Solve the equation. ㏒₃(1/9)=2x-1
Enter your answer in the box. Enter a fractional answer as a simplified fraction.

The given set of questions are solved under the condition of  parametric equations x(t)=sin(3t) and y(t)=cos(3t) .

Hence, the length of the curve from t= 0 to t= π is 3π.

Now,

A. To evaluate the velocity, we need to perform the derivative of x(t) and y(t) concerning t.

x'(t) = 3cos(3t)

y'(t) = -3sin(3t)

Therefore,  the velocity vector is

v(t) = <3cos(3t), -3sin(3t)>

B. To define the acceleration, we need to evaluate  the derivative of v(t) concerning t.

a(t) = v'(t) = <-9sin(3t), -9cos(3t)>

C. To describe  the speed, we need to calculate  the magnitude of the velocity vector.

|v(t)| = √((3cos(3t))² + (-3sin(3t))²)

= 3

D. In order to find the number of times at which the particle stops, to find when the speed is equal to zero.

|v(t)| = 0 when cos(3t) = 0

sin(3t) = 0.

Therefore,

cos(3t) = 0 when t = (π/6) + (nπ/3),

here n = integer.

sin(3t) = 0 when t = (nπ/3),

here n = integer.

E. To calculate the length of the curve from t=0 to t=π by performing  calculus

L = ∫[a,b] √((dx/dt)² + (dy/dt)²) dt

Therefore, a=0 and b=π.

L = ∫[0,π] √((3cos(3t))² + (-3sin(3t))²) dt

 = ∫[0,π] 3 dt

 = 3π

The given set of questions are solved under the condition of  parametric equations x(t)=sin(3t) and y(t)=cos(3t) .

Hence, the length of the curve from t=0 to t=π is 3π.

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The complete question is

A particle moves along a path in the xy-plane. the path is given by

the parametric equations x(t)=sin(3t) and y(t)=cos(3t), help with

steps A-E

a. Find the velocity

b. Find the acceleration.

c. Find the speed and simplify your answer completely.

d. Find any times at which the particle stops. Thoroughly explain your answer.

e. Use calculus to  find the length of the curve from t=0 to t = π , show your work.

The values that satisfy the inequality -1/2x + 5>7 are -8 and -6.

To determine which values from the set {-8, -6, -4, -1, 0, 2} satisfy the inequality -1/2x + 5 > 7, we first need to isolate the variable x. Start by subtracting 5 from both sides of the inequality:

-1/2x > 2

Now, multiply both sides by -2 to solve for x. Remember that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign:

x < -4

Now we can see that the inequality is asking for all values of x that are less than -4. Looking at the given set {-8, -6, -4, -1, 0, 2}, we can identify the values that satisfy this condition:

-8 and -6 are the values that are less than -4.

Therefore, the values from the set {-8, -6, -4, -1, 0, 2} that satisfy the inequality -1/2x + 5 > 7 are -8 and -6.

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The number of bracelets that can be made using all the colors one time only is 720.

Given that Diana is making bracelet with 6 different colors we need to find the number of bracelets that can be made using all the colors one time only,

Since there are 6 beads so, the number of bracelets can be made = 6!

= 720

Hence the number of bracelets that can be made using all the colors one time only is 720.

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Answer:

89

Step-by-step explanation:

construct the formula to obtain the total of marks in first 9 tests:

79 = x/9

79 * 9 = 711

x = 711

so to get 80% in the next test we construct this formula:

80 = 711 + y / 9 + 1

the value we get for y is 89

to confirm the answer, 711+89/10 gives us 80, which is the correct value we want

To calculate the total dinner cost, we need to add the bill amount to the tip amount.

The tip amount is 18% of the bill amount:

Tip = 0.18 x $57.50 = $10.35

Therefore, the total dinner cost is:

Total Cost = Bill Amount + Tip Amount

Total Cost = $57.50 + $10.35

Total Cost = $67.85

So, the total dinner cost including the 18% tip is $67.85.

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Answer: 7

Step-by-step explanation:

Answer:

y=|x-2|-2

Step-by-step explanation:

Go onto desmos and you can ask it to graph an equation to test your answers.

