In a circle with an 8-inch radius, a central angle has a measure of 60°. How long is the segment joining the endpoints of the arc cut off by the angle?

The length and width of the given rectangle are 23 and 20 inches respectively.

Suppose the width of the given rectangle=x inch

So, the length of the given rectangle = x+3 inch

The area of a rectangle with length l and width b is lb.

So, the area of the given rectangle = x(x+3)

Given that area of the rectangle = 460 squared inches

So, x(x+3)=460

[tex]x^{2} +3x=460[/tex]

[tex]x^{2} +x-460=0[/tex]

[tex]x^{2} +23x-20x-460=0[/tex]

[tex]x(x+23)-20(x+23) =0[/tex]

[tex](x+23) (x-20)=0[/tex]

[tex]x=-23[/tex]...Not possible

[tex]x=20[/tex]

The width of the rectangle =20 inches

So, the length of the rectangle =20+3 = 23 inches

Therefore, the length and width of the given rectangle are 23 and 20 inches respectively.

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Answer:

the correct answer is option D.

Step-by-step explanation:

average rate of change in the graph in interval x = 0 and x = pi/2

when we look into the graph between interval x = 0 and x = pi/2

we can see that on y - axis  

at  x = 0                value of   y = 3   and

at  x = π/2             value of   y = 5  

rate of change = [tex]\frac{change\ in y-axis}{change\ in\ x-axis}[/tex]

                        = [tex]\dfrac{y_2-y_1}{x_2- x_1}[/tex]

                        =[tex]\dfrac{5-3}{\frac{\pi}{4} - 0}[/tex]

                        = 4/π

hence, the correct answer comes out to be 4/π

hence, the correct answer is option D.

Answer:

a + a - 5b - b + 2c - c

2a - 6b + c

the answer is 2

Answer:A

Step-by-step explanation:

To solve this problem, we make use of the formula for Confidence Interval:

Confidence Interval = X ± z * σ / sqrt (n)

where X is the mean value, z is the z score which is taken from the standard tables, σ is the standard deviation, and n is the number of samples

z = 1.645 (at 90% Confidence Level)

Substituting the values into the equation:

Confidence Interval = 94 ± 1.645 * 12 / sqrt (70)

Confidence Interval = 94 ± 2.36

Confidence Interval = 91.64, 96.36

Therefore at 90% confidence level, the blood pressure reading ranges from 91.64 mmHg to 96.36 mmHg.

Final answer:

The remainder when 5k is divided by 3 is 2. This is because k can be represented as 6n + 1 where n is the quotient, and when 5 is multiplied by k and then divided by 3, after simplifying, only the term +5 contributes to the remainder.

Explanation:

When a positive integer k is divided by 6, the remainder is 1. Thus, we can write k as 6n + 1 where n is a quotient. To find the remainder when 5k is divided by 3, we must first multiply k by 5, resulting in 5(6n + 1) = 30n + 5.

Breaking down 30n + 5, we see that 30n is divisible by 3 since 30 is a multiple of 3. Hence, it will not contribute to the remainder when divided by 3. All that is left to consider is the +5 part. The closest multiple of 3 to 5 is 3 itself, meaning 5 divided by 3 will leave a remainder of 2. Therefore, the remainder when 5k is divided by 3 is 2.

Density , Speed and Mileage(Fuel Economy) are the three common measurements which can be expressed as a ratio of two units.

What are Fundamental Units?

A fundamental unit is a tool for measuring a base quantity. Any physical quantity that cannot be expressed in terms of another physical quantity is referred to as a basic quantity. They are

Three Common Measurements be Density , Speed and Mileage where

Density = [tex]\frac{kg}{m^{3} }[/tex]

Speed = [tex]\frac{m}{s}[/tex]

Mileage = [tex]\frac{km}{L}[/tex]

Hence , the three common measurements which can be expressed as a ratio of two units are Density , Speed and Mileage

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2w+2(2w+3)=60

2w+4w+6=60

6w+6=60

6w=54

w=54/6 =9

 the width is 9 feet

Answer:

9 is a solution.

Step-by-step explanation:

The replacement set is the set of values that can be substituted in the equation (for the variables).

Given a replacement set, we need to substitute every value of the set in our equation and see if the equation holds for that value. If it does, then the value is a solution of the equation.

In this problem we have the equation 2w + 2(2w + 3) = 60 and the replacement set is {6,7,8,9}

We are going to start replacing the different values of the set.

