(07.03 MC)
Given the function h(x)=-2√x +3-1, which statement is true about h(x)?

Answer: A. 251.3 in.

Step-by-step explanation:

        We can use this formula to solve for the circumference of a circle. r is equal to the radius, or half of the circle's width.

                 C = 2πr

C = 2πr

C = 2π(40)

C ≈ 251.3

                 A. 251.3 in.

Answer:

a. Plot the points on the graphing calculator, then generate a logarithmic regression equation.

y = 34.17860108 - 5.682632245ln(x)

b. Setting this equation equal to 10, we have x = 70.44 mph

c. If x = 75 mph, then y = 9.64.

Answer: C. 3/4 of a cup of corn syrup

Step-by-step explanation:

    We will set up a proportion to help us solve.

[tex]\displaystyle \frac{1/4\text{ cup corn syrup}}{6\text{ cups of soap and water}} =\frac{x\text{ cups corn syrup}}{18\text{ cups of soap and water}}[/tex]

    Next, we will cross-multiply.

1/4 * 18 = 6 * x

9/2 = 6x

    Lastly, we will divide both sides of the equation by 6.

x = 3/4 of a cup of corn syrup

                       C. 3/4 of a cup of corn syrup

According to the given information, if 5 cups of flour are used in the recipe, we should use 5/3 cups of sugar.

Multiplication is an arithmetic operation that involves combining two or more numbers to get a product. It is a way of repeatedly adding a number to itself a certain number of times. The number being multiplied is called the multiplicand, while the number of times it is being multiplied is called the multiplier. The result of the multiplication operation is called the product

If the recipe requires 2 cups of flour for every 2/3 cups of sugar, we can write this relationship as:

2 cups of flour : 2/3 cups of sugar

To find out how many cups of sugar should be used if 5 cups of flour are included, we can set up a proportion:

2 cups of flour / 2/3 cups of sugar = 5 cups of flour / x cups of sugar

where x is the number of cups of sugar we want to find.

To solve for x, we can cross-multiply:

2 cups of flour * x cups of sugar = 2/3 cups of sugar * 5 cups of flour

2x = (2/3) * 5

2x = 10/3

x = 5/3

Therefore, According to the given information,  if 5 cups of flour are used in the recipe, we should use 5/3 cups of sugar.

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The description that correctly represents g(x) is g of x is a piecewise function, which equals negative 2, when x is less than or equal to negative 1, and it equals the quantity x minus 1 end quantity squared, when negative 1 is less than x is less than or equal to 0, and it equals negative x plus 1, when x is greater than 0 (third option).

Looking at the graph, we can see that:

Thus, the correct piecewise function that represents g(x) is:

g(x) = {-2 if x ≤ -1

{ (x-1)^2 if -1 < x ≤ 0

{ -x+1 if x > 0

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Answer: a = 5 and k = 1/2

Step-by-step explanation:

We are given that the function is defined by:

f(x) = 16 + a(3^(kx))

We need to find the real numbers a and k such that f(0) = 21 and f(4) = 61. Using the given values of f(0) and f(4), we can form a system of two equations:

f(0) = 16 + a(3^(k(0))) = 21

f(4) = 16 + a(3^(k(4))) = 61

Simplifying the first equation, we get:

16 + a(3^0) = 21

16 + a = 21

a = 5

Substituting this value of a into the second equation, we get:

16 + 5(3^(4k)) = 61

5(3^(4k)) = 45

3^(4k) = 9

We know that 3^2 = 9, therefore

4k = 2

=> k = 2/4

=> k = 1/2

Therefore, the real numbers a and k that satisfy the given conditions are a = 5 and k = 1/2. So the function is:

f(x) = 16 + 5(3^(x/2))

The solution to the equation give us an approximate that x = 1.55.