To find the vector a = AB, we subtract the coordinates of point A from the coordinates of point B:

a = B - A = (-2,4,3) - (4,-6,-3) = (-2-4, 4+6, 3+3) = (-6, 10, 6)
The vector a can be written as a column vector with angle brackets: a = < -6, 10, 6 >.
To find the vector AB (a), we need to subtract the coordinates of point A from the coordinates of point B. Here's the calculation:

a = B - A
a = (-2, 4, 3) - (4, -6, -3)
Now, subtract each corresponding coordinate:

a = (-2 - 4, 4 - (-6), 3 - (-3))
a = (-6, 10, 6)
So, the vector AB (a) is <-6, 10, 6>.

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Sure, I'd be happy to help you with these questions!

a) To calculate the total number of possible 6-digit security numbers, we can use the permutation formula:

nPr = n! / (n-r)!

where n is the total number of digits available (9) and r is the number of digits we are selecting (6).

So, the number of possible 6-digit security numbers without any restrictions is:

9P6 = 9! / (9-6)! = 9! / 3! = 9 x 8 x 7 x 6 x 5 x 4 = 60,480

Therefore, there are 60,480 possible 6-digit security numbers that can be made with the digits 1-9 without repeating any digits.

b) If the first digit cannot be a one, we are left with 8 choices for the first digit (since we cannot use 1) and 8 choices for the second digit (since we have already used one digit). For the remaining 4 digits, we still have 7 choices for each digit, since we cannot repeat any digits.

Using the permutation formula again, the number of possible 6-digit security numbers with the first digit not being one is:

8 x 8 x 7 x 7 x 7 x 7 = 1,322,496

Therefore, there are 1,322,496 possible 6-digit security numbers that can be made with the digits 1-9 without repeating any digits, where the first digit is not one.

c) To create an odd number, the last digit must be an odd number, which means we have 5 choices for the last digit (1, 3, 5, 7, or 9). For the first digit, we cannot use 0 or 1, so we have 7 choices. For the remaining 4 digits, we still have 8 choices for each digit (since we can use any digit).

Using the permutation formula again, the number of possible 6-digit security numbers with the last digit being odd is:

7 x 8 x 8 x 8 x 8 x 5 = 7,1680

Therefore, there are 7,1680 possible 6-digit security numbers that can be made with the digits 1-9 without repeating any digits, where the last digit is odd.

d) To create a number greater than 300,000, the first digit must be 3, 4, 5, 6, 7, 8, or 9. If the first digit is 3, we have 7 choices for the first digit (3, 4, 5, 6, 7, 8, or 9). For the remaining 5 digits, we still have 8 choices for each digit.

If the first digit is not 3, we have 6 choices for the first digit (since we cannot use 1 or 2). For the remaining 5 digits, we still have 8 choices for each digit.

Using the permutation formula again, the number of possible 6-digit security numbers greater than 300,000 is:

7 x 8 x 8 x 8 x 8 x 8 + 6 x 8 x 8 x 8 x 8 x 8 = 2,526,720

Therefore, there are 2,526,720 possible 6-digit security numbers that can be made with the digits 1-9 without repeating any digits, where the number is greater than 300,000.

e) To create a number greater than 750,000, the first digit must be 8 or 9. If the first digit is 8, we have 2 choices for the first digit (8 or 9). For the remaining 5 digits, we still have 8 choices for each digit.

If the first digit is 9, we only have one choice for the first digit (9). For the remaining 5 digits, we still have 8 choices for each digit.

Using the permutation formula again, the number of possible 6-digit security numbers greater than 750,000 is:

2 x 8 x 8 x 8 x 8 x 8 + 1 x 8 x 8 x 8 x 8 x 8 = 262,144

Therefore, there are 262,144 possible 6-digit security numbers that can be made with the digits 1-9 without repeating any digits, where the number is greater than 750,000.

If the surface area of a can with a particular volume is minimized when the height of the can is 22 cm, we would expect the radius of the can to be the same as the height, given that a cylinder has the smallest surface area when its height and radius are equal.