[tex]2w + 2(2w + 3) = 60\\2(6) + 2(2(6)+3) = 60\\12 + 2(12+3) = 60\\12 + 2(15) = 60\\12 + 30 \neq 60[/tex]

Thus, 6 is not a solution.

[tex]2w + 2(2w + 3) = 60\\2(7) + 2(2(7) +3) = 60\\14 + 2 (14+3) =60\\14 + 2(17)=60\\14+34=60\\48 \neq 60[/tex]

Thus, 7 is not a solution.

[tex]2w + 2(2w + 3) = 60\\2(8) + 2(2(8)+3) = 60\\16 + 2(16+3)=60\\16 +2 (19) =60\\16 + 38 \neq 54[/tex]

Thus, 8 is not solution.

[tex]2w + 2(2w + 3) = 60\\2(9) + 2(2(9) + 3) =60\\18 + 2 (18+3) =60\\18 +2(21)=60\\60=60[/tex]

Thus, 9 is a solution.

An equation is formed when two equal expressions. The total investment that Billy made in stocks is $135,000.

An equation is formed when two equal expressions are equated together with the help of an equal sign '='.

Let the amount that is invested in stocks be represented by x, while the amount that is been invested in bonds is represented by y.

Given that Bily invested 4.5 times as much in stocks as he did in bonds. Therefore, the equation to represent the given situation can be written as,

x = 4.5y

Also, given that the total amount that Billy invested is $165,000. Therefore, the total invested amount can be written as,

$165,000 = x + y

Substitute the value of x in the equation,

$165,000 = 4.5y + y

$165,000 = 5.5y

y = $165,000 / 5.5

y = $30,000

Now substitute the value of y in the first equation to know Billy's investment in stocks.

x = 4.5y

x = 4.5($30,000)

x = $135,000

Hence, the total investment that Billy made in stocks is $135,000.

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The distance between the friends changing with the rate of 7√15/4 meter per sec.

Let B represents the position of runner, A represents the position of the friend and C represents the position of centre of the circular track. ( shown  below ),

We need to find: dc/dt

By the cosine law,

[tex]c^2 =a^2+b^2+2abcosC[/tex]

Differentiating with respect to t ( time ),

[tex]2c \frac{dc}{dt}=2absinC \frac{dc}{dt}[/tex]

[tex]\frac{dc}{dt}=\frac{2absinC\frac{dc}{dt}}{2c}[/tex]..(1)

Now, by arc length formula,

Radius( say r ) × angle = arc length ( say l )

r × ∠C= l

Differentiating w. r. t. t,

[tex]r \times \frac{dC}{dt} + \angle C \times \frac{dr}{dt} = \frac{dl}{dt}[/tex]

Here dl/dt=7 m per sec

dr/dt=0

r=110

dC/dt=7/110

Now again , [tex]C^2= a^2+b^2-2abcosC[/tex]

[tex]220^2 = 110^2 + 220^2 -2(110)(220)cosC[/tex]

[tex]48400=12100+48400-48400cosC[/tex]

[tex]-12100=-48400cosC[/tex]

[tex]1/4=cosC[/tex]..(2)

[tex]sinC=\frac{\sqrt{15}}{4}[/tex] ...(3)

From equation (1), (2) and (3),

[tex]dC/dt = \frac{(110)(220)\sqrt{15}/4. 7/110}{220}[/tex]

=7√15/4 meter per second.

Hence, the distance between the friends changing with the rate of 7√15/4 meter per sec.

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When the runner's distance from his friend is same as the distance from the friend to center of the track, the runner is moving in a way that the distance between him and his friend isn't changing. Hence, at those specific moments, their relative speed is zero m/s.

This problem involves the concept of relative velocities in a plane. Since the runner is moving along a circular path with a constant speed of 7 m/s, the runner's speed towards or away from his friend depends on the runner's location on the track. However, when the runner is at a distance of 220 m from his friend (which is also the distance from the friend to the center of the track), the runner is moving perpendicular to the line joining the runner and his friend. This is because, in this particular case, the runner is moving along a tangent to the circle that intersects the friend's location. Therefore, the distance between the friends isn't changing at all during those moments - their relative speed is zero m/s.

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The third side of the triangle is AC = 10 centimeters

A triangle is a plane figure or polygon with three sides and three angles.