To solve this equation, we can start by subtracting 4 from both sides to isolate the exponential term:

3.5^x + 4 - 4 = 11 - 4

This simplifies to:

3.5^x = 7

Next, we can take the logarithm of both sides with base 3.5 to eliminate the exponent:

log(3.5^x) = log(7)

Using the property of logarithms that log(a^b) = b*log(a), we can rewrite the left side as:

x*log(3.5) = log(7)

Now, we can solve for x by dividing both sides by log(3.5):

x = log(7) / log(3.5)

Plugging in the values, we get:

x = 0.84509804001 / 0.54406804435

x = 1.55329475566

x ≈ 1.55.

Answered question "Solve the equation 3.5^x + 4 = 11"

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The rate of the plane in still air is 810 km/h and the rate of the wind is 90 km/h.

Explain the term rate

Rate is a measure of the speed or frequency of change of a quantity over time or space. It is expressed as a ratio between two quantities, such as distance travelled per unit of time or number of occurrences per unit of time. Rates are commonly used in mathematics, science, and economics to analyze and compare different phenomena.

According to the given information

Let's let x be the rate of the plane in still air and y be the rate of the wind. When flying against the wind, the plane's effective speed is (x - y) km/h, and when flying with the wind, its effective speed is (x + y) km/h.

From the information given in the problem, we can set up two equations to represent the plane's distance travelled when flying against and with the wind:

6390 = 9(x - y)

2970 = 3(x + y)

Solving for x and y in these equations, we find that x = 810 and y = 90. So the rate of the plane in still air is 810 km/h and the rate of the wind is 90 km/h.

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Step-by-step explanation:

To calculate the probability of getting at least four hearts when dealing five cards from a standard deck, we can use the concept of combinations and probability.

There are a total of 52 cards in a standard deck, and 13 of them are hearts (assuming no jokers). So the probability of drawing a heart on the first card is 13/52, or 1/4.

Now, there are two possible scenarios that would result in at least four hearts:

Four hearts and one non-heart: This can happen in C(13,4) * C(39,1) ways, where C(n, k) is the number of combinations of n items taken k at a time. The first part C(13,4) represents choosing 4 hearts out of 13, and the second part C(39,1) represents choosing 1 non-heart out of the remaining 39 cards. The total number of ways to choose 5 cards from 52 is C(52,5).

Five hearts: This can happen in C(13,5) ways, where C(13,5) represents choosing all 5 hearts out of 13.

So, the total number of favorable outcomes is C(13,4) * C(39,1) + C(13,5), and the total number of possible outcomes is C(52,5). Therefore, the probability of getting at least four hearts is:

P(at least 4 hearts) = (C(13,4) * C(39,1) + C(13,5)) / C(52,5)

Plugging in the values and simplifying, we get:

P(at least 4 hearts) = (C(13,4) * C(39,1) + C(13,5)) / C(52,5)

= (715 * 39 + 1287) / 2,598,960

= 27,885 / 2,598,960

≈ 0.0107566

So, the probability of getting at least four hearts when dealing five cards from a standard deck is approximately 0.0107566 or about 1.08%.

The coordinates of the fourth vertex are (-2.2, 1.5).

To find the coordinates of the fourth vertex, we can use the fact that opposite sides of a rectangle have equal length and are parallel to each other.

First, we can find the length of one of the sides of the rectangle. Let's take the side connecting the points (-2.2, -2.3) and (-2.2, 1.5):

length = |1.5 - (-2.3)| = 3.8

Since this is a rectangle, the length of the side opposite to this one must also be 3.8. We can find this side by taking the points (-2.2, 1.5) and (1.5, 1.5):

width = |1.5 - (-2.2)| = 3.7

Now we know the length and width of the rectangle, so we can find the coordinates of the fourth vertex. Let's call this point (x, y).

Since the fourth vertex is opposite to the point (-2.2, -2.3), the x-coordinate of the fourth vertex must be the same as the x-coordinate of (-2.2, -2.3):

x = -2.2

Similarly, since the fourth vertex is opposite to the point (1.5, 1.5), the y-coordinate of the fourth vertex must be the same as the y-coordinate of (1.5, 1.5):

y = 1.5

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The algebraic expression that represents the age of Ethan's dad is 4a - 5

Let's start by defining a variable to represent Ethan's age. We'll call this variable "a".

