What is the radius of a circle that has a circumference of 68 cm

Answer:

centre = (5, - 6 ) , radius = 7

Step-by-step explanation:

the equation of a circle in standard form is

(x - h)² + (y - k)² = r²

where (h, k ) are the coordinates of the centre and r is the radius

given

x² + y² - 10x + 12y + 12 = 0 ( subtract 12 from both sides )

x² + y² - 10x + 12y = - 12 ( collect terms in x/ y )

x² - 10x + y² + 12y = - 12

using the method of completing the square

add ( half the coefficient of the x/ y terms )² to both sides

x² + 2(- 5)x + 25 + y² + 2(6)y + 36 = - 12 + 25 + 36

(x - 5)² + (y + 6)² = 49 = 7² ← in standard form

with centre (5, - 6 ) and radius = 7

Answer:

Center = (5, -6)

Radius = 7

Step-by-step explanation:

To find the center and the radius of the circle represented by the given equation, rewrite the equation in standard form by completing the square.

To complete the square, begin by moving the constant to the right side of the equation and collecting like terms on the left side of the equation:

[tex]x^2-10x+y^2+12y=-12[/tex]

Add the square of half the coefficient of the term in x and the term in y to both sides of the equation:

[tex]x^2-10x+\left(\dfrac{-10}{2}\right)^2+y^2+12y+\left(\dfrac{12}{2}\right)^2=-12+\left(\dfrac{-10}{2}\right)^2+\left(\dfrac{12}{2}\right)^2[/tex]

Simplify:

[tex]x^2-10x+(-5)^2+y^2+12y+(6)^2=-12+(-5)^2+(6)^2[/tex]

       [tex]x^2-10x+25+y^2+12y+36=-12+25+36[/tex]

       [tex]x^2-10x+25+y^2+12y+36=49[/tex]

Factor the perfect square trinomials on the left side:

[tex](x-5)^2+(y+6)^2=49[/tex]

The standard equation of a circle is:

[tex]\boxed{(x-h)^2+(y-k)^2=r^2}[/tex]

where:

Comparing this with the rewritten given equation, we get

Therefore, the center of the circle is (5, -6) and its radius is r = 7.

The correct statement regarding the translation in this problem is given as follows:

A. The graph of g(x) is the graph of f(x) shifted up 3 units.

A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.

The four translation rules for functions are defined as follows:

In this problem, we have an addition by 3, hence there is a translation up 3 units.

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The function that matches the graph is of the form:

4cos((pi x)/7) + 1

Graphs of trigonometric functions are graphs used in representing trigonometric functions.

From these graphs, some basic properties such as Amplitude, phase difference, period and vertical shift can be deduced.

From the given graph in the question, it can be seen that the graph crosses the y-axis at it's amplitude (highest point), so its easier to use the cosine relation.

To calculate the midline M:

Use the formula,

M = (maximum + minimum)/2

= (5 + -3)/2 = 2/2 = 1

Vertical shift: It can be seen from the graph that there is a vertical upward shift of 1 unit. C = 1

Amplitude: Maximum value - vertical shift is:

A = 5 - 1 = 4

Period = spacing between repeating patterns. There are 14 units between each peak (peak when x = -14, next peak when x = 0).

k = 2pi/Period;

So: k = 2pi/14 = pi/7

Therefore y = 4cos(pix/7) + 1 is the function that matches the given graph.

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Answer:

[tex]x = 5.3 \: or \: 5 \frac{1}{3} [/tex]

Step-by-step explanation:

[tex]2 \times 2 + 3x - 20 = 0 \: \: \: \: \: \: 4 + 3x - 20 = 0 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:3x = 20 - 4 = 16 \: \: \: \: \: \: \: \: 3x = 16 \: \: divide \: both \: side \: by \: 3 = \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: x = 5.3[/tex]

The value of the algebric  expression [tex]2^{-3} \cdot 3^{-2}$ is $\frac{1}{72}[/tex].

To compute the expression [tex]2^{-3} \cdot 3^{-2}[/tex], we can simplify each term separately and then multiply the results.

First, let's simplify [tex]2^{-3}[/tex]. The exponent -3 indicates that we need to take the reciprocal of the base raised to the positive exponent 3. Therefore, [tex]2^{-3} = \frac{1}{2^3} = \frac{1}{8}[/tex].

Next, let's simplify 3^{-2}. Similar to before, the exponent -2 means we need to take the reciprocal of the base raised to the positive exponent 2. So, [tex]3^{-2} = \frac{1}{3^2} = \frac{1}{9}[/tex].