The surface area of a can with height h and radius r can be given by the formula:

A = 2πr² + 2πrh

The volume of the can is given by:

V = πr²h

If we differentiate the surface area with respect to r and equate it to zero to find the critical point, we get:

dA/dr = 4πr + 2πh(dr/dr) = 0

Simplifying this expression, we get:

2r + h = 0

Since we know that the height of the can is 22 cm, we can substitute h = 22 in the equation to get:

2r + 22 = 0

Solving for r, we get:

r = -11

Since the radius of the can cannot be negative, we discard this solution. Therefore, the radius of the can should be equal to its height, which is 22 cm.

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The benchmark fractions are the most common fraction.

Such as 1/2, 0, 3/8 etc.

Mixed fractions are a type of fraction in which there is a whole number part and a fractional part. for example 17/3 would be 5 2/3 as a mixed fraction

sin 2u = -1/2, cos 2u = -1/2, tan 2u = 1, because cot u = sqrt(2) and the range of u is between pi and 3pi/2.

Given cot u = sqrt(2) and the range of trigonometric of u, we can determine the values of sine, cosine, and tangent of 2u using the double-angle formulas. First, we can find the value of cot u by using the fact that cot u = 1/tan u, which gives us tan u = 1/sqrt(2). Since u is in the third quadrant (i.e., between pi and 3pi/2), sine is negative and cosine is negative.

Using the double-angle formulas, we can express sin 2u and cos 2u in terms of sin u and cos u as follows:

sin 2u = 2sin u cos u

cos 2u =[tex]cos^2[/tex] u - [tex]sin^2[/tex] u

Substituting the values of sine and cosine of u, we get:

sin 2u = 2*(-sqrt(2)/2)*(-sqrt(2)/2) = -1/2

cos 2u = (-sqrt(2)/2[tex])^2[/tex] - (-1/2[tex])^2[/tex] = -1/2

To find the value of tangent of 2u, we can use the identity:

tan 2u = (2tan u)/(1-[tex]tan^2[/tex] u)

Substituting the value of tan u, we get:

tan 2u = (2*(1/sqrt(2)))/(1 - (1/sqrt(2)[tex])^2[/tex]) = 1

Therefore, sin 2u = -1/2, cos 2u = -1/2, and tan 2u = 1.

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The probability of the event of having ace of diamonds and ace of clubs is 1/2704

A probability event can be defined as a set of outcomes of an experiment. In other words, an event in probability is the subset of the respective sample space.

In a standard deck of cards, we have 52 cards of which 4 are aces. The probability of drawing the first ace of diamonds will be 1/52. Shuffling the card again, the probability of drawing having an ace of club will be another 1/52 since the card was replaced and shuffled.

To determine the probability of the two events occurring will be

P = (1/52 * 1/52) = 1 / 2704 = 0.0003698

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The expression that is equivalent to 1/4(8 - 6x + 12) is 2 - 3x/2 + 6

Algebraic expressions are simply defined as those mathematical expressions that are composed of terms, variables, their coefficients, their factors and constants.

These mathematical expressions are also comprised of arithmetic operations.

These operations are listed thus;

From the information given, we have that;

1/4(8 - 6x + 12)

expand the bracket, we have;

8 - 6x + 12/4

Divide in group, we have;

8/4 - 6x/4 + 12/4

Divide the values

2 - 3x/2 + 6

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Answer:

j: 0, m: (-4)

Step-by-step explanation:

RECALL:

Rational function is the func. expressed by polynomials p(x) and q(x) as:

p(x)/q(x) where q(x) is non-zero

j(m+4) must be non zero, or

j(m+4)≠0

j≠0 and m+4≠0

j≠0 and m≠(-4)

The function that represents the relationship between x and y is y = 3/20 x

Since y varies directly with x, we can write the relationship between x and y as

y = kx

where k is the constant of proportionality.

y = 18/5 when x = 24

Substituting these values into the equation, we get:

18/5 = k(24)

Simplifying this equation, we get:

k = (18/5) / 24

k = (18/5 × 24)

k = 18/120

We can simplify this expression to:

k = 3/20

Therefore, the function that represents the relationship between x and y is y = 3/20 x

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The average rate of change is -0.1295, under the condition the given exponential decay function is f (t) = 2(0. 95).