According to the problem, Ethan's dad is five years younger than four times Ethan's age. So, if we multiply Ethan's age by 4 and then subtract 5 years, we'll have the age of Ethan's dad.

Algebraically, we can represent this as:

4a - 5

Therefore, the algebraic expression that represents the age of Ethan's dad is 4a - 5.

Solve for x by simplifying both sides of the equation and isolating the variable.

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

Convert mixed numbers to improper fraction :

[tex]3\frac{2}{3} =\frac{11}{3}[/tex]

[tex]\frac{11}{3}x=2\frac{2}{5}[/tex]

Convert mixed numbers to improper fractions:

[tex]2\frac{2}{5} = \frac{12}{5}[/tex]

[tex]\frac{11}{3}x=\frac{12}{5}[/tex]

Multiply both sides by 3

[tex]11x=\frac{36}{5}[/tex]

Divide both sides by 11

[tex]x=\frac{36}{55}[/tex]

Therefore, [tex]3\frac{2}{3} x=2\frac{2}{5}[/tex] = [tex]x=\frac{36}{55}[/tex]

Steve sold 252 fruit baskets for the fundraiser. Therefore, we can find how many baskets Evie sold by using the ratio of her sales to Steve's sales, which is given as 25 baskets for each 100 baskets Steve sold.

We can set up a proportion to solve for the number of baskets Evie sold:

25/100 = x/252

Here, x represents the number of fruit baskets Evie sold.

To solve for x, we can cross-multiply and simplify:

25 * 252 = 100 * x

6300 = 100x

x = 63

Therefore, Evie sold 63 fruit baskets for the fundraiser.

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Answer:

If the point (1,5) lies on a circle with center (0,0), then the radius of the circle is the distance from the center to the point (1,5), which is:

r = sqrt((1-0)^2 + (5-0)^2) = sqrt(26)

So the equation of the circle is:

x^2 + y^2 = 26

To find a point in quadrant 2 that lies on the circle, we need to find a point with a negative x-coordinate and a positive y-coordinate that satisfies this equation.

Among the given options, only (a) (-5,1) and (e) (-2,3) have a negative x-coordinate. Let's check if they satisfy the equation of the circle:

For point (a) (-5,1):

x^2 + y^2 = 26

(-5)^2 + 1^2 = 26

25 + 1 = 26

This point does not lie on the circle.

For point (e) (-2,3):

x^2 + y^2 = 26

(-2)^2 + 3^2 = 26

4 + 9 = 26

This point also does not lie on the circle.

Therefore, among the given options, there is no point in quadrant 2 that lies on the circle.

Step-by-step explanation:

since the center of the circle is at the origin, that means that the circle is evenly going through all four quadrants - one quarter arc after the other.

the point (1, 5) is in the first quadrant (upper right one, positive x, positive y).

the circle arc in the second quadrant (upper left one, negative x, positive y) is the mirrored image of the arc in the first quadrant across the y-axis.

so, the point on it gets mirrored too.

that means, it's coordinates are then the same y and the x is converted to its negative value :

(-1, 5)

that is the easily identifiable point in the second quadrant that has to be on the circle.

You have to find what 0.2 of 132 is

132 x 0.2 = 26.40

The answer [tex]\boxed{\bold{\$26.40}}[/tex]

Answer: $3.00

Step-by-step explanation:

To figure this out, we can use the following formula:

[tex]\textsf{Cost per pound x number of pounds we want to buy}[/tex]

In this case, we want to buy 2.5 pounds of apples, so we can multiply 2.5 by $1.20 to get the total cost:

[tex] \textsf{2.5 x 1.20 = 3} [/tex]

So, 2.5 pounds of apples will cost $3.00

Answer:

Yes, the figure is line symmetric.

A snowflake has six lines of symmetry. Therefore, the correct answer is: 6.

Answer:

The question is asking about the line of reflection that would transform polygon ABCDE to its image A' B' C' D' E'.

To determine the line of reflection, we need to identify the axis or line that reflects each point of polygon ABCDE to its corresponding point on polygon A' B' C' D' E'.

