Find the distance between the points on a number line. -8,9

a)  If the tank is initially full, it will take  14.31 min. long to  tank to empty

b) If the height of the water is initially 10 feet, it will take 1.67 min for the tank to empty.

For the first case, we  have the differential  equation governing the height of the water, where the volume of a cone  is  V= 1/3 πr^2h by the similar triangles  we can conclude that  what the  value  will be  for the height, the cone  shape will always have the same relation  with the  radius of the water r/h=8/20⟹r=2/5h⟹r^22=4/25h^2, so V= 4/75πh^3, we will  take the derivative of the  height  of with respect to time t

dV/dt=4/25πh2 dh/dt, now converting into inches  we get  :

dV/dt=−cAh√2gh=−3/5(π(1/6)^2)√64h=−24π/5⋅36 √h=−2π/15√h.

now adding  both the  equations of   dV/dt,

425πh^2dhdt=−2π15√h⟹dhdt=−56h−3/2.

since the tank is empty m  it will happen in at h=0, so for the value of t  we will solve for h(t). after  solving  this differential equation using  the separation of  variables : dh h^3/2=−5/6dt, after  integration of both sides, we get: 25h5/2=−5/6t+C0⟹h5/2=−25/12t+C, since  here the intial height is  20 feet , so h(0)=20, Therefore  20^5/2=−25/12(0)+C⟹C=20^5/2, so h(t)=(−25/12t+20^5/2)^2/5.so  when  h  will be 0 , then −25/12t+20^5/2=0⟺25/12t=205/2⟺t=12/25 20^5/2≃858.65 s=14.31 min.

For the second case, the relationship between the height and radius is different, here the angle between the side of the tank and the vertical is  30 degrees, and the ratio of the radius and height of the tank is √3:1, which is also the ratio of height and radius of the water.  

1/√3=r/h⟹r=h/√3⟹r^2=h^2/3.

V=1/3πr^2h=π/9h^3, to solve it  we need to for dVdt, to  find the height and  rate of  change of height :dV/dt=π/3h^2dhdt here c is 0.6 and Ah=1/9π, dVdt=−35⋅19π⋅8√h, adding up those equations we get dV/dt, we have   π/3h^2dh/dt=−8π/15√h⟹dh/dt=−8/5h^−3/2,  now apply the separation of variables:

h^3/2dh=−8/5dt⟹2/5h^5/2=−8/5t+C, here the initial was at t=0, h is 11, C will be 2/5 11^5/2, therefore h(t)=−8/5t+2/5 11^5/2, so the time when the tank is empty will be  0=−8/5t+2/5 11^5/2⟺85t=25 11^5/2⟺t=1/4 11^5/2= 100.33 s=1.67 min.

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Suppose water is leaking from a tank through a circular hole of area at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to , where is an empirical constant.

A tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom.

a) Suppose the tank is high and has radius and the circular hole has radius . The differential equation governing the height h in feet of water leaking from a tank after t seconds is . In this model, friction and contraction of the water at the hole are taken into account with , and is taken to be . See the figure below. If the tank is initially full, how long will it take the tank to empty? (Round your answer to two decimal places.)

b) Suppose the tank has a vertex angle of and the circular hole has radius . Determine the differential equation governing the height h of water. Use and If the height of the water is initially , how long will it take the tank to empty? (Round your answer to two decimal places.)

The required area of the given triangle is given as 15 cm².

The triangle is a geometric shape that includes 3 sides and the sum of the interior angle should not be greater than 180°

To calculate the area of a triangle with its vertices A(2, 5), B(3, -1), and C(-2, -1),

Evaluate the absolute value of the expression 1/2
= 1/2|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|

Substitute the value in the above expression
= 1/2|2(-1 + 1) + 3(-1 - 5) -2(5 + 1)|
= 1/2[30]
= 15 cm²

Thus, the required area of the given triangle is given as 15 cm².

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The equation of the surface of the sphere is

z = √(100 - (x - 4)² - (y + 3)²). The solution has been obtained by using equation of sphere.

What is a sphere?

A sphere has a round shape and symmetrical arrangement. This three-dimensional solid's surface points are evenly spaced apart from the centre. It has both a volume and a surface area, depending on the radius. It has no faces, corners, or edges.

We know that general form of an equation of sphere is

(x - a)² + (y - b)² + (z - c)² = r²

where a, b, c are the coordinates and r is the radius.

The equation of the sphere is

(x - 4)²+ (y + 3)² + z² = 10²

So, the equation of the bottom hemisphere of sphere is

z = √(100 - (x - 4)² - (y + 3)²)

Hence, the equation of the surface of the sphere is

z = √(100 - (x - 4)² - (y + 3)²).

