need to follow these steps:

Calculate the mean (average) of the data set.

Subtract the mean from each data point to find the deviation.

Take the absolute value of each deviation.

Calculate the mean of the absolute deviations.

Let's calculate the MAD for the given data set: 18, 13, 6, 14, 11, 22.

Calculate the mean.

Mean = (18 + 13 + 6 + 14 + 11 + 22) / 6 = 84 / 6 = 14.

Calculate the deviation for each data point.

Deviation = data point - mean.

Deviation for 18 = 18 - 14 = 4.

Deviation for 13 = 13 - 14 = -1.

Deviation for 6 = 6 - 14 = -8.

Deviation for 14 = 14 - 14 = 0.

Deviation for 11 = 11 - 14 = -3.

Deviation for 22 = 22 - 14 = 8.

Take the absolute value of each deviation.

Absolute deviation = |deviation|.

Absolute deviation for 4 = |4| = 4.

Absolute deviation for -1 = |-1| = 1.

Absolute deviation for -8 = |-8| = 8.

Absolute deviation for 0 = |0| = 0.

Absolute deviation for -3 = |-3| = 3.

Absolute deviation for 8 = |8| = 8.

Calculate the mean of the absolute deviations.

Mean absolute deviation = (4 + 1 + 8 + 0 + 3 + 8) / 6 = 24 / 6 = 4.

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The graph that shows the solution to the system of linear inequalities is the Graph D.

Here, we are given first inequality as:

Y > -1/3x + 2

The graph of this inequality is a dotted straight line( since the inequality is strict) that passes through (6,0) and (0,2) with shaded region away from the origin ( since it fails zero test)

Second inequality is given by:

y > 2x - 3

The graph of this inequality is a dotted straight line( since the inequality is strict) that passes through (3/2,0) and (0,-3) with shaded region away towards the origin ( since it passes zero test). On plotting it's graph, we get the solution.

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The right Riemann sum of [tex]\int_1^5f(x).dx[/tex] with 4 rectangles of the function f(x) is 6 square units.

we have to find the area of the function from 1 to 5.

we have to use 4 rectangles.

So, the width of the each rectangle is = (5 - 1)/4 = 4/4 = 1 unit

The figure will be

The right Riemann sum with 4 rectangles will be given by

= [1*5 + 1*4 + 1*0 + 1*(-3)] square units

= (5 + 4 + 0 - 3) square units

= 6 square units.

Hence, the required right Riemann sum is 6 square units.

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The question is incomplete. The complete question will be -

"Given the graph of the function f(x) below, use a right Riemann sum with 4 rectangles to approximate the integral [tex]\int_1^5f(x).dx[/tex]."

The term 4.59  is the y-intercept of the linear equation and represents the flat rate (or flat cost).

We know that the linear function:

f(x)=0.75x+4.59

models the cost of shipping x packages.

Remember that the general line is:

y = ax + b

Where a is the slope and b the y-intercept.

if x ≈ 0, then you ship no packages, and in that case the value of the function becomes equal to the second term:

f(x ≈ 0) = 4.59

So the term 4.59 represents the flat rate (or flat cost).

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It would take Mike approximately 0.989 minutes (or about 59 seconds) to wash the car alone.

We have,

Let's start by assigning variables to unknown quantities.

Let m be the time it takes for Mike to wash the car alone (in minutes).

We know that Stephanie takes 17 minutes to wash the car alone, so her rate of work is 1/17 car per minute.

Working together, their combined rate of work is 1/8.24 car per minute.

Using the formula:

(rate of work) = (amount of work) / (time)

we can set up the following equation to represent the work they do together:

1/8.24 = (1 car) / t

where t is the time it takes them to wash the car together.

We can also set up a similar equation for Stephanie's work alone:

1/17 = (1 car) / (t + x)

where x is the extra time it takes Mike to wash the car alone.

Since they are working on the same car, the amount of work done in both cases is the same, so we can set the two equations equal to each other and solve for x:

1/8.24 = 1/(t + x) + 1/17

Multiplying both sides by the least common multiple of the denominators (8.2417(t+x)).

17*(t+x) + 8.24*(t+x) = 8.24*17

Simplifying.

25.24t + 17x = 139.68

We also know that Mike's rate of work is 1/m car per minute.

