From a point P on the circumference of circle O, three chords are drawn meeting the circle at points A, B, and C. Prove that the three points of intersection of the three circles with PA, PB, and PC as diameters, are collinear.

The function g(x) increases faster than the function f(x).

We know, the quadratic function f(x) will be

f(x) = ax² + bx + c

At x = 1, f(x) will be 3

So,  a + b + c = 3 ...(1)

and x = 2, f(x) will be 6

4a + 2b + c = 6 ...  (2)

and, x = 3, f(x) will be 11

9a + 3b + c = 11 ...  (3)

On solving equations we get

a = 1, b = 0, and c = 2

So, the quadratic function will be

f(x) = x² + 2

Now, The exponential function g(x) will be given as

g(x) = abˣ

At x = 1, g(x) will be 3

ab = 3 .....(4)

and, x = 2, g(x) will be 9

ab² = 9 ...(5)

From equations 4 and 5 we get

a = 1 and b = 3

So, the exponential function will be

g(x) = [tex]3^x[/tex]

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200 + 50x = y

this works because you have to add the original cost (200) and then 50 per hour (x) if you do a letter and a number it represents multiplication, then = y  because y is the total cost

The projection of vector u in the direction of vector v is equal to P = - 6.354 i - 5.718 j.

In this problem we need to determine the expression of the projection of vector u in the direction of vector v, whose formula is now introduced:

P = [(u • v) / ||v||²] · v

Where:

If we know that u = - 7 i - 5 j and v = - 10 i - 9 j, then the projection of the vector is:

u • v = 70 + 45

u • v = 115

||v||² = 100 + 81

||v||² = 181

P = (115 / 181) · (- 10 i - 9 j)

P = - (1150 / 181) i - (1035 / 181) j

P = - 6.354 i - 5.718 j

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Answer:

1431

Step-by-step explanation:

The number of unique connections between two phones can be found using the formula for combinations.

Since we want to find the number of ways to choose two phones out of 54 phones, we can use the following formula:

nCk = n! / (k! * (n-k)!)

where n is the total number of phones (54), and k is the number of phones we want to choose (2).

nCk = 54! / (2! * (54-2)!)

= 54! / (2! * 52!)

= (54 * 53) / 2

= 1,431

Therefore, there are 1,431 unique connections that can be made between two phones in the office building.

I swear I hope I did this correctly t-t

Answer: 1431

Step-by-step explanation: i took the quiz

The function f(x) is [tex]\frac{2}{3}x^3 - 76x^2 + 150x + 2.67[/tex].

To find f(x), we need to integrate the given slope (x-1)(2x-150) with respect to x, because the slope of a tangent line to a function is the derivative of that function. A line's slope is a gauge of its steepness. Between any two points on the line, it is calculated as the ratio of the change in the vertical coordinate (rise) to the change in the horizontal coordinate (run).

So, we have:

f'(x) = (x-1)(2x-150)

Integrating both sides with respect to x:

[tex]f(x) = ∫(x-1)(2x-150) dx[/tex]

[tex]f(x) = \int (2x^2 - 152x + 150) dx[/tex]

[tex]f(x) = \frac{2}{3}x^3 - 76x^2 + 150x + C[/tex]

where C is an arbitrary constant of integration.

To determine the value of C, we can use the given condition f(2) = 2:

[tex]f(2) = \frac{2}{3}(2)^3 - 76(2)^2 + 150(2) + C = 2[/tex]

Simplifying:

C = 2 - (8/3) + 304 - 300 = 2.67

Therefore, the function f(x) is:

f(x) = [tex]\frac{2}{3}x^3 - 76x^2 + 150x + 2.67[/tex].