In order to find the average rate of change from x=0 to x=4 for the given exponential decay function [tex]f(t) = 2(0.95)^{t}[/tex], we need to find the slope of the line that passes through the points (0,f(0)) and (4,f(4)).

f(0) = 2(0.95)⁰ = 2

f(4) = 2(0.95)⁴ ≈ 1.482

The slope of the line passing through these two points is:

(f(4) - f(0))/(4 - 0)

= (1.482 - 2)/4

≈ -0.1295

Therefore, the average rate of change from x=0 to x=4 is approximately -0.1295.

An exponential decay function is a form of a function that reduces at a constant rate over time. It is a type of  mathematical model used to present many real-world phenomena such as radioactive decay, population growth, and the depreciation of assets.

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There are 26 zeros in this number when it is written in standard notation. The correct answer is option (A). The mass of the sun is estimated to be 1.9891 x 10³⁰kg. To determine the number of zeros in this number when written in standard notation, we need to first convert it to standard form.

In standard form, the number is expressed as a decimal between 1 and 10 multiplied by a power of 10. To convert the given number to standard form, we move the decimal point 30 places to the right because the exponent is positive 30. This gives us 1989100000000000000000000000000. As we can see, there are 27 digits in this number. Therefore, there are 27-1=26 zeros in this number when it is written in standard notation.

In conclusion, the answer is A, 26. This type of question is commonly asked in science and engineering, where large or small numbers are expressed in scientific notation for convenience. Understanding how to convert between scientific notation and standard form is important for anyone studying or working in these fields.

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Based on the information, the three numbers are 14, 34, and 70.

Based on the information, the second number = 3x - 8

The third number is five times the first number, which can be written as:

third number = 5x

The sum of the three numbers is 118, so we can write an equation:

x + (3x - 8) + 5x = 118

9x - 8 = 118

Adding 8 to both sides:

9x = 126

x = 236 / 914

Now we can use this value of x to find the other two numbers:

second number = 3x - 8 = 3(14) - 8 = 34

third number = 5x = 5(14) = 70

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The planes intersect at the point (-19/21, -11/14, 1).

To determine how, if at all, the planes intersect, we need to solve the system of equations given by the three planes:

[2T/3A] 2x + 2y + z - 10 = 0

5x + 4y - 4z = 13

3x – 2z + 5y - 6 = 0

We can use elimination to solve this system. First, we can eliminate z from the second and third equations by multiplying the second equation by 2 and adding it to the third equation:

5x + 4y - 4z = 13

6x - 4z + 10y - 12 = 0

11x + 14y - 12 = 0

Next, we can eliminate z from the first and second equations by multiplying the first equation by 2 and subtracting the second equation from it:

4x + 4y + 2z - 20 = 0

-5x - 4y + 4z = -13

9x - y - 6z - 20 = 0

Now we have two equations in three variables. To eliminate y, we can multiply the second equation by 14 and subtract it from the first equation:

11x + 14y - 12 = 0

-70x - 56y + 56z = -182

-59x - 42z - 12 = 0

Finally, we can substitute this expression for x into one of the previous equations to find z:

3(59/42)z - 12/42 - 2y - 10 = 0

177z - 60 - 84y - 420 = 0

177z - 84y - 480 = 0

Now we have two equations in two variables, z and y. We can solve for y in terms of z from the second equation:

y = (177/84)z - (480/84)

Substituting this expression for y into the third equation, we can solve for z:

177z - 84[(177/84)z - (480/84)] - 480 = 0

177z - 177z + 480 - 480 = 0

This equation simplifies to 0=0, which means that z can be any value. Substituting z=1 into the expression for y, we get:

y = (177/84)(1) - (480/84) = -11/14

Substituting z=1 and y=-11/14 into the expression for x, we get:

x = (59/42)(1) - (12/42) + 2(-11/14) + 10 = -19/21

Therefore, the planes intersect at the point (-19/21, -11/14, 1).

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Yes. The null hypothesis is that the changes in the course did not have a significant impact on the DFW rate. The chi-squared statistic is 44.63, with 2 degrees of freedom, and the p-value is less than 0.001.

1. Yes, it is likely that the changes in the gateway course had an impact on the DFW rate, as the rate decreased from 42.1% in Year 1 to 19.4% in Year 3.