A Triangle has three vertices and the sum of the interior angles add up to 180°

Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

The area of the triangle = ( 1/2 ) x Length x Base

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

if a² + b² = c² , it is a right triangle

if a² + b² < c² , it is an obtuse triangle

if a² + b² > c² , it is an acute triangle

Given data ,

Let the triangle be represented as ΔABC

Now , the measures of the sides of the triangle are

The measure of side AB = 6 cm

The measure of side BC = 8 cm

Now , For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

On simplifying , we get

The measure of side AC = √ ( AB )² + ( BC )²

The measure of side AC = √ ( 6 )² + ( 8 )²

The measure of side AC = √ ( 36 + 64 )

The measure of side AC = √ ( 100 )

The measure of side AC = 10 cm

Therefore , the value of AC is 10 cm

Hence , the right triangle is solved and the side AC = 10 cm

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The 8th term of this geometric sequence 5,-10,20,-40 = -640.

A geometric sequence is one in which every number is the product of its previous number and a constant ratio.

The first term is taken as a, the constant ratio is taken as r. The n-th term of such a series is given as aₙ = a.rⁿ⁻¹.

In the question, we are asked to find the 8th term of the geometric sequence 5, -10, 20, -40.

The first term (a) = 5.

The constant ratio (r) = -10/5 = -2.

We will find the 8th term using the formula of the n-th term aₙ = a.rⁿ⁻¹.

Therefore, 8th term = 5(-2)⁸⁻¹ = 5 * (-2)⁷ = 5 * (-128) = -640.

The 8th term of this geometric sequence 5,-10,20,-40 = -640.

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perimeter = 4S = 16*4 = 64 units

area = S^2 = 16^2 = 256 square units

Answer:

The perimeter of a square with side lengths of 16 units is 64 units.

The area of a square with side lengths of 16 units is [tex]256 unit^2[/tex].

Step-by-step explanation:

Length of the square = a = 16 units

Perimeter of the square = P = [tex\]4\times a[/tex]

[tex]P=4\times 16 units  = 64 units[/tex]

Area of the square = A = [tex]a^2[/tex]

[tex]A=a^2=(16 unit)^2=256 unit^2[/tex]

The perimeter of a square with side lengths of 16 units is 64 units.

The area of a square with side lengths of 16 units is [tex]256 unit^2[/tex].

Answer:

9 feet

Step-by-step explanation:

Answer (APEX):

- Radius

Answer:

Using [tex]|X-a|=b[/tex]

where a is the center or average and b is the distance from the center or the range

Here, v represents the wind speed.

As per the statement:

Minimum sustained wind speed(A)= 74 miles per hour.

Maximum sustained wind speed(B) = 95 miles per hour.

Use the center and distance to write an absolute value equation:

[tex]\text{Center} = \frac{\text{A+B}}{2}[/tex]

then;

[tex]\text{Center (a)} = \frac{74+95}{2}=\frac{169}{2}=84.5[/tex]

Distance from the center(b)= 95-84.5 = 10.5

then;

[tex]|v-84.5| =10.5[/tex]

Therefore, an absolute value equation that represents the minimum and maximum speeds is,  [tex]|v-84.5| =10.5[/tex]

Answer: 15 feet

Step-by-step explanation:

The length of other diagonal is 15 feet.

Both diagonals bisect opposite angles of the quadrilateral. Both diagonals form symmetry lines for the quadrilateral. The diagonals divide the quadrilateral into four congruent right triangles. The intersection point of the diagonals is the incenter of the quadrilateral

Given:

width= 9 feet

length= 12 feet

Diagonal length = 15 feet

We know the diagonals of a rectangle are equal and bisect each other.

So,

The length of other diagonal is 15 feet.

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you have a 1/6 probability of rolling a 6

 so for 3 6's it would be

1/6 x 1/6 x 1/6 = 1/216

The coach's predictions cannot be entirely correct because the sum of the probabilities given by the coach does not equal 1. Therefore, the coach's predictions are not consistent with the rules of probability.

The coach's predictions cannot be entirely correct because the sum of the probabilities of all possible outcomes must equal 1. In this case, the sum of the probabilities given by the coach is 2/9 + 1/2 + 2/5 = 49/90, which is not equal to 1. Therefore, the coach's predictions are not consistent with the rules of probability.

To justify this, we can add up the probabilities of winning, tying, and losing the next game. The correct probabilities should add up to 1 since these are the only possible outcomes.

If we add these probabilities together: 2/9 + 1/2 + 2/5 = 49/90, which is not equal to 1. Therefore, the coach's predictions cannot be entirely correct.

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