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The equation of a line is : y = -5x + 11

The slope of a line is the measure of the tangent of the angle made by the line with the x-axis. The slope is constant throughout a straight line. The slope-intercept form of a straight line can be given by y = mx + b. The slope is represented by the letter m, and is given by, m = tan θ = (y2 - y1)/(x2 - x1)

The slope of line is:

y = 1/5x -2 , m = 1/5

The slope of the line perpendicular to line is m ' = -1/m , -5

The equation of line having slope m and passing through the point (a, b) is given by y − b = m (x − a)

Therefore, the equation of line having slope m ′ = -5and passing through the point (3,-4) is given by:

y - (-4) = -5(x -3)

y + 4 = -5(x -3)

y + 4 = -5x + 15

y = -5x + 15 -4

y = -5x + 11

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The value of angle x for the two chords intersecting at the center is 50⁰.

The value of angle x is calculated by applying intersecting chord theorem as shown below;

For a vertex inside angle, the angle formed by the intersection of two chords inside a circle is equal to half of the sum of the two arc angles.

The value of angle x for the two chords intersecting at the center is calculated as;

x = ¹/₂ (arc DC + arc AB )

x = ¹/₂ (56⁰ + 44⁰ )

x = 50⁰

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A vector space is a collection of objects, called vectors, that can be added together and multiplied by scalars (usually real numbers or complex numbers) to produce new vectors, while satisfying a set of axioms or rules, such as associativity, commutativity, distributivity, and the existence of a zero vector and additive inverses.

The set of all triples of real numbers with the standard vector addition but with scalar multiplication defined by k(x, y, z) = (k+x, k*y, k*z) is not a vector space. This is because Axiom 8 fails to hold.

Axiom 8 states that 1 * (x, y, z) should equal (x, y, z) for any vector (x, y, z). However, with the given scalar multiplication, 1 * (x, y, z) = (1+x, 1*y, 1*z) = (x+1, y, z), which is not equal to (x, y, z). Therefore, this set does not form a vector space due to the violation of Axiom 8.

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The median wait time for the speed slide is 2 times longer than the median wait time for the wave machine.

No, There is a lot of overlap between the two data sets.

We have,

Box plots:

Speed side

Median = 11

First quartile = 6

Third quartile = 12

IQR = 12 - 6 = 6

Wave machine

Median = 9

First quartile = 8

Third quartile = 14

IQR = 14 - 8 = 6

Now,

Difference between the median.

= 11 - 9

= 2

Now,

From the box plots, the wait time for the wave machine is longer than the speed side.

Thus,

The median wait time for the speed slide is 2 times longer than the median wait time for the wave machine.

No, There is a lot of overlap between the two data sets.

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The surface area of the given cylinder is 200.96 square centimeters.

Given that the radius of the cylinder is 4 cm and the height of the cylinder is also 4 cm,

The surface area of the cylinder can be found using the formula:

SA = 2πr² + 2πrh, where r is the radius of the circular base and h is the height of the cylinder.

Substitute given values into the formula to get:

SA = 2π(4)² + 2π(4)(4)

= 2π(16) + 2π(16)

= 32π + 32π

= 64π

= 64(3.14)

= 200.96

Therefore, the surface area of the cylinder is 200.96 square centimeters.

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The complete question is as follows

What is the surface area of the cylinder?

Here, the radius of the cylinder is 4 cm and the height of the cylinder is 4 cm

Apply the formula SA = 2πr² + 2πrh.

Answer: 39

Step-by-step explanation:

    First, we will list some multiples of 3.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, [tex]\boxed{39}[/tex], 42, 45, 48, 51, 54, etc.

    Next, we will list some multiples of 13.

13, 26, [tex]\boxed{39}[/tex], 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, etc.

    We see that the least common multiple of 3 and 13 is 39. This is the smallest value that shows up in both lists.

    What is a multiple?

A multiple is a number that can be divided with that number without a reminder.

60 ounces of iodine worth 20 cents per ounce must be mixed with 40 ounces of iodine worth 15 cents per ounce to create a mixture worth 18 cents per ounce.

We have,

Let x be the number of ounces of iodine worth 20 cents per ounce that must be mixed.

The total amount of iodine after mixing is x + 40 ounces, and the total value of the mixture is (20x + 15(40)) cents.

