Find the perimeter of a sector whose radius is 4 unit and arc length is 16π​

Answer:

d ≈ 8.1

Step-by-step explanation:

calculate the distance d using the distance formula

d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]

with (x₁, y₁ ) = R (5, 7 ) and (x₂, y₂ ) = S (- 2, 3 )

d = [tex]\sqrt{(-2-5)^2+(3-7)^2}[/tex]

  = [tex]\sqrt{(-7)^2+(-4)^2}[/tex]

  = [tex]\sqrt{49+16}[/tex]

  = [tex]\sqrt{65}[/tex]

  ≈ 8.1 ( to 1 decimal place )

The vertex of h(x) is (-3, 3), and the range is y ≥ 3.

To find the vertex of the quadratic function h(x), we can use the formula x = -b/2a, where the quadratic function is in the form [tex]ax^2 + bx + c[/tex].

From the given table, we can observe that the x-values of the vertex correspond to the minimum points of the function.

The minimum point occurs between -4 and -3, which suggests that the x-coordinate of the vertex is -3. Therefore, x = -3.

To find the corresponding y-coordinate of the vertex, we look at the corresponding h(x) value in the table, which is 3. Hence, the vertex of the function h(x) is (-3, 3).

To determine the range of h(x), we need to consider the y-values attained by the function.

From the table, we see that the lowest y-value is 3 (the y-coordinate of the vertex), and there are no other y-values lower than 3. Therefore, the range of h(x) is all real numbers greater than or equal to 3.

The vertex of h(x) is (-3, 3), and the range is y ≥ 3.

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The vertex of the quadratic function is (-4, 12).

The range of h(x) is [3, ∞).

To find the vertex and range of the quadratic function h(x) based on the given table, we can use the properties of quadratic functions.

The vertex of a quadratic function in the form of f(x) = ax² + bx + c can be determined using the formula:

x = -b / (2a)

The domain of h(x) is all real numbers, we can assume that the quadratic function is of the form h(x) = ax² + bx + c.

Looking at the table, we can see that the x-values are increasing from left to right.

Additionally, the y-values (h(x)) are increasing from -6 to -4, then decreasing from -4 to -1.

This indicates that the vertex of the quadratic function lies between x = -4 and x = -3.

To find the exact x-coordinate of the vertex, we can use the formula mentioned earlier:

x = -b / (2a)

Based on the table, we can choose two points (-4, 4) and (-3, 3).

The difference in x-coordinates is 1, so we can assume that a = 1.

Plugging in the values of (-4, 4) and a = 1 into the formula, we can solve for b:

-4 = -b / (2 × 1)

-4 = -b / 2

-8 = -b

b = 8

The equation of the quadratic function h(x) can be written as h(x) = x² + 8x + c.

Now, let's find the y-coordinate of the vertex.

We can substitute the x-coordinate of the vertex, which we found as -4, into the equation:

h(-4) = (-4)² + 8(-4) + c

12 = 16 - 32 + c

12 = -16 + c

c = 28

The equation of the quadratic function h(x) is h(x) = x² + 8x + 28.

The range of the quadratic function can be determined by observing the y-values in the table.

From the table, we can see that the minimum y-value is 3.

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Answer: 18.84

Step-by-step explanation : To find the circumference you use the formula:

2πr

Since we have the diameter (6), divide by 2 to find the radius, or r.

So (2)(3.14)(3)

Answer:

c=25

Step-by-step explanation:

Since you are given [tex]x^{2}[/tex]+10x+c

We know that in an equation of [tex]ax^{2}+bx+c[/tex], when a = 1, c can be found by [tex](\frac{b}{2})^{2}[/tex]

So c = [tex](10/2)^{2}[/tex]=[tex]5^{2}[/tex]=25

[tex] \sf \hookrightarrow \: {8}^{2} + {b}^{2} = {17}^{2} [/tex]

[tex] \sf \hookrightarrow \: 8 \times 8 + {b}^{2} = 17 \times 17[/tex]

[tex] \sf \hookrightarrow \: 8 \times 8 + {b}^{2} = 289[/tex]

[tex] \sf \hookrightarrow \: 64 + {b}^{2} = 289[/tex]

[tex] \sf \hookrightarrow \: {b}^{2} = 289 - 64[/tex]

[tex] \sf \hookrightarrow \: {b}^{2} = 225[/tex]

[tex] \sf \hookrightarrow \: b = \sqrt{225} [/tex]

[tex] \sf \hookrightarrow \: b = \sqrt{15 \times 15} [/tex]

[tex] \sf \hookrightarrow \: b = 15[/tex]

Answer:

x = - 2 , x = 4

Step-by-step explanation:

the x- intercepts are the points on the x- axis where the graph crosses

the graph crosses the x - axis at - 2 and 4 , then

x- intercepts are x = - 2 , x = 4

Answer:

  (c)  61°

Step-by-step explanation:

You want the measure of the external angle formed by a tangent and secant that intercept arcs of 73° and 195° of a circle.