2. The null hypothesis is that the changes in the course did not have a significant impact on the DFW rate, while the alternative hypothesis is that the changes did have a significant impact on the rate.

3. The chi-squared statistic is 44.63, with 2 degrees of freedom, and the p-value is less than 0.001. This indicates that there is a significant relationship between the year the course was given and the DFW rate, providing evidence to reject the null hypothesis in favor of the alternative hypothesis that the changes made to the course had a significant impact on the DFW rate.

The p-value of less than 0.001 indicates strong evidence against the null hypothesis, as it is less than the significance level of 0.1.

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The point of intersection between the two given equations is (3, -2).

The problem is asking to find the point of intersection between the two given equations:

y = (1/3)x - 3 ............... (equation 1)

y = -x + 1 ............... (equation 2)

To solve for the intersection point, we can set the two equations equal to each other:

(1/3)x - 3 = -x + 1

Simplifying and solving for x:

(1/3)x + x = 1 + 3

(4/3)x = 4

x = 3

Now that we know x = 3, we can substitute it into either of the two original equations to find y:

Using equation 1: y = (1/3)x - 3 = (1/3)(3) - 3 = -2

Using equation 2: y = -x + 1 = -(3) + 1 = -2

Therefore, the intersection point is (3, -2).

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After plugging the derivatives of f(x) we get, dx f-'(4) = f'(√(2/3)) = 4 + cos(Ф)

To find dx f-'(4), we need to take the derivative of f(x) and then solve for x when f'(x) equals 4.

First, let's find the derivative of f(x):

f'(x) = 6x² + cos(Ф)

Next, we need to solve for x when f'(x) equals 4:

6x² + cos(Ф) = 4

cos(Ф) = 4 - 6x²

Now, we can use the given value of πα d to solve for x:

πα d = -1/2

α = -1/2πd

α = -1/2π(-1)

α = 1/2π

d = -1/2πα

d = -1/2π(1/2π)

d = -1/4

So, we have:

cos(Ф) = 4 - 6x²

cos(πα d) = 4 - 6x²   (substituting in the given value of πα d)

cos(-π/2) = 4 - 6x²    (evaluating cos(πα d))

0 = 4 - 6x²

6x² = 4

x² = 2/3

x = ±√(2/3)

Since we're looking for the derivative at x = 4, we can only use the positive root:

x = √(2/3)

Now, we can plug this value of x back into the derivative of f(x) to find dx f-'(4):

f'(√(2/3)) = 6(√(2/3))² + cos(Ф)

f'(√(2/3)) = 4 + cos(Ф)

dx f-'(4) = f'(√(2/3)) = 4 + cos(Ф)

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The FICA tax will be $3595.5 on an income of $47,000.

Given that the principal amount = $47,000

Given that the FICA is taxed at the percentage of 7.65%

To findout the FICA tax we have to findout the 7.65% of money from the principal money $47,000.

The formula for finding the Y% of money from Z amount is = [tex]\frac{y}{100}[/tex] * Z

From the above formula, we can find the FICA tax.

FICA tax = [tex]\frac{7.65}{100}[/tex] * 47000 = 0.0765 * 47000 = 3595.5.

From the above solution, we can conclude that the FICA tax on an income of $47,000 is $3595.5

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The car depreciated at an annual rate of approximately 45.81%.

In 2016, Dave bought a new car for $15,500, and its current value is $8,400. To find the annual depreciation rate, we'll use the formula A(t) = P(1 ± r)t, where A(t) is the future value, P is the initial value, r is the annual rate, and t is the time in years.

Here, A(t) = $8,400, P = $15,500, and t = 1 (one year). We are solving for r, the annual depreciation rate.

$8,400 = $15,500(1 - r)¹

To isolate r, we'll first divide both sides by $15,500:

$8,400/$15,500 = (1 - r)

0.541935 = 1 - r

Now, subtract 1 from both sides:

-0.458065 = -r

Finally, multiply both sides by -1 to find r:

0.458065 = r

To express r as a percentage, multiply by 100:

0.458065 x 100 = 45.81%

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The car depreciated at an annual rate of 12.2%.

