Solve for the lengths of the missing sides in the triangle. Leave your answer in radical form. Show your work and
explain the steps you used to solve.
30°
18
b
60°
B

Answer:

0

Step-by-step explanation:

If we have the function f(x)=sqrt((x-8)/3), where 8 is less than or equal to x which is less than or equal to 16, we can evaluate f(8) as follows:

f(8)=sqrt((8-8)/3)

=sqrt(0/3)

=0

Hope this helps!

Answer:

Step-by-step explanation:

To solve the differential equation:

y′′′ + y = 3(y′ − 1)

we first find the characteristic equation:

r^3 + 1 = 0

The roots of this equation are:

r = -1, 0 ± i

So, the general solution of the homogeneous equation (y′′′ + y = 0) is:

y_h(t) = c1 e^(-t) + c2 cos(t) + c3 sin(t)

To find a particular solution of the non-homogeneous equation (3(y′ − 1)), we assume a solution of the form:

y_p(t) = a t + b

Taking the first, second and third derivatives of y_p(t), we get:

y′_p(t) = a

y′′_p(t) = 0

y′′′_p(t) = 0

Substituting these into the differential equation, we get:

0 + (a t + b) = 3(a - 1)

Solving for a and b, we get:

a = 1

b = 2

Therefore, the particular solution of the non-homogeneous equation is:

y_p(t) = t + 2

So, the general solution of the non-homogeneous equation is:

y(t) = y_h(t) + y_p(t) = c1 e^(-t) + c2 cos(t) + c3 sin(t) + t + 2

To find the values of c1, c2 and c3, we use the initial conditions:

y(0) = c1 + c2 + 2 = 1

y′(0) = -c1 + c2 + c3 = 0

y′′(0) = c1 - c2 = 0

Solving these equations simultaneously, we get:

c1 = c2 = 1/2

c3 = -1/2

So, the solution of the differential equation with initial conditions (0) = 1, ′(0) = 0, ′′(0) = 0 is:

y(t) = 1/2 e^(-t) + 1/2 cos(t) - 1/2 sin(t) + t + 2

There are 5,245,786 possible winning number combinations in a 6/42 lottery, and for every ticket purchased, there is a 1 in 5,245,786 chance of winning the jackpot prize.

What is the combinations?

Combinations refer to the different ways of selecting a subset of items from a larger set without regard to the order in which the items are selected. In other words, combinations are the different ways of choosing a group of items without taking into account their arrangement or order.

In a 6/42 lottery, there are 6 numbers drawn from a pool of 42 numbers. To determine the number of possible winning number combinations, we can use the formula for combinations:

C(n,r) = n! / (r! * (n-r)!)

where n is the total number of items in the set and r is the number of items being chosen.

In this case, we have 42 numbers in the pool and we need to choose 6 numbers, so the formula becomes:

C(42,6) = 42! / (6! * (42-6)!)

which simplifies to:

C(42,6) = 5,245,786

Therefore, there are 5,245,786 possible winning number combinations in a 6/42 lottery.

To calculate the probability of winning the jackpot prize, we need to divide the number of ways to win by the total number of possible outcomes. Since there is only one winning combination, the probability of winning the jackpot is:

1 / 5,245,786

which simplifies to approximately:

0.000019% or 1 in 5,245,786.

This means that for every ticket purchased, there is a 1 in 5,245,786 chance of winning the jackpot prize.

hence,  there are 5,245,786 possible winning number combinations in a 6/42 lottery, and for every ticket purchased, there is a 1 in 5,245,786 chance of winning the jackpot prize.

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The value of m + n for the given quadratic equation is 8.

What is a quadratic equation?

A quadratic equation is a polynomial equation of the form:

[tex]ax^{2} +bx+c=0[/tex]

where x is the unknown variable, and a,b and c are constants. The highest power of the variable x in a quadratic equation is 2, hence the term "quadratic". The constants a,b and c are known as the coefficients of the quadratic equation.

According to the given information:

Let's denote the roots of the quadratic equation [tex]2x^{2} -mx+n=0[/tex] as r1 and r2, with r1 and r2 being the roots of the equation.

Given that the sum of the roots is 6 and the product of the roots is 10. This gives us the following equations:

r1 + r2 = 6 ------ (1)

r1*r2 = 10 ------- (2)

We can use these equations to solve for the values of m and n.

The sum of the roots r1 + r2 is also equal to [tex]\frac{-m}{2}[/tex] by Vieta's formulas, which state that the sum of the roots of a quadratic equation [tex]ax^{2} +bx+c=0[/tex] is given by [tex]\frac{-b}{a}[/tex]. Therefore, we can write:

[tex]\frac{-m}{2}[/tex] = 6

Solving for m, we get:

m = -2*6 = -12  ----------(3)

The product of the roots r1*r2 is also equal to n/2 by Vieta's formulas, which states that the product of the roots of a quadratic equation [tex]ax^{2} +bx+c=0\\[/tex] is given by [tex]\frac{c}{a}[/tex]. Therefore, we can write:

n/2 = 10

Solving for n, we get:

n = 2 * 10 = 20  ------------(4)

Now, we can find the value of m+n by substituting the values of m and n from equations (3) and (4) respectively:

m + n = -12 + 20 = 8

So, the value of m+n is 8.

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The questions have been answered below based on the concept of apportionment.