The measure of the angle at F is half the difference of intercepted arcs HE and EG.

  (195° -73°)/2 = 122°/2 = 61°

The measure of angle F is 61°.

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The break-even sales (units) when the variable costs are decreased by $4 is approximately 47,714 units.

To find the break-even sales (units) when the variable costs are decreased by $4, we need to calculate the new unit variable costs and then use the break-even formula.

Fixed costs (F) = $334,000

Unit selling price (P) = $22

Unit variable costs (V) = $19

Change in unit variable costs = $4

New unit variable costs (V') = V - Change in unit variable costs

= $19 - $4

= $15

Now, let's calculate the break-even sales (units) using the formula:

Break-even sales (units) = Fixed costs / (Unit selling price - Unit variable costs)

Break-even sales (units) = $334,000 / ($22 - $15)

= $334,000 / $7

= 47,714.29

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The area of quadrilateral QUAD is 2.5 square units.

To find the area of quadrilateral QUAD with vertices Q (-4, 3), U (3, 6), 1 (6, 3), and D (1, -4), we can use the Shoelace formula (also known as Gauss's area formula or the surveyor's formula).

The Shoelace formula states that the area of a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn) can be calculated as:

[tex]Area = 1/2 * |(x1y2 + x2y3 + ... + xny1) - (x2y1 + x3y2 + ... + x1yn)|[/tex]

Using this formula, we can calculate the area of quadrilateral QUAD as follows:

Area = [tex]1/2 * |(-46 + 33 + 6*(-4) + 13) - (33 + 6*(-4) + 1*(-4) + (-4)*3)|[/tex]

Simplifying the expression, we get:

[tex]Area = 1/2 * |(-24 + 9 - 24 + 3) - (9 - 24 - 4 - 12)|Area = 1/2 * |(-36) - (-31)|Area = 1/2 * |-36 + 31|Area = 1/2 * |-5|Area = 1/2 * 5Area = 5/2[/tex]

The area of quadrilateral QUAD is 2.5 square units.

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The area of quadrilateral QUAD is 17.5 square units.

The area of quadrilateral QUAD, we can use the Shoelace Formula, also known as the Gauss's Area Formula.

The formula states that if the coordinates of the vertices of a polygon are given in order, then the area of the polygon can be calculated using the following formula:

Area = 1/2 × |(x1y2 + x2y3 + ... + xn-1yn + xny1) - (y1x2 + y2x3 + ... + yn-1xn + ynx1)|

Let's apply this formula to find the area of quadrilateral QUAD:

Q (-4, 3)

U (3, 6)

A (6, 3)

D (1, -4)

Area = 1/2 × |(-4 × 6 + 3 × 3 + 6 × (-4) + 3 × (-1)) - (3 × 3 + 6 × (-4) + (-4) × (-1) + (-1) × (-4))|

Area = 1/2 × |(-24 + 9 - 24 - 3) - (9 - 24 + 4 + 4)|

Area = 1/2 × |(-42) - (-7)|

Area = 1/2 × |-42 + 7|

Area = 1/2 × |-35|

Area = 1/2 × 35

Area = 17.5

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Answer:182.12 meters of fencing required.

Step-by-step explanation:

The z - score z = (x - μ)/σ equals z = (p' - p)/[√(pq/n)]

The z-score is the statical value used to determine probability in a normal distribution

Given the z-score z = (x - μ)/σ where

We need to show that

z = (p' - p)/√(pq/n)

We proceed as follows

Now, the z-score

z = (x - μ)/σ

Substituting in the values of μ and σ into the equation, we have that

So, z = (x - μ)/σ

z = (x - np)/[√(npq)]

Now, dividing both the numerator and denominator by n, we have that

z = (x - np)/[√(npq)]

z = (x - np) ÷ n/[√(npq)] ÷ n

z = (x/n - np/n)/[√(npq)/n]

z = (x/n - p)/[√(npq/n²)]

z = (x/n - p)/[√(pq/n)]

Now p' = x/n

So, z = (x/n - p)/[√(pq/n)]

z = (p' - p)/[√(pq/n)]

So, the z - score is z = (p' - p)/[√(pq/n)]

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Answer:

There were 126 student tickets sold and 252 adult ticket sold.