Looking at the coordinates of the vertices, we can see that the x-coordinates of the corresponding points are the same, but the y-coordinates are opposite in sign. This suggests that the line of reflection is parallel to the x-axis.

Furthermore, we can see that the reflected image is below the original polygon, so the line of reflection must be the x-axis itself, reflecting the polygon downward.

Therefore, the correct answer is "Reflection across the x-axis."

Answer:

$3.87 for the 18% tip.

Step-by-step explanation:

the area of a right-angled triangle is

leg1 × leg2 / 2 =

sin(angle)×baseline × cos(angle)×baseline / 2 =

sin(angle)×cos(angle) × baseline² / 2

4 = sin(30)×baseline = 0.5 × baseline

baseline = 4/0.5 = 8 units

Pythagoras gives us the third side = cos(30)×baseline :

8² = 4² + side²

64 = 16 + side²

48 = side²

side = sqrt(48) = sqrt(16×3) = 4×sqrt(3) units

so, the area of the triangle is again

sin(30)×baseline × cos(30)×baseline / 2 =

= 4 × 4×sqrt(3) / 2 = 2×4×sqrt(3) = 8×sqrt(3) units² =

= 13.85640646... units²

Since this is about the angle theorem and the angle theorem is a theorem that helps us know the relationship between angles. Being a corresponding angle, x = 105°. Hence option A is correct.

A corresponding angle is an angle that occupies the same relative position as another angle elsewhere in the diagram. The corresponding angle in planar geometry occurs when a transverse line crosses his two straight lines. Two angles correspond or relate to each other by being on the same side of the horizontal line. One is the exterior angle (outside the parallel lines) and the other is the interior angle (inside the parallel lines).

You can see that the transversal line intersects two parallel lines DE and XY. From the transversal theorem of angles in mathematics, we know that when a transversal line intersects two parallel lines, the corresponding angles formed are congruent and therefore equal to each other.

In this question we see that one of the angles is 105°. Therefore x° is congruent to it, x = 105°.

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The complete question is as follows:

Answer:

206 inches

Step-by-step explanation:

1 feet = 12 inches

17 feet = 17 x 12 = 204 inches

17 feet 2 inches = 204 + 2 = 206 inches

So, 17 feet 2 inches = 206 inches

The work done in stretching the spring from 10 cm to 40 cm is 10,837.5 J.

The work done in stretching a spring is given by the formula:

W = (1/2)k[tex]x^{2}[/tex]

where W is the work done, k is the spring constant, and x is the displacement of the spring from its original position.

To find the spring constant k, we can use the given information:

F = kx

where F is the force applied to the spring. Rearranging this equation, we get:

k = F/x

Substituting the given values, we get:

k = 350 N / 30 cm = 11.67 N/cm

Now, we can find the work done in stretching the spring from 10 cm to 40 cm:

W = (1/2)k([tex](x_{2})^{2} -(x_{1} )^{2}[/tex])

where [tex]x_{1}[/tex] is the initial displacement of the spring (10 cm) and [tex]x_{2}[/tex] is the final displacement (40 cm).

Substituting the values, we get:

W = (1/2) * 11.67 N/cm * ([tex](40 cm)^{2} - (10 cm)^{2}[/tex])

W = (1/2) * 11.67 N/cm * (1600  [tex]cm^{2}[/tex] - 100  [tex]cm^{2}[/tex])

W = (1/2) * 11.67 N/cm * 1500 [tex]cm^{2}[/tex]

W = 10,837.5 joules (J)

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The journal entry to record the sale of the equipment would be:Cash $105,800,Loss on Sale of Equipment $35,650,Equipment $212,000

Accumulated Depreciation $66,000

How to solve the question?

a. To calculate the book value of the equipment at December 31 of the fifth year, we need to determine the accumulated depreciation for the first five years. The annual depreciation expense is calculated as follows:

Depreciation Expense = (Cost - Residual Value) / Useful Life

Depreciation Expense = ($212,000 - $14,000) / 15

Depreciation Expense = $13,200 per year

The accumulated depreciation for five years is therefore:

Accumulated Depreciation = Depreciation Expense x Number of Years

Accumulated Depreciation = $13,200 x 5

Accumulated Depreciation = $66,000

The book value of the equipment at December 31 of the fifth year is calculated as follows:

Book Value = Cost - Accumulated Depreciation

Book Value = $212,000 - $66,000

Book Value = $146,000

b.