The problem can be expressed as the equation:

(20x + 15(40))/(x + 40) = 18

Multiplying both sides by (x + 40) gives:

20x + 600 = 18(x + 40)

Expanding the right side gives:

20x + 600 = 18x + 720

Subtracting 18x and 600 from both sides gives:

2x = 120

Dividing both sides by 2 gives:

x = 60

Therefore,

60 ounces of iodine worth 20 cents per ounce must be mixed with 40 ounces of iodine worth 15 cents per ounce to create a mixture worth 18 cents per ounce.

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The calculated value of the probability P(yellow) is 0.5 i.e. one half

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

Sections = 2

Color = yellow, and green

Using the above as a guide, we have the following:

Yellow = 1

When the yellow section is selected, we have

P(yellow) = yellow/section

The required probability is

P(yellow) = 1/2

Evaluate

P(yellow) = 0.5

Hence, the value is 0.5

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We can conclude that for all integers a and b, if a|b, then a ≤ b.

To prove that for all m and n, if m ≠ n, then m+n ≠ Q, we will use proof by contradiction.

Assume that for some m and n, m ≠ n, and m+n = Q, where Q is a rational number. By the definition of a rational number, Q can be expressed as the ratio of two integers, p and q, where q ≠ 0.

Thus, we have:

m + n = p/q

Multiplying both sides by q, we get:

mq + nq = p

Rearranging, we get:

mq = p - nq

Since p, n, and q are integers, p - nq is also an integer. Therefore, mq is an integer.

But we know that m and n are integers and m ≠ n, which implies that m and n have different prime factorizations. Therefore, mq cannot be an integer, as it would require m and q to have a common factor, which is not possible.

This contradicts our assumption that m+n = Q, and hence, we can conclude that for all m and n, if m ≠ n, then m+n ≠ Q.

To prove that for all integers a and b, if a|b, then a ≤ b, we will use direct proof.

Assume that a and b are integers such that a|b, i.e., there exists an integer k such that b = ak.

To prove that a ≤ b, we need to show that a is less than or equal to k times a, i.e., a ≤ ka.

Dividing both sides of the equation b = ak by a (which is possible as a ≠ 0 since it is a divisor of b), we get:

b/a = k

Since k is an integer, we know that b/a is also an integer. Therefore, a must be less than or equal to b/a.

Multiplying both sides of the inequality a ≤ b/a by a (which is a positive number since a > 0), we get:

[tex]a^2 ≤ ab[/tex]

Since a and b are both positive integers, we know that [tex]a^2 ≤[/tex] ab implies that [tex]a ≤ b[/tex].

Therefore, we can conclude that for all integers a and b, if a|b, then a ≤ b.

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The formula for the nth partial sum of the series n(n-1) is Sn = n(n+1)(2n-1)/6 and the series diverges.

Part 1: The formula for the nth partial sum, Sn, for the series n(n-1), can be found by using the formula for the sum of the first n natural numbers, which is given by Sn = n(n+1)/2. To find the sum of n(n-1), we can rewrite it as n^2 - n and use the formula for the sum of the first n squares, which is given by n(n+1)(2n+1)/6. Therefore, the formula for Sn is Sn = n(n+1)(2n-1)/6.

Part 2: To determine whether the given series converges or diverges, we need to evaluate the limit of Sn as n approaches infinity. Taking the limit of the formula for Sn, we get lim Sn = lim [n(n+1)(2n-1)/6] = lim (2n^3/6) = lim (n^3/3) = inf. Since the limit of Sn is infinity, the series diverges.

In conclusion, the formula for the nth partial sum of the series n(n-1) is Sn = n(n+1)(2n-1)/6 and the series diverges as the limit of Sn approaches infinity. This means that the sum of the series does not converge to a finite value and grows without bound as n increases.

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The effective interest method of amortization is a method used to allocate the cost of a bond over the bond's life, in order to determine the amount of interest expense to be recorded each period.

In the case of Bentley Corporation, since they issued $1,000,000 of 10-year, 8% bonds at 105, this means that they received $1,050,000 in cash from investors.