Since we know the combined rate of work when they work together, we can set up another equation:

1/m + 1/17 = 1/8.24

Multiplying both sides by the least common multiple of the denominators (8.2417m).

178.24m + 8.2417m = 8.2417m + m8.24m

Simplifying.

141.08m = 139.68

Solving for m.

m = 0.989

Therefore,

It would take Mike approximately 0.989 minutes (or about 59 seconds) to wash the car alone.

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Answer:

Center= -4,8

Radius=7

Step-by-step explanation:

The correct option is- I, IV. Fixed Annuity and Traditional IRA are the plans are  not considered a defined contribution plan.

In contrast, a defined contribution plan does not guarantee a certain level of benefits at retirement. Under these plans, either the employee or even the employer (or both) may make contributions to the owner's individual account under the plan.

Traditional IRA:

An individual retirement account (IRA) of this type allows your income to increase tax-free. Only when you take withdrawals in retirement do you have to pay taxes on the investment earnings.

Fixed Annuity:

A fixed annuity is a financial contract that offers a retirement income stream and guarantees a certain rate of return, such as 2%. You can predict how much your annuity will increase and how much money it will produce by using a set interest rate.

Thus, Fixed Annuity and Traditional IRA are the plans are  not considered a defined contribution plan.

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Adding 5 bonus points to each test grade in Mrs. Matthew's class will increase the mean, median, and mode of the scores by 5 points each.

The addition of 5 bonus points to each test grade will affect the mean, median, and mode in the following ways:

So, in conclusion, adding 5 bonus points to each student's test score will raise the mean, median, and mode of the scores by 5 points each.

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Juan's claim is plausible based on the given function because the maximum height of the ball is 62.875 feet, which is higher than the 50-foot light tower.

To prove that Juan hit the ball over the 50-foot light tower, we need to find the maximum height of the ball using the given function h(x) = -16x^2 + 60x + 4 and show that the maximum height is greater than 50 feet.

The maximum height of the ball occurs at the vertex of the parabolic function h(x), which is given by the formula x = -b/2a, where a = -16 and b = 60.

Substituting these values into the formula, we get:

x = -b/2a = -60/(2*(-16)) = 1.875

So the ball reaches its maximum height after 1.875 seconds. To find the maximum height, we can substitute this value of x into the original function h(x):

h(1.875) = -16(1.875)^2 + 60(1.875) + 4 = 62.875

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Therefore , the solution of the given problem of percentage comes out to be 15 gallons of the first brand and 75 gallons of the second brand must be utilised

In statistics, a figure or metric than may be presented as a percentage or 100 is denoted by the abbreviation "a%". Another unusual spelling is "pct," "pct," and "pc." The percent symbol ("%") is the method that is most usually used for this. Additionally, there are no indications or predetermined ratios of any component to the whole. Numbers are effectively integers since they frequently add up to 100.

Here,

Let's write "x" for the first brand's (55% pure antifreeze) number of gallons and "y" for the second brand's (85% pure antifreeze) number of gallons.

Given:

90 gallons of mixture total are required.

Antifreeze content in the mixture should be 80%.

Based on the information provided, we can construct the following system of equations:

Formula 1: x + y = 90

Formula 2: 0.55x + 0.85y = 0.80 * 90

=> x = 90 - y

=> 0.55(90 - y) + 0.85y = 0.80 * 90

=> 49.5 - 0.55y + 0.85y = 72

=> 0.30y = 22.5

=> y = 75

=> x = 90 - 75

=> x = 15

In order to get 90 gallons of a mixture that contains 80% pure antifreeze, 15 gallons of the first brand and 75 gallons of the second brand must be utilised.

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Answer: 10 pounds                                               .                                    

Answer:

The mean of the numbers 5, 3, 6, x, 9, and 10 is given as 7.

To find the value of x, we can use the formula for the mean:

mean = (sum of all the numbers) / (number of numbers)

In this case, we have:

7 = (5 + 3 + 6 + x + 9 + 10) / 6

Multiplying both sides of the equation by 6, we get:

42 = 33 + x

Subtracting 33 from both sides of the equation, we get:

x = 9

Therefore, the value of x that makes the mean of 5, 3, 6, x, 9, and 10 equal to 7 is 9.