Step-by-step explanation:

Let x be the number of adult tickets sold

y be the number of students tickets sold

Twice as many student tickets as adult tickets were sold

a.

x = 2y  ---equation 1

6x + 4y = 2016   ---equation 2

b.

Substitute equation 1 to equation 2

6(2y) + 4y = 2016

12y + 4y = 2016

16y = 2016

Divide both sides of the equation by 16

16y/16 = 2016/16

y = 126

Substitute y = 126 to equation 1

x = 2y

x = 2(126)

x = 252

The surface area and the volume of the rectangular prism are 280 and 300

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

The rectangular prism

The surface area is caculated as

Surface area = 2 * (10 * 5 + 10 * 6 + 5 * 6)

Evaluate

Surface area = 280

For the volume, we have

Volume = 10 * 5 * 6

Evaluate

Volume = 300

Hence, the surface area and the volume are 280 and 300

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The perimeter of the figure is 30 cm.

The perimeter of the figure is the sum of the whole sides of the figure. Therefore, the perimeter of the figure can be found as follows:

perimeter of the figure = sum of the whole sides

Therefore,

perimeter of the figure = 6 cm + 9 cm + 2 cm + 3cm + 2cm + 3cm + 2cm + 3cm

Hence,

perimeter of the figure = 15 cm + 5 cm + 5cm + 5 cm

perimeter of the figure = 30 cm

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Answer:

80

Step-by-step explanation:

Just average the two numbers to get (70+90)/2 = 160/2 = 80

Answer:

Step-by-step explanation:

To find the midpoint between two numbers, you add them together and divide the sum by 2.

In this case, the midpoint between 70 and 90 would be:

(70 + 90) / 2 = 160 / 2 = 80.

Therefore, the midpoint between 70 and 90 is 80.

The population at 4:00 P.M., 27 minutes earlier, is[tex]3^3[/tex] times the initial population P.

To determine the population of a bacteria culture 27 minutes earlier, we need to calculate the population growth from 4:00 P.M. to 4:27 P.M. given that the bacteria culture triples every 5 minutes.

Let's break down the time period into intervals of 5 minutes:

From 4:00 P.M. to 4:05 P.M., the population triples once.

From 4:05 P.M. to 4:10 P.M., the population triples again.

From 4:10 P.M. to 4:15 P.M., the population triples for the third time.

From 4:15 P.M. to 4:20 P.M., the population triples for the fourth time.

From 4:20 P.M. to 4:25 P.M., the population triples for the fifth time.

From 4:25 P.M. to 4:27 P.M., the population undergoes partial growth.

Since the population triples every 5 minutes, we can express the population at 4:27 P.M. as 3^5 times the initial population at 4:00 P.M.

Let's denote the initial population at 4:00 P.M. as P. Then, the population at 4:27 P.M. is [tex]3^5[/tex] * P.

To find the population 27 minutes earlier, we need to reverse the growth from 4:27 P.M. to 4:00 P.M. Since the population triples every 5 minutes, we need to divide the population at 4:27 P.M. by [tex]3^{(27/5).[/tex]

Therefore, the population at 4:00 P.M., 27 minutes earlier, can be calculated as:

Population at 4:00 P.M. = (Population at 4:27 P.M.) / [tex]3^{(27/5)[/tex]

[tex]= (3^{5} * P) / 3^{(27/5)\\\\\\\\= 3^{(25/5)} * P\\= 3^5 * P / 3^2\\= 3^3 * P[/tex]

Hence, the population at 4:00 P.M., 27 minutes earlier, is 3^3 times the initial population P.

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The solution for k in the equation 10 - 10|-8k + 4| = 10, expressed in set notation, is {1/2}.