Since the market rate of interest was 7%, the bonds were sold at a premium, which means that the effective interest rate is less than the stated interest rate of 8%. The effective interest rate is the rate at which the present value of the bond's future cash flows equals the amount of cash received at the time of issuance.

Using the effective interest method of amortization, the premium of $50,000 will be amortized over the life of the bond, reducing the effective interest rate each year. The interest expense recorded on December 31, 2021, the first interest payment date, will be calculated as follows:

$1,050,000 x 7% = $73,500 (effective interest)
$73,500 - $80,000 (stated interest) = -$6,500 (amortization of premium)
$80,000 - $6,500 = $73,500 (interest expense)

The premium of $50,000 will be reduced by $6,500, leaving a balance of $43,500 at the end of the first year. This process will continue each year until the bond matures in 2031.

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The area of the circle with a circumference of 6pi m is 9π square meters.

A circle is simply a closed 2-dimensional curved shape with no corners or edges.

The area of a circle is expressed mathematically as;

A = πr²

The circumference of a circle is expressed as:

C = 2πr

Where r is radius and π is constant pi.

We are given that the circumference of the circle is 6π m, so we can set up the equation and solve for the radius.

C = 2πr

6π = 2πr

Simplifying, we get:

r = 6π/2π

r = 3

Now that we know the radius of the circle, we can use the formula for the area of a circle:

A = πr²

Substituting the value of r, we get:

A = π(3)²

Simplifying, we get:

A = 9π

Therefore, the area is 9π m.

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Answer:

13.65

Step-by-step explanation:

25% + 15% = 35%

35% of 21 = 7.35

21 - 7.35 = 13.65

hope this helps :)

The surface area of the triangular prism is 75 ft squared.

The prism is a triangular base prism. The surface area of the prism can be found as follows:

surface area of the prism = (a + b + c)l + bh

where

Therefore,

a = 2 ft

b = 1.5 ft

c = 2.5 ft

l = 12 ft

Hence,

Surface area of the triangular prism = (2 + 1.5 + 2.5)12 + 2(1.5)

Surface area of the triangular prism = 6(12) + 3

Surface area of the triangular prism = 72 + 3

Surface area of the triangular prism = 75 ft²

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The solution of the given inequality is x less than 3/23.

The given inequality is 8>23x+5.

Subtract 5 on both the sides of an inequality, we get

8-5>23x+5-5

3>23x

Divide 23 on both the sides of an inequality, we get

3/23>x

x<3/23

Therefore, the solution of the given inequality is x less than 3/23.

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The student who cut his time by the greatest percentage is

Student B.

We have,

Students A's time decreased by 0.125

= 0.125 x 100

= 12.5%

Student B's decreased by 1/6 which in decimal is

= 1/6

= 0.16667

= 16.66667%

and, Students C's time decreased by 15%.

= 15/100

= 0.15

Thus, the student who cut his time by the greatest percentage is

Student B.

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The exact area of the figure is 85.54 m².

The coordinates endpoints of the semicircle are (-3,-4) and (6,2).

∴ Length of the diameter of the circle = distance between the end points of diameter = √[6 -(-3))²+(-4-2)²] = 10.81 m

⇒ The radius of the semicircle = 10.81/2 = 5.45 m

∴ The area of the semicircle = πr²/2 = 3.14 × (5.45)² /2= 46.63 m²

Now one side of the rectangle = diameter of the semicircle

                                                   = 10.81 m

The endpoints of another side of the rectangle are (-3,-4) and (-1,-7)

∴  The length of this side = √[-3 -(-1))²+(-4-(-7)²] = 3.6 m.

∴ The area of the rectangle = product of sides = 3.6 × 10.81 = 38.91 m²

The total area of the figure = area of the semicircle + area of the rectangle

                                       = 46.63 + 38.91 =85.54 m²

Hence, the exact area of the figure is 85.54 m².