To record depreciation for the three months until the sale date, we need to calculate the depreciation expense for the sixth year. The annual depreciation expense remains the same at $13,200, but since the equipment was sold on April 1, we need to prorate the depreciation for the first three months of the year.

Depreciation Expense for the Sixth Year = Depreciation Expense x (Months Remaining / 12)

Depreciation Expense for the Sixth Year = $13,200 x (9 / 12)

Depreciation Expense for the Sixth Year = $9,900

The journal entry to record depreciation for the three months until the sale date would be:

Depreciation Expense $9,900

Accumulated Depreciation $9,900

To record the sale of the equipment, we need to compare the sale price to the book value of the equipment at the time of the sale.

The book value of the equipment at the time of the sale is calculated as follows:

Book Value = Cost - Accumulated Depreciation

Book Value = $212,000 - ($13,200 x 5.25)

Book Value = $141,450

Since the sale price of $105,800 is less than the book value of $141,450, we need to record a loss on the sale of the equipment. The journal entry to record the sale of the equipment would be:

Cash $105,800

Loss on Sale of Equipment $35,650

Equipment $212,000

Accumulated Depreciation $66,000

The cash received from the sale is debited, and the difference between the sale price and the book value is recorded as a loss on the sale of the equipment. The equipment and its accumulated depreciation are removed from the books.

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we can see that the growth rate increases as x increases. However, since we are only interested in the average growth rate over the interval [1, 3], we can calculate it using the formula mentioned above.

How to solve the question?

To determine the average growth rate of each sample over the interval [1, 3], we need to calculate the ratio of the change in bacteria population to the change in time for each sample, and then take the average of these ratios over the given interval.

For Sample A, the change in bacteria population over the interval [1, 3] is 480 - 120 = 360, and the change in time is 3 - 1 = 2. So the average growth rate of Sample A over this interval is 360/2 = 180 bacteria per hour.

For Sample B, the change in bacteria population over the interval [1, 3] is 480 - 240 = 240, and the change in time is 3 - 1 = 2. So the average growth rate of Sample B over this interval is 240/2 = 120 bacteria per hour.

For Sample C, the change in bacteria population over the interval [1, 3] is (1.2600)(1.2*1.2 - 1) = 345.6, and the change in time is 3 - 1 = 2. So the average growth rate of Sample C over this interval is 345.6/2 = 172.8 bacteria per hour.

For Sample A, the growth rate is the highest, followed by Sample C and then Sample B. Therefore, the order of the samples by their average growth rate over the interval [1, 3] from least to greatest is Sample C, Sample A, and Sample B.

It's important to note that the growth rate of Sample C is not constant but increases over time due to the 20% increase in the initial bacteria population. The exponential function f(x) = 200(3/2)in power x represents this growth, and we can see that the growth rate increases as x increases. However, since we are only interested in the average growth rate over the interval [1, 3], we can calculate it using the formula mentioned above.

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Your complete question is :-A scientist is studying the growth rates of three samples of bacteria in different conditions. The following three functions represent the number of bacteria in the three samples after x hours.

Sample A Sample B Sample C

x g(x)

0 60

1 120

2 240

3 480

The line that goes through the specified points has the equation into point-slope form is  [tex]y+1=6/11(x+4)[/tex].

X-axis points equal (7, -4).

Y-axis points are equal to (5, -1).

To determine the equation of the line passing through the specified points in point-slope form:

A straight line equation's direction and steepness are commonly described by its slope, which is also known as a line's gradient.

In mathematics, the following formula yields the slope of a line:

Slope(m) = [tex]y2-y1/x2-x1[/tex]

slope= [tex]\frac{-1-5}{-4-7}[/tex]

slope= 6/11

To find point slope:

[tex]y-y1=m(x-x1)\\y-(-1)=6/11(x-(-4)\\y+1=6/11(x+4)[/tex]

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A graph of the data set is shown in the image attached below.