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The coordinates of point c on the coordinate plane are (4, 1)

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

B = (10, 7)

M = (7, 4)

The midpoint formula for two points (x1, y1) and (x2, y2) is:

(x1 + x2) / 2 = x_midpoint

(y1 + y2) / 2 = y_midpoint

So, for points B and C, we have:

x_midpoint = 7

y_midpoint = 4

Also, we have

x_B = 10

y_B = 7

The coordinates of C are calculated as

x_C = 2 * x_midpoint - x_B = 2 * 7 - 10 = 4

y_C = 2 * y_midpoint - y_B = 2 * 4 - 7 = 1

The coordinates of point C are (4, 1).

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The linear equation that represents the given literal problem is  3x+12= 28.50 (letter D) and Ava pays $5.5 (letter B)  for each pumpkin.

An equation can be represented by a linear function. The standard form for the linear equation is: y= mx+b , for example, y=9x+5. Where:

m= the slope.

b= the constant term that represents the y-intercept.

For the given example: m=9 and b=5.

The exercise presents two questions, before solving them, you should convert the given text information into equations.

Data question

     1) Question 1

     Here you should write a linear equation that represents the given problem. Thus, you can write

    3x+12= 28.50

    Since 3x represents the payment of pumpkins and the value 12 is the difference between the total of Ava´s money and the total cost of pumpkins.  

    Therefore, the answer to question 1 is the letter D ( 3x+12= 28.50)

   2) Question 2

As you know the equation that represents the problem, for solving this question you need to find the value of x.

    3x+12= 28.50

     3x=28.5-12

    3x=16.5

      x=16.5/3

      x=5.5 (letter B)

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Answer:y = (5/4)x - 3/2

Step-by-step explanation:

Answer:0.628571428571- or 63% and in fraction form it would be 5/8

Step-by-step explanation:

Answer: 22/35 (Result in decimals: 0.62)

Step-by-step explanation:

13/15 - 5/21

= 13 x 7/15 x 7 - 5 x 5/21 x 5

= 91/105 - 25/105

= 91 - 25/105

= 66/105

= 66 divided by 3 / 105 divided by 3

= 22/35

The factors of the quadratic function  p(x) is equal to

(x + 5 + √(65)/2)(x - 5 + √(65)/2).

We know that if x = a is one of the roots of a given polynomial x - a = 0 is a factor of the given polynomial.

To confirm if x - a = 0 is a factor of a polynomial we replace f(x) with f(a) and if the remainder is zero then it is confirmed that x - a = 0 is a factor.

Given, The zeros of the quadratic function p(x) = x² - 5x - 10 are,

(5 + √(65)/2, 5 - √(65)/2).

Therefore, The factors of p(x) = [x + (5 + √(65)/2)][x + (5 - √(65)/2)]

p(x) = (x + 5 + √(65)/2)(x - 5 + √(65)/2).

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The instantaneous rate of change of a function at a point is given by the derivative of the function at that point.

The instantaneous rate of change of a function at a particular point x is equal to its derivative at that point. To find the derivative of the function g(x) = 3/x, we can use differentiation rules.

the function g(x) = 3/x describes the relationship between x and 3/x. The derivative of this function, which is -3/x^2, gives us the instantaneous rate of change at any given value of x.

The derivative of g(x) is given by:

d/dx (3/x) = -3/x^2

The derivative at a given x tells us exactly how quickly the value of the function is increasing at that particular x.

So, the instantaneous rate of change of g(x) at a point x is given by -3/x^2.

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The height of the vase is 45 inches. To the nearest tenth of an inch, this is 45.0 inches.

A hyperbola is a type of conic section that is the result of intersecting a right circular cone with a plane that is perpendicular to one of its sides, and is oblique to the other. It is defined by two curves that are mirror images of each other and are each a set of all points such that the difference of their distances from two fixed points, called the foci, is a constant value.

A hyperbola can be represented in standard form as an equation, where the x and y terms are squared and the constant terms have opposite signs. It can also be graphed in the coordinate plane, where it appears as a set of two open curves, each going off to infinity in opposite directions.

The height of the vase can be found by using the formula for the height of a hyperbolic paraboloid, which is given by:

h = e × c × √(1 + (2a/c)²)

where h is the height of the vase, e is the eccentricity, a is the width at the narrowest point (4 inches), and c is the average width of the opening and base (6 inches).