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Therefore, the rate of change of revenue per unit when the cost per unit is changing by $12 and the revenue is $4500 is approximately 12.444 dollars per unit of revenue per dollar increase in cost.

The rate of change of revenue per unit when the cost per unit is changing by $12 and the revenue is $4500, we need to find dR/dC when C changes by 12 and R is 4500.

From the given equation, we know that:

C = [tex]R^2/112,000 + 5807[/tex]

Taking the derivative of both sides with respect to R, we get:

dC/dR = 2R/112,000

Solving for dR/dC, we get:

dR/dC = 1/(dC/dR) = 112,000/(2R)

When the cost per unit changes by $12, the new cost is:

C + dC = [tex]R^2/112,000 + 5807 + 12[/tex]

And when the revenue is $4500, we have:

R = 4500

Values into the expression for dR/dC, we get:

dR/dC = 112,000/(2 * 4500)

= 12.444

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The graph of the given line  2 x + y = 4 is a straight line and the coordinates present above the line will be ( 0, 4) and ( 2, 0) hence option (C) will be correct.

We know, that,

A line section that can connect two places is referred to as a segment.

In other words, a line segment is just part of a big line that is straight and going unlimited in bdirections.

The line is here! It extends endlessly in both directions and has no beginning or conclusion.

A coordinate that is present in the line will always satisfy the equation of the line.

In the given option the coordinate  (0, 4) and (2, 0) is satisfying the given equation by substituting the value  2 x + y = 4 hence option (C) will correct.

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The value of x in the given triangle is 24.

The value of x in the given triangle is calculated as follows;

30 + 4x + 10 + x + 20 = 180 ( sum of angles in a triangle )

Collect similar terms together as shown below;

4x + x = 180 - 30 - 10 - 20

5x = 120

divide both sides of the equation by 5;

5x/5 = 120/5

x = 24

Thus, the value of x is determined from the principle of sum of angles in a triangle.

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Step-by-step explanation:

See image:

The area of the donut ta that is left is 16.485 inches².

We have,

Donuts:

Diameter = 5 inches

Radius = 5/2 inches

The area of the donuts.

= πr²

= 3.14 x 5/2 x 5/2

= 19.625 inches²

Now,

The diameter of the hole in the donut = 2 inches

Radius = 2/2 = 1 inch

Area of the donut hole.

= 3.14 x 1 x 1

= 3.14 inches²

Now,

The area of the donut ta that is left.

= 19.625 - 3.14

= 16.485 inches²

Thus,

The area of the donut ta that is left is 16.485 inches².

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The critical value is -1.645 and we reject the null hypothesis at the 0.05 level of significance.

a. To find the critical value(s), we need to use a z-table. Since the alternative hypothesis is one-tailed (p<0.57), we will use the one-tailed z-table. At a significance level of 0.05, the critical value is the z-score that corresponds to an area of 0.05 in the tail of the distribution. From the z-table, we find that the critical value is -1.645.

b. To determine whether to reject or fail to reject the null hypothesis (H), we compare the test statistic (z=-2.12) to the critical value (-1.645).

Since the test statistic is smaller (more negative) than the critical value, we reject the null hypothesis at the 0.05 level of significance. This means that we have sufficient evidence to conclude that the true proportion (p) is less than 0.57.

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Answer:

(3×2−∣∣∣−5 +12∣∣∣ )+(−3) 2 = -3/1 = -3

Step-by-step explanation:

Multiple: 3 * 2 = 6

Absolute value: abs(the result of step No. 1) = abs(6) = 6

Subtract: the result of step No. 2 - 5 = 6 - 5 = 1

Absolute value: abs(12) = 12

Multiple: the result of step No. 3 * the result of step No. 4 = 1 * 12 = 12

Absolute value: abs(the result of step No. 6) = abs(12) = 12

Exponentiation: (-3) ^ 2 = 9

Add: the result of step No. 8 + the result of step No. 9 = -12 + 9 = -3

Also yeah and just trust ig  Hope it helps if it did tell me cause if it fully did