Now that we have simplified both terms, we can multiply them together: [tex]\frac{1}{8} \cdot \frac{1}{9}[/tex]. When multiplying fractions, we multiply the numerators together and the denominators together. So, [tex]\frac{1}{8} \cdot \frac{1}{9} = \frac{1 \cdot 1}{8 \cdot 9} = \frac{1}{72}[/tex].

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Answer:

39°

Step-by-step explanation:

the sum of the 3 angles in a triangle = 180°

let the other angle be x , then

x + 54° + 87° = 180°

x + 141° = 180° ( subtract 141° from both sides )

x = 39°

that is the other angle is 39°

In a triangle, the sum of all angles is always 180°. To find the third angle in a scalene triangle where two angles are known, subtract the known angles from 180°. In this case, subtracting 54° and 87° from 180° gives a third angle of 39°.

The question refers to finding the third angle in a scalene triangle, where we know two of the angles. A scalene triangle is a triangle where all three sides are of a different length, and therefore all three angles are also different. The sum of the angles in any triangle is always 180°.

To find the third angle in the triangle, you can use the equation: Angle C = 180° - Angle A - Angle B.

So, we subtract the known angles from 180°: Angle C = 180° - 54° - 87° = 39°.

Therefore, the third angle in this scalene triangle is 39°.

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The cost of Donna's club membership exhibits a non-proportional relationship over time.

The cost of Donna's club membership can be analyzed to determine whether it exhibits a proportional or non-proportional relationship over time.

In this scenario, Donna pays a monthly fee of $80, along with a yearly membership fee of $60.50. To assess the proportionality, we can examine how the cost changes relative to time.

In a proportional relationship, the cost would increase or decrease at a constant rate. For example, if the monthly fee remained constant, the total cost would be directly proportional to the number of months of membership.

However, in this case, the presence of a yearly membership fee indicates a non-proportional relationship.

The yearly membership fee of $60.50 is a fixed cost that Donna incurs only once per year, regardless of the number of months she remains a member.

As a result, the cost is not directly proportional to time. Instead, it has a fixed component (the yearly fee) and a variable component (the monthly fee).

In summary, the cost of Donna's club membership exhibits a non-proportional relationship over time. While the monthly fee is a constant amount, the yearly membership fee introduces a fixed cost that is independent of the duration of her membership.

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The equation of a parabola with its vertex at the origin includes a negative coefficient 'a', the parabola opens downward. option B.

The equation y = ax² represents a parabola with its vertex at the origin. In this case, if the coefficient 'a' is negative, it determines the direction in which the parabola opens.

When 'a' is negative, the parabola opens downward. This means that the vertex, which is at the origin (0, 0), represents the highest point on the graph, and the parabola curves downward on both sides.

To understand this concept, let's consider the basic equation y = x², which represents a standard upward-opening parabola. As 'a' increases, the parabola becomes narrower. Conversely, when 'a' becomes negative, it flips the parabola upside down, resulting in a downward-opening parabola.

For example, if we have the equation y = -x², the negative coefficient causes the parabola to open downward. The vertex remains at the origin, but the shape of the parabola is now inverted.

In summary, when the equation of a parabola with its vertex at the origin includes a negative coefficient 'a', the parabola opens downward. This can be visually represented as a U-shape curving downward from the origin. So Optyion B is correct.

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Answer:

9.5 years

Step-by-step explanation:

P(t) = P(t)

106.67t+4763.67=308.8t+2844.18

Minus 106.67t on both sides

4763.67=202.13t+2844.18

Minus 2844.18 on both sides

1919.49=202.18t

Solve for t

t=9.4963...

t=9.5 years

The exact values of csc(theta) and cot(theta) are: csc(theta) = 4√7/7

cot(theta) = -3√7/7.

To find the exact values of csc(theta) and cot(theta), given that cos(theta) = -3/4 and theta is an angle in quadrant two, we can use the trigonometric identities and the Pythagorean identity.