1. Start with the equation: 10 - 10|-8k + 4| = 10.

2. Simplify the expression inside the absolute value brackets: -8k + 4.

3. Remove the absolute value brackets by considering two cases:

  Case 1: -8k + 4 ≥ 0 (positive case):

    -8k + 4 = -(-8k + 4)  [Removing the absolute value]

    -8k + 4 = 8k - 4     [Distributive property]

    -8k - 8k = -4 + 4    [Group like terms]

    -16k = 0             [Combine like terms]

    k = 0               [Divide both sides by -16]

  Case 2: -8k + 4 < 0 (negative case):

    -8k + 4 = -(-8k + 4)  [Removing the absolute value and changing the sign]

    -8k + 4 = -8k + 4     [Simplifying the expression]

    0 = 0                [True statement]

4. Combine the solutions from both cases: {0}.

5. Check if the solution satisfies the original equation:

  For k = 0: 10 - 10|-8(0) + 4| = 10

             10 - 10|4| = 10

             10 - 10(4) = 10

             10 - 40 = 10

             -30 = 10 [False statement]

6. Since k = 0 does not satisfy the equation, it is not a valid solution.

7. Therefore, the final solution expressed in set notation is {1/2}.

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Due to the presence of spheres X and Y, the electric field strength at point B is [tex]1.01 * 10^6 N/C[/tex] and [tex]-4.05 * 10^6 N/C[/tex], respectively.

Given that two spheres X and Y are carrying charges of 72mC and -72mC respectively, and they are located 4.0 m apart from each other. The electric field strength at points A and B due to the presence of each sphere is to be determined.

Let's begin by calculating the electric field strength at point A due to sphere X. Electric field strength is given by E=kq/r², where k is Coulomb's constant, q is the charge and r is the distance between the two charges. The electric field strength at point A due to sphere X, E₁=kq₁/r₁² [tex]= (9*10^9Nm^2/C^2) * (72mC) / (4.0m)^2 = 4.05 * 10^6 N/C[/tex] (approx.)

Similarly, the electric field strength at point A due to sphere Y can be calculated as follows, E₂=kq₂/r₂² [tex]= (9*10^9Nm^2/C^2) * (72mC) / (4.0m)^2 = 4.05 * 10^6 N/C[/tex] (approx.). Here, the negative sign indicates that the electric field due to sphere Y is in the opposite direction to the electric field due to sphere X. Now, let's calculate the electric field strength at point B. The electric field strength at point B due to sphere X, E₁=kq₁/r₁² [tex]= (9*10^9Nm^2/C^2) * (72mC) / (8.0m)^2 = 1.01 * 10^6 N/C[/tex] (approx.)

Similarly, the electric field strength at point B due to sphere Y can be calculated as follows, E₂=kq₂/r₂² [tex]= (9*10^9Nm^2/C^2) * (-72mC) / (4.0m)^2 = -4.05 * 10^6 N/C[/tex] (approx.). Therefore, the electric field strength at point A due to the presence of sphere X is [tex]4.05 * 10^6 N/C[/tex] and due to the presence of sphere Y is [tex]-4.05 * 10^6 N/C[/tex]. The electric field strength at point B due to the presence of sphere X is [tex]1.01 * 10^6 N/C[/tex] and due to the presence of sphere Y is [tex]-4.05 * 10^6 N/C[/tex].

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Answer:

AB is not tangent to the circle.

Step-by-step explanation:

A tangent is a straight line that touches a circle at only one point.

The tangent of a circle is always perpendicular to the radius.

Therefore, if AB is tangent to the circle, it will form a right angle with the radius, CA.

To determine if AB is tangent, we can use Pythagoras Theorem.

[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]

If AB is tangent, then angle CAB will be a right angle. So AC and AB would be the legs of the right triangle, and BC would be the hypotenuse.

Therefore:

[tex]AC^2+AB^2=BC^2[/tex]

Substitute the values into the equation:

[tex]7^2+12^2=15^2[/tex]

[tex]49+144=225[/tex]

[tex]193 = 225 \; \leftarrow\; \sf not\;true[/tex]

As 193 ≠ 225, the equation does not hold, hence proving that AB is not tangent to the circle.

Answer:she will actually need 1 dollar because all of that would be 31 dollars.

Step-by-step explanation:

3 pounds of lemons= $6

1 box of rice= $4

7 frozen diners= $21

6+4=10

10+21=31

The new required rate of return for Happy Corp. can be calculated using the Capital Asset Pricing Model (CAPM). The formula for CAPM is:

Required rate of return = Risk-free rate + Beta * Market risk premium

Since the analyst believes that the market risk premium will be unchanged, the only factor that will affect the new required return is the risk-free rate.