In Mathematics and Geometry, a number line simply refers to a type of graph with a graduated straight line which comprises both positive and negative numbers that are placed at equal intervals along its length.

This ultimately implies that, a number line primarily increases in numerical value towards the right from zero (0) and decreases in numerical value towards the left from zero (0).

In this scenario and exercise, we would use an online graphing calculator to graphically represent the given data set on a number line as shown in the image attached below.

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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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A function's codomain is a subset of its range. Although it is not a true subset, it is conceivable for the range to be equal to the codomain, in which case it is also a subset of the codomain.

A set that includes all or a portion of another set's elements is known as a subset. For instance, because every even integer is also an integer, the set of even integers is a subset of the set of integers. A subset that contains some, but not all, of the members of another set is said to be a proper subset. Because it contains some, but not all, of the positive integers, the set of positive integers less than 10 is a valid subset of the set of positive integers.

A function's codomain is a subset of its range. Although it is not a true subset, it is conceivable for the range to be equal to the codomain, in which case it is also a subset of the codomain.

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a) The mean of the mileage is 90.4 miles per day

b) The median of the mileage is 255.5 miles

Given data ,

Let the data be represented as A

Now , the value of A is

A = { 248 176 379 263 202 }

And , Mean = (248 + 176 + 379 + 263 + 202) / 5 = 452 / 5 = 90.4 miles per day (rounded to one decimal place)

Therefore, the mean of the data is 90.4 miles per day.

To find the median of the data, we need to first put the distances in order from least to greatest:

176, 202, 248, 263, 379

There are 5 data points, so the median is the middle value.

Median = (248 + 263) / 2 = 511 / 2 = 255.5 miles

Hence , the median of the data is 255.5 miles

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The complete question is attached below :

The Swamis family went on a vacation. They recorded the number of miles they drove each day. The mileage is shown in the table. Day 1 2 3 4 5 Distance (mi) 248 176 379 263 202 What is the mean of the data? ____miles What is the median of the data? ______miles

Approximately 1.18 hours the concentration is highest.

The given function is c(t)=100t/t² +85.

When the concentration is highest, the function becomes

100t/t² +85 =0

100/t +85=0

100+85t=0

85t=100

t=100/85

t=1.18 hours

Therefore, approximately 1.18 hours the concentration is highest.

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The length of FG chord is 5 units.

What bout chord?

A chord is a line segment that joins two points on the circumference of a circle. More specifically, a chord is a straight line that intersects a circle at two points, and the segment of the line between these two points is called a chord.

The length of a chord can be calculated using the Pythagorean theorem, given the radius of the circle and the distance between the two points on the circumference. The length of a chord is always less than or equal to the diameter of the circle.

Chords have various applications in geometry, trigonometry, and calculus. For example, in trigonometry, the chord function is defined as the ratio of the length of a chord to the radius of the circle. In calculus, chords are used in the definition of the derivative, which is the slope of the tangent line to a curve at a given point.

According to the given information:

2(x + 2 )=  ( 5 x - 5 )

2x + 4 = 5x -5

9 = 3x

3 = x

The length of FG is = x+2 = 3+2 = 5 units

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The length of FG chord is 5 units.

What bout chord?

A chord is a line segment that joins two points on the circumference of a circle. More specifically, a chord is a straight line that intersects a circle at two points, and the segment of the line between these two points is called a chord.

The length of a chord can be calculated using the Pythagorean theorem, given the radius of the circle and the distance between the two points on the circumference. The length of a chord is always less than or equal to the diameter of the circle.

Chords have various applications in geometry, trigonometry, and calculus. For example, in trigonometry, the chord function is defined as the ratio of the length of a chord to the radius of the circle. In calculus, chords are used in the definition of the derivative, which is the slope of the tangent line to a curve at a given point.

According to the given information:

2(x + 2 )=  ( 5 x - 5 )

2x + 4 = 5x -5

9 = 3x

3 = x

The length of FG is = x+2 = 3+2 = 5 units

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