We know that cos(theta) = adjacent/hypotenuse, and in quadrant two, the adjacent side is negative. Let's assume the adjacent side is -3 and the hypotenuse is 4. Using the Pythagorean identity, we can find the opposite side:

[tex]opposite^2 = hypotenuse^2 - adjacent^2opposite^2 = 4^2 - (-3)^2opposite^2 = 16 - 9opposite^2 = 7[/tex]

opposite = √7

Now we have the values for the adjacent side, opposite side, and hypotenuse. We can use these values to find the values of the other trigonometric functions:

csc(theta) = hypotenuse/opposite

csc(theta) = 4/√7

To rationalize the denominator, we multiply the numerator and denominator by √7:

csc(theta) = (4/√7) * (√7/√7)

csc(theta) = 4√7/7

cot(theta) = adjacent/opposite

cot(theta) = -3/√7

To rationalize the denominator, we multiply the numerator and denominator by √7:

cot(theta) = (-3/√7) * (√7/√7)

cot(theta) = -3√7/7

Therefore, the exact values of csc(theta) and cot(theta) are:

csc(theta) = 4√7/7

cot(theta) = -3√7/7

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Answer:

Step-by-step explanation:

add it then subtract the value

The shaded fraction of the shape is 2/3.

To determine the fraction of the shape that is shaded, we need to compare the shaded area to the total area of the shape.

1. Identify the shaded region in the shape. In this case, we have a shape with some part shaded.

2. Calculate the area of the shaded region. Given the dimensions provided, the area of the shaded region is determined by multiplying the length and width of the shaded part. In this case, the dimensions are 18 mm and 10 mm, so the area of the shaded region is (18 mm) × (10 mm) = 180 mm².

3. Calculate the total area of the shape. The total area of the shape is determined by multiplying the length and width of the entire shape. In this case, the dimensions are 18 mm and 12 mm, so the total area of the shape is (18 mm) × (12 mm) = 216 mm².

4. Determine the fraction. To find the fraction, divide the area of the shaded region by the total area of the shape: 180 mm² ÷ 216 mm². Simplifying this fraction gives us 5/6.

5. Convert the fraction to its simplest form. By dividing both the numerator and denominator by their greatest common divisor, we get the simplified fraction: 2/3.

Therefore, the fraction of the shape that is shaded is 2/3.

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The value of the expression (-2)(3)º(4) - 2 is -164.

Based on the answer choices provided, none of the options matc.

To solve the expression (-2)(3)º(4)-2, we need to follow the order of operations, which is parentheses, exponents, multiplication, and subtraction.

Let's break down the expression :

(-2)(3)º(4) -2

First, we calculate the exponent:

(-2)(81) - 2

Next, we perform the multiplication:

-162 - 2

Finally, we subtract:

-164

Therefore, the value of the expression (-2)(3)º(4) - 2 is -164.

Based on the answer choices provided, none of the options match the value of -164.

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The future value of the loan, or the total amount due at the end of 6 months, is $9,450.

We can use the following formula to calculate the future value of a loan:

[tex]A = P + P * r * t[/tex]

Given: $9,000 principal (P).

10% interest rate (r) = 0.10

6 months is the time period (t).

When we enter these values into the formula, we get:

A=9,000+9,000*0.10*6/12

First, compute the interest portion:

Interest is calculated as = 9,000*0.10*6/12=450

We may now calculate the future value:

A=9,000+450=9,450

As a result, the loan's future value, or the total amount payable in 6 months, is $9,450.

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Answer:

B.     [tex] 9^{\frac{1}{8}x} [/tex]

Step-by-step explanation:

[tex] \sqrt[4]{9}^{\frac{1}{2}x} = [/tex]

[tex] = ({9}^{\frac{1}{4}})^{\frac{1}{2}x} [/tex]

[tex] = 9^{\frac{1}{4} \times \frac{1}{2}x} [/tex]

[tex] = 9^{\frac{1}{8}x} [/tex]

The system has no solution. Option C is correct.

To solve the given system of equations using row operations, we can write the augmented matrix and perform Gaussian elimination. The augmented matrix for the system is:

1  1 -1 |  6

3 -1  1 |  2

1  4  2 | -34

We'll use row operations to transform the augmented matrix into row-echelon form or reduced row-echelon form. Let's proceed with the row operations:

R2 = R2 - 3R1:

1  1 -1 |  6

0 -4  4 | -16

1  4  2 | -34

R3 = R3 - R1:

1  1 -1 |  6

0 -4  4 | -16

0  3  3 | -40

R3 = R3 + (4/3)R2:

1  1 -1 |  6

0 -4  4 | -16

0  0  0 |  -4

Now, we can rewrite the augmented matrix in equation form:

x +  y -  z =  6

      -4y + 4z = -16

              0 = -4

From the last equation, we can see that it leads to a contradiction (0 = -4), which means the system is inconsistent. Therefore, the system has no solution.

The correct answer is (C) This system has no solution.

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Answer: y = 12.6

Step-by-step explanation:

Since x = 2 and y = 6.3 * x, y = 6.3 * 2.

6.3 * 2 is equal to 12.6, so y is 12.6.