Given that the analyst predicts a 2.0% increase in inflation, the risk-free rate will also increase by that amount. Therefore, the new required rate of return for Happy Corp. will be the current risk-free rate plus the product of Happy Corp.'s beta and the market risk premium.

To plot the new Security Market Line (SML) on the graph, we would use the new required return calculated above and plot it against the corresponding beta values. The SML represents the relationship between risk (beta) and return (required rate of return).

By incorporating the new required return, we can determine the new expected returns for various levels of beta and create the updated SML.

It is important to note that without specific values provided for the risk-free rate, market risk premium, and Happy Corp.'s beta, it is not possible to calculate the exact new required return or plot the new SML accurately.

These values are crucial in determining the precise position of the SML on the graph.

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The distance around the shape after the smaller semicircle is removed is 29 cm.The correct answer is option D.

To find the distance around the shape after the smaller semicircle is removed, we need to calculate the circumference of the larger semicircle and subtract the circumference of the smaller semicircle.

The circumference of a semicircle is given by the formula:

Circumference = π * radius + diameter/2

For the larger semicircle:

Radius = diameter/2 = 28/2 = 14 cm

Circumference of the larger semicircle = π * 14 + 28/2 = 22/7 * 14 + 14 = 44 + 14 = 58 cm

For the smaller semicircle:

Radius = diameter/2 = 14/2 = 7 cm

Circumference of the smaller semicircle = π * 7 + 14/2 = 22/7 * 7 + 7 = 22 + 7 = 29 cm

Therefore, the distance around the shape after the smaller semicircle is removed is:

58 cm - 29 cm = 29 cm

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The Probable question may be:
A metalworker cuts out a large semicircle with a diameter of 28 centimeters. Then the metalworker cuts a smaller semicircle out of the larger one and removes it. The diameter of the semicircular piece that is removed is 14 centimeters. What will be the distance around the shape after the smaller semicircle is removed? Use 22/7​ as an approximation for π.

A. 80cm

B. 82cm

C. 85cm

D. 86cm

m∠BCD = 31.57° (approx). Hence, the answer of the angle is 31.57 degrees.

In the given diagram, BD is extended through point D to point E, m∠CDE = (9x - 12)°, m∠BCD = (2x + 3)°, and m∠DBC = (3x + 5)°. We need to find m∠BCD.

       We know that m∠BCD = (2x + 3)°

       Substituting x = 92/7, we get:

       m∠BCD = (2 × 92/7 + 3)° = (184/7 + 3)° = 221/7°

       Therefore, m∠BCD = 31.57° (approx). Hence, the answer is 31.57.

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Step-by-step explanation:

The formula to calculate the circumference (C) of a circle is C = 2πr, where r represents the radius of the circle.

In this case, the given circumference is 68 cm. Plugging this value into the formula, we can solve for the radius (r):

68 = 2πr

To find the radius, we can divide both sides of the equation by 2π:

r = 68 / (2π)

Using an approximate value of π ≈ 3.14159, we can calculate the radius:

r ≈ 68 / (2 × 3.14159) ≈ 10.8419 cm

Therefore, the radius of the circle, which has a circumference of 68 cm, is approximately 10.8419 cm.

Therefore, The Radius of the Circle is 10.8 cm.

The equation (-14) + x = 14 is solved by adding 14 to both sides of the equation, resulting in x = 28. This means that 28 is the value of x that satisfies the equation and makes it true.

To solve the equation (-14) + x = 14, we need to isolate the variable x on one side of the equation. Let's go through the steps:

Step 1: Add 14 to both sides of the equation to eliminate the -14 on the left side.

(-14) + x + 14 = 14 + 14

x = 28

The solution to the equation (-14) + x = 14 is x = 28.

In this equation, we start with (-14) on the left side, and we want to determine the value of x that makes the equation true. To do that, we need to isolate x. By adding 14 to both sides of the equation, we cancel out the -14 on the left side, leaving us with just x. On the right side, 14 + 14 simplifies to 28.

Therefore, the solution to the equation is x = 28. This means that if we substitute 28 for x in the original equation, (-14) + 28 will indeed equal 14. Let's verify this:

(-14) + 28 = 14

14 = 14

The left side of the equation simplifies to 14, and the right side is also 14. Since both sides are equal, it confirms that x = 28 is the correct solution to the equation.

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