The car depreciated in value over time, so we want to find the rate of decrease. We can use the formula:

A(t) = P(1 - r)t

where A(t) is the current value of the car, P is the original price of the car, r is the annual rate of depreciation, and t is the time elapsed in years.

We can plug in the given values and solve for r:

$8,400 = $15,500(1 - r)⁵

Dividing both sides by $15,500, we get:

0.54 = (1 - r)⁵

Taking the fifth root of both sides, we get:

(1 - r) = 0.878

Subtracting 1 from both sides, we get:

-r = -0.122

Dividing both sides by -1, we get:

r = 0.122

Multiplying by 100 to express as a percentage, we get:

r = 12.2%

Therefore, the car depreciated at an annual rate of 12.2%.

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The law of sines is solved and the triangle is given by the following relation

Given data ,

From the law of sines , we get

a / sin A = b / sin B = c / sin C

a)

C = 135° C = 45₁ B = 10°

So , the measure of triangle is

A/ ( 180 - 35 - 10 ) = A / 35

And , a/ ( sin 135/35 ) = sin 35 / a

On simplifying , we get

a = 36.50

Hence , the law of sines is solved

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Answer:  In bold

Step-by-step explanation:

The formula they gave is a rate

Let's solve for the rate first.

This equation is done for 3 years 2018-2021  that's why ^3

3.55 =  2.90(1+x)³                 >divide both sides by 2.90

1.224 = (1+x)³                         > take cube root of both sides

1.0697 =  1+x

x= .0697

so let's make our generic formula

[tex]y = 2.90(1+.0697)^{t}[/tex]        let t be years and let y=  price  

Let's calculate 2018, so this would be year 0

[tex]y = 2.90(1+.0697)^{0}[/tex]

y=$2.90   this is for 2018

They already gave you 2021 price

y=$3.55   this is for 2021

Rate of increase is .0697 

In 2025

That's 7 years=t

[tex]y = 2.90(1+.0697)^{7}[/tex]

y=$4.65    for 2025

Answer:

13 units

Step-by-step explanation:

To find the distance between the two points, use the distance formula.

[tex]\sqrt{(x-x)^{2}+(y-y)^{2} }[/tex]

Plug in the point values.

[tex]\sqrt{(-8--3)^{2}+(-6-6)^{2} }[/tex]

Simplify the parenthesis.

[tex]\sqrt{(5)^2+(-12)^2}[/tex]

Get rid of the parenthesis.

[tex]\sqrt{25+144}[/tex]

Simplify.

[tex]\sqrt{169}[/tex]

Solve.

13 units

The measure of the radius of the hexagon rounded to the nearest inch is 14 inches.

The problem presents a hexagon with a central angle of 60º, and the task is to calculate its radius. To do so, we can use the trigonometric relationship between the radius, apothem, and an angle. The apothem is a line segment from the center of a polygon perpendicular to one of its sides. For a regular hexagon, the apothem length is equal to the radius, which we want to find.

The trigonometric relationship for this case is cos(30) = a/c, where a is the apothem and c is the radius. By rearranging the equation to solve for c, we get c = a/cos(30).

Substituting the value of 12 inches for the apothem, we get c = 12/cos(30). Using a calculator, we can find that cos(30) = 0.866, so c = 12/0.866 = 13.855 inches.

To round to the nearest whole number, we get c = 14 inches.

Correct Question :

A regular hexagon is shown. What is the measure of the radius, c, rounded to the nearest inch? use the appropriate trigonometric ratio to solve. 6 in. 10 in. 14 in. 24 in.

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The least-squares regression line of height versus age will have a slope of 0.8 .  Was true statement option (2)

The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables, in this case, height and age. A correlation coefficient of 0.8 indicates a strong positive linear relationship between height and age. The slope of the least-squares regression line represents the change in the height of a child for each one-unit increase in age.

Therefore, a slope of 0.8 indicates that for each one-year increase in age, the expected increase in height is 0.8 units. The other options are not correct or relevant based on the given information.

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Full Question: the height and age of each child in a random sample of children was recorded. the value of the correlation coefficient between height and age for the children in the sample was 0.8 . based on the least-squares regression line created from the data to predict the height of a child based on age, which of the following is a correct statement?

responses on average,