Answer:

y = 12.6

Step-by-step explanation:

y = 6.3x                     x = 2

Solve for y.

y = 6.3(2)

y = 12.6

So, the answer is 12.6

118 be the median of a positively skewed distribution with a mean of 122. Option D.

To determine which of the given values could be the median of a positively skewed distribution with a mean of 122, we need to consider the relationship between the mean, median, and skewness of a distribution.

In a positively skewed distribution, the tail of the distribution is stretched towards higher values, meaning that there are more extreme values on the right side. Consequently, the median, which represents the value that divides the distribution into two equal halves, will typically be less than the mean in a positively skewed distribution.

Let's examine the given values in relation to the mean:

A. 122: This value could be the median if the distribution is perfectly symmetrical, but since the distribution is positively skewed, the median is expected to be less than the mean. Thus, 122 is less likely to be the median.

B. 126: This value is higher than the mean, and since the distribution is positively skewed, it is unlikely to be the median. The median is expected to be lower than the mean.

C. 130: Similar to option B, this value is higher than the mean and is unlikely to be the median. The median is expected to be lower than the mean.

D. 118: This value is lower than the mean, which is consistent with a positively skewed distribution. In such a distribution, the median is expected to be less than the mean, so 118 is a plausible value for the median.

In summary, among the given options, (118) is the most likely value to be the median of a positively skewed distribution with a mean of 122. So Option D is correct.

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Note the complete question is

If the mean of a positively skewed distribution is 122, which of these values could be the median of the distribution?

A. 122

B. 126

C. 130

D. 118

Answer:

Step-by-step explanation:

Consider functions fand g below g(x)=-x^2+2x+4 is option D.As x approaches infinity, the value of f(x) decreases and the value of g(x) increases.

The limit of a function, as x approaches infinity, is defined as a certain value if the function approaches the same value as x approaches infinity from both sides. The behavior of a function, as x approaches infinity, is determined by the function's rate of increase or decrease and the value of the function at x = 0.

The value of f(x) and g(x) will both increase as x approaches infinity in situation C. This implies that the functions are continuously increasing without bound, i.e., the function's value at any given point will always be greater than the previous point. Consider the example of f(x) = x² and g(x) = 2x. As x approaches infinity, f(x) and g(x) will both continue to increase indefinitely.

This is because x² and 2x are both monotonically increasing functions.As x approaches infinity, the value of f(x) decreases and the value of g(x) increases in situation D. As the value of f(x) approaches infinity, it will eventually reach a point where its rate of increase slows and the function will start to decrease.

On the other hand, g(x) will continue to increase because its rate of increase is faster than f(x) and does not slow down as x approaches infinity. Consider the example of f(x) = 1/x and g(x) = x². As x approaches infinity, f(x) decreases towards zero while g(x) continues to increase without bound.The correct answer is d.

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The equation of the function is y = 1/(x + 3) - 1

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

The reciprocal function shifted down one unit and left three units

The equation of the reciprocal function is represented as

y = 1/x

When shifted down one unit, we have

y = (1/x) - 1

When shifted left three units, we have

y = 1/(x + 3) - 1

Hence, the equation of the function is y = 1/(x + 3) - 1

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Answer:

I pass my classes.

Step-by-step explanation:

I had to add this sentence or else it wouldnt allow me to send it.

Answer:

In math, the word "if" can be used for piecewise functions. In piecewise functions, you can see equations where f(x) = x+3 IF x>0 and f(x) = -x IF x<0.

Answer:

|5x - 1| < 1

-1 < 5x - 1 < 1

0 < 5x < 2

0 < x < 2/5

Answer:

X=11 trust me on my mom

In triangle PAD, using the Pythagorean theorem, we find AD = 5√2. Given that ∠PAQ's opposite side is PQ, which equals 3, we have sin∠PAQ = PQ/AQ = √2/10.

In square ABCD, we are given that points P and Q are on lines BC and CD respectively such that BP=4 and PC=1, DQ=4 and QC=1. Considering triangle PAD, it is a right triangle in the given square, and, using the Pythagorean theorem, we can find the hypotenuse AD as AD = √(5² + 5²) = 5√2. The same reasoning, AD = AQ.

Because ∠PAQ is the angle we are interested in finding the sine of, we know that sin∠PAQ = opposite/hypotenuse. In this case, the opposite side is PQ which we determine is 3 using the given distances (PC+QC). So, sin∠PAQ = PQ/AQ = 3/(5√2) = √2/10. Thus, the sine of angle PAQ is √2/10.