Plugging in the values, we get:

h = 2.5 × 6 × sqrt(1 + (2 × 4 / 6)²)

h = 2.5 × 6 × sqrt(1 + (2 × 2)²)

h = 2.5 × 6 × sqrt(1 + 8)

h = 2.5 × 6 × sqrt(9)

h = 2.5 × 6 × 3

h = 45

The height of the vase is 45 inches. To the nearest tenth of an inch, this is 45.0 inches.

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Answer:x=8

Step-by-step explanation:

The interior angles of a triangle add up to 180 so

2x+39 + 2x+70 + x+31 = 180

2x + 2x +x=180-39-70-31

5x= 40

x=40/5

x=8

x+31 is in there because one of the int angles of the triangles is congruent with the angle of x+31

Answer:

155

Step-by-step explanation:

see attached.

Answer:

ksjsvsbsvsdjsjsbwbwbwbwushs

Using proportional relationships, it is found that:

1. A = 15.8, B = 5, C = 63.2.

2. A table with a rate of 0.82 represents the data.

3. Candidate B received 54 votes.

A direct proportional relationship is a function in which the output variable is given by the multiplication of the input variable and the constant of proportionality k, also called unit rate, as follows:

y = kx

here, we have,

For item 1, the constant is given as follows:

k = 31.6/4 = 7.9.

Hence the equation is:

y = 7.9x.

Then the value of A is found as follows:

A = 7.9(2)

= 15.8.

The value of B is found as follows:

7.9B = 39.5

B = 39.5/7.9

B = 5.

The value of C is:

C = 7.9(8) = 63.2.

For item 2, the rate is given as follows:

k = 35/42.5 = 0.82.

Hence a table with a rate of 0.82 represents the data.

For item 3, we have that the relation is:

B = kA.

Hence the constant is:

k = 18/15

k = 1.2.

Then the number of votes received by Candidate B is:

B = 1.2 x 45 = 54 votes.

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Hello :3

Answer:

C. what is m<ACB

Explanation:

Because you found the angles of ABC and BAC which is 35 and 115, it would only be appropriate to find ACB as well! :)

Hope this helps! ;)

Answer:

3

Step-by-step explanation:

The degree of a polynomial in two or more variables is the highest of the sum of the degrees of each variable

Here there is only one term with variables x and y: xy²

Degree of x = 1, degree of y = 2

Sum of the individual degrees = 1 + 2 = 3

The lowest score for the normally distribution with mean 72 and standard deviation 8 is equal to 82.

Mean of the normally distributed data 'μ' = 72

Standard deviation 'σ ' = 8

Lowest score with 10% ( 90percentile )

z = InvNorm(0.90)

 = 1.28

Let 'X' be the lowest score for the normal distribution

z = ( X - μ ) / σ

Substitute the values we get,

⇒ 1.28 = ( X - 72 ) / 8

⇒ X - 72 = 1.28 × 8

⇒ X = 10.24 + 72

⇒ X = 82.24

⇒ X = 82 ( round to an integer )

Therefore, the lowest score for normally distribution which will place manage on 90th percentile with given mean and standard deviation is equal to 82.

The above question is incomplete, the complete question is:

Scores on a management aptitude examination are normally distributed with a mean of 72 and a standard deviation of 8.we want to find the lowest score that will place a manager in the top 10% (90th percentile) of the distribution. Your answer is Please round to an integer number.

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Therefore , the solution of the given problem of relative frequency comes out to be  nonfiltered cigarettes have a higher relative frequency of cigarettes with higher tar grades.

A class interval's frequency is the amount of observations that take place during a specific predetermined interval. In other words, the frequency for such 5–9 age range is 20 if there are 20 children between the ages of 5 and 9 in the data from our study.

Here,

Tar(mg) Rel. freq.(Non-filtered) Rel. freq.(filtered)

4 to 9              0                            0.12

10 to 15               0                    0.08

16 to 21            0.04                       0.2

22 to 27           0.08                        0.6

28 to 33          0.52                           0

34 to 39            0.32                           0

40 to 45           0.04                            0

and

Yes, since nonfiltered cigarettes have a higher relative frequency of cigarettes with higher tar grades.

Therefore , the solution of the given problem of relative frequency comes out to be  nonfiltered cigarettes have a higher relative frequency of cigarettes with higher tar grades.

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Answer:

y = 4/5 x + 5

Step-by-step explanation:

Slope

Change in y over the change in x

[tex]\frac{1-5}{-5-0}[/tex] = [tex]\frac{-4}{-5}[/tex] = [tex]\frac{4}{5}[/tex]

The slope is [tex]\frac{4}{5}[/tex]

The y-intercept is 5.  It comes from the point (0,5).

Given the problem: Linear equation that passes through the points of (0,5) and (-5,1).

First, whenever you are asked to find a linear equation that passes through two points, remember its going to be slope intercept equation, or y=mx+b. m is the slope, b is the y-intercept.

The y-intercept is where the line crosses the y-axis. The slope can be written as rise/run. Rise and run right is positive, drop and run left is negative.

So in this case, looking at the first point: (0,5), we can see that the y-intercept is 5. Now let's find the slope. We will start at (-5,1) and go to (0,5). Count how many units we rise/drop and how many units we run. Rise 4, run right 5. So the slope is 4/5. Now put all the information into y=mx+b.

Your answer is y=4/5x+5

Answer:0.794

Step-by-step explanation:

Answer:

the answer would be 0.794. but hundredths would be 0.79

hope this helped!

Step-by-step explanation:

JL is MK congruent by CPCT.

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

Given that, a figure, in which JK ≅ ML, ∠ JKL ≅ ∠ MLK,

We need to prove, JL ≅ MK,

The proof of the same is follows;

     STATEMENT                                                                     REASON

1) JK ≅ ML                                                                                  Given

2) ∠ JKL ≅ ∠ MLK                                                                      Given

3) KL ≅ KL                                                                          Reflexive property

4) Δ JKL ≅ Δ MLK                                                                   By SSA rule

5) JL ≅ MK                                                                                By CPCT

Hence, we get JL is MK congruent by CPCT.

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The length of the third side of the given right triangle is 6.32.

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”. The formula for the same is;

c² = a² + b², where a, b, c are sides of a right triangle.

Given that, a right triangle, with two sides 7 and 3, since, the figure is unavailable, let us assume that 7 is the hypotenuse and 3 is the other side, let the unknown side be x,

According to the Pythagoras theorem,

7² = 3² + x²

x² = 7² - 3²

x² = 49-9

x² = 40

x = √40

x = 6.32

Hence, the length of the third side of the given right triangle is 6.32.

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The equation of the profit function is P(x) = 269x - 4x^2 - 128

The profit function P(x) can be found by subtracting the cost function C(x) from the revenue function R(x), where R(x) = p * x:

So, we have

P(x) = R(x) - C(x)

Substitute the known values in the above equation, so, we have the following representation

P(x) = (285 - 4x) * x - (128 + 16x)

So, we have

P(x) = 285x - 4x^2 - 128 - 16x

Evaluate the like terms

P(x) = 269x - 4x^2 - 128

To find the profit for making and selling 3 million fans, we substitute x = 3 into the profit function P(x):

P(3) = 269(3) - 4(3)^2 - 128

P(3) = 643

For other number of fans, we have

P(5) = 269(5) - 4(5)^2 - 128

P(5) = 1117

P(9) = 269(9) - 4(9)^2 - 128

P(9) = 1969

P(12) = 269(12) - 4(12)^2 - 128

P(12) = 2524

Hence, the profits are 643, 1117, 1969 and 2524

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Answer:

Slope-intercept form: y=3/4x+3

Slope:3/4

y-intercept: (0,3)

Slope of line perpendicular: -4/3

Slope of line parallel: 3/4

Hope this helps!!

y=3/4x+3

the slope for the line perpendicular is the inverse negative slope which would be -4/3x

put in the original slope for the slope and the y-i is (0,3)

any line with the same slope would be parallel (i.e 3/4)

Answer:

Step-by-step explanation:

If the limit of f(x) as x approaches 6 is 2, then 3f(x) is 3 times 2 which is 6. If the limit of g(x) as x approaches 6 is 8, then 5g(x) is 5 times 8 which is 40. Subtract the two results:

6 - 40 = -34

The exact value of is x  [tex]e^{4}[/tex] - 6/5 and the value of x will be 9.720.

What is exponential?

The exponential is an example of a mathematical function that is useful in determining if something is increasing or decreasing exponentially is the exponential function. As implied by its name, an exponential function uses exponents. But take note that an exponential function does not have a variable as its exponent and a constant as its base (if a function has a variable as the base and a constant as the exponent then it is a power function but not an exponential function).

a) In(5x + 6) = 4 exact value decimal approximation XE

ln (5x+6) = [tex]e^{4}[/tex]

5x = [tex]e^{4}[/tex] - 6

x =  [tex]e^{4}[/tex] - 6/5

5x = e^-6 x = 5 (r) x = 9.720

Hence, The exact value of is x  [tex]e^{4}[/tex] - 6/5 and the value of x will be 9.720.

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Answer:

5/13 chance of getting blue marble

Step-by-step explanation:

5+6+2=13

5 blue marbles so

5/13