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The surface area and volume of the square pyramid is 96 squared centimeter and 48 cubic centimeters respectively.

The surface area of a square pyramid is expressed as:

SA = [tex]a^2 + 2a \sqrt{\frac{a^2}{4}+h^2 }[/tex]

The volume of a square pyramid is expressed as:

Volume = [tex]a^2*\frac{h}{3}[/tex]

Where a is the base edge and h is the height.

From the figure a = 6cm

First, we determine the h, using pythagorean theorem:

h² = 5² - (6/2)²

h² = 5² - 3²

h² = 25 - 9

h² = 16

h = √16

h = 4 cm

Solving for surface area:

SA = [tex]a^2 + 2a \sqrt{\frac{a^2}{4}+h^2 }[/tex]

[tex]= a^2 + 2a \sqrt{\frac{a^2}{4}+h^2 }\\\\= 6^2 + 2*6 \sqrt{\frac{6^2}{4}+4^2 }\\\\= 36 + 12 \sqrt{\frac{36}{4}+16 }\\\\= 36 + 12 (5)\\\\= 36 + 60\\\\= 96 cm^2[/tex]

Solving for the volume:

Volume = [tex]a^2*\frac{h}{3}[/tex]

[tex]= a^2*\frac{h}{3}\\\\= 6^2*\frac{4}{3}\\\\= 36*\frac{4}{3}\\\\=\frac{144}{3}\\\\= 48 cm^3[/tex]

Therefore, the volume is 48 cubic centimeters.

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Answer:

The marginal pdf of X is fX(x) = x + 1/2

We need to integrate the joint pdf over the given region. This can be done as follows:

P(X < 1/2, Y > 1/4) = ∫∫[x + y] dx dy over the region 0 ≤ x ≤ 1/2 and 1/4 ≤ y ≤ 1

= ∫[x + y] dy from y = 1/4 to 1 ∫ dx from x = 0 to 1/2

= ∫[x + y] dy from y = 1/4 to 1 (1/2 - 0)

= ∫[x + y] dy from y = 1/4 to 1/2 + ∫[x + y] dy from y = 1/2 to 1 (1/2 - 0)

= ∫[x + y] dy from y = 1/4 to 1/2 + ∫[x + y] dy from y = 1/2 to 1/2

= ∫[x + y] dy from y = 1/4 to 1/2

= [(x + y)y] evaluated at y = 1/4 and y = 1/2

= [(x + 1/2)(1/2) - (x + 1/4)(1/4)]

= (1/2 - 1/4)(1/2) - (1/4 - 1/8)(1/4)

= (1/4)(1/2) - (1/8)(1/4)

= 1/8 - 1/32

= 3/32

Therefore, P(X < 1/2, Y > 1/4) = 3/32.

The marginal pdfs of X and Y can be done as follows:

For the marginal pdf of X:

fX(x) = ∫[x + y] dy over the range 0 ≤ y ≤ 1

= [xy + (1/2)y^2] evaluated at y = 0 and y = 1

= (x)(1) + (1/2)(1)^2 - (x)(0) - (1/2)(0)^2

= x + 1/2

Therefore, the marginal pdf of X is fX(x) = x + 1/2.

For the marginal pdf of Y:

fY(y) = ∫[x + y] dx over the range 0 ≤ x ≤ 1

= [xy + (1/2)x^2] evaluated at x = 0 and x = 1

= (y)(1) + (1/2)(1)^2 - (y)(0) - (1/2)(0)^2

= y + 1/2

Therefore, the marginal pdf of Y is fY(y) = y + 1/2.

To determine if X and Y are independent, we need to check if the joint pdf factors into the product of the marginal pdfs.

fX(x) * fY(y) = (x + 1/2)(y + 1/2)

However, this is not equal to the joint pdf f(x, y) = x + y. Therefore, X and Y are not independent.

To derive the conditional pdf of X given Y = y, we can use the formula:

f(xy) = f(x, y) / fY(y)

Here, we have f(x, y) = x + y from the joint pdf, and fY(y) = y + 1/2 from the marginal pdf of Y.

Therefore, the conditional pdf of X given Y = y is:

f(xy) = (x + y) / (y + 1/2)

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The side length that prove a right angle triangle is √2, √3 and √5.

A right angle triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.

Therefore, a right angle triangle can be proved by using the Pythagoras's theorem as follows:

Hence,

c² = a² + b²

where

Therefore,

(√2)² + (√3)² = (√5)²

Hence, the right angle triangle is the triangle with sides √2, √3 and √5.

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