Here, we have,

Apportionment is the challenge of distributing a certain number of items among groups of varying sizes. The Mathematics of Apportionment discusses the mathematical concepts and methods used to allocate identical things fairly among parties with differing claims.

In question 1, the quantity that does not vary from state to state when we apportion congress representatives to different states is called the standard quota.

In question 2, the standard quota is found by dividing the state's population by the standard divisor.

In question 3, the modified divisor in Webster's method is found by dividing the state's population by the modified divisor.

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The correct option- C: 75T + 70S + 160, represent their combined earnings of Terry and Susan  for the week.

Every feature, number, or amount that can be gauged or quantified is referred to as a variable. An amount that can fluctuate depending on the mathematical issue is referred to as a variable. The generic letters x, y, and z are frequently employed in mathematical expressions and equations. A variable, then, is a symbol denoting a number whose value is unknown.

Given data:

T - number of cars detailed by Terry (T)

S - number of cars detailed by Susan (S).

Terry's charge - $75 for each car .

Total Terry's charge for T number of cars - 75T

Additional tip - $65

Susan's charge - $70 for each car.

Total Susan's charge for S number of cars - 70T

Additional tip - $95

Combined earnings = 75T + 65 + 70T + 95

Combined earnings = 75T + 70T + 160

Thus,  75T + 70S + 160, represent their combined earnings of Terry and Susan  for the week.

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Complete question:

Terry and Susan both detail cars. Terry chargers $75 for each car and Susan charges $70 for each car. This week Terry made an additional $65 in tips and Susan made an additional $95 in tips. Given that T and S represent the number of cars detailed by Terry (T) and Susan (S), which expression can be used to represent their combined earnings for the week?​

A: 145(T+S) + 160

B: 65T + 70S + 160

C: 75T + 70S + 160

D: 145TS + 160

Answer: 11 miles

Step-by-step explanation: 1/5280 = x/58,080

All the arcs are derived as given below.

5)

∡LO = 180°

so

∡KL = ∡ON = 180-90-67

∡KL = ∡ON = 23°
∡OM = 90 + 23 = 113°
∡ NL = 67 + 90 = 157°


7)

Where

(9x + 23) + 31 = 180
9x + 54 = 180
9x = 180-54

9x = 126
x = 126/9
x = 14

9.

(21x-9) = 90+27
21x - 9 = 117
21x = 117 + 9
x = 126/21
x = 6

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The number line where we plotted -1 1/6 and 7/3 is added as an attachment

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

-1 1/6 and 7/3

To start with, we convert both numbers to the same form

i.e. decimal or fraction

When converted to fractions, we have

-7/6 and 7/3

Rewrite the denominators of the fractions

So, we have

-7/6 and 14/6

This means that we can plot -7/6 at -7 and 7/3 at point 14 where the difference in each interval is 1/6

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

The number line is attached

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The equation of the function g(x) is g(x) = f(x+3) + 4 and g(x) = log₂(x+3) + 4

To shift the function 3 units left, we replace x with x+3. To shift it 4 units up, we add 4 to the function.

Therefore, the expression for g(x) in terms of f(x) is:

g(x) = f(x+3) + 4

Substituting f(x) = log₂(x), we get:

g(x) = log₂(x+3) + 4

The equation of the function g(x) is g(x) = log₂(x+3) + 4.

The domain of g(x) is all x such that x+3 > 0, since the argument of the logarithm must be positive.  This gives x > -3.

The range of g(x) is all real numbers, since the logarithm takes on all real values.

The vertical asymptote is x = -3, since this is where the function becomes undefined (the argument of the logarithm becomes 0).

The horizontal asymptote is y = 4, since as x approaches infinity or negative infinity, the function approaches 4.

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The framers of the Constitution believed that as the country grew, more courts would be needed to meet its needs.

Explanation :

The statement that best explains that Article III of the constitution gives congress the ability to create lower courts inferior to the supreme court from "time to time" is that the framers of the Constitution believed that as the country grew, more courts would be needed to meet its needs.

As there will be more framers of the constitution which gave them as a  congress the power to create lower courts as they wanted that the congress should have authority over the courts and to control them.

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complete question:

Which statement best explains why Article lll of the constitution gives Congress the ability to create lower courts inferior to the Supreme Court “from time to time”?

Answer:

The domain is (-infinity, infinity), or all real numbers.

The range is (-infinity, infinity), or all real numbers.

Answer:

One pair of vertical angles is a and c.

Answer:

(a) To write g as a set of ordered pairs, we can use the given function values as follows:

g = {(-2, -5), (0, 0), (2, 2), (4, -5)}

(b) The domain of g is the set of all input values for which the function is defined. In this case, the domain is {-2, 0, 2, 4}.

The range of g is the set of all output values that the function can produce. In this case, the range is {-5, 0, 2}.

The expressions using scratch addition is 516

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

45 + 58 + 29 + 59 + 67 + 37 + 55 + 79 + 87

Using the Scratch method of Addition. we add the numbers one after the other

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

45 + 58 + 29 + 59 + 67 + 37 + 55 + 79 + 87 = 103 + 29 + 59 + 67 + 37 + 55 + 79 + 87

45 + 58 + 29 + 59 + 67 + 37 + 55 + 79 + 87 = 132 + 59 + 67 + 37 + 55 + 79 + 87

45 + 58 + 29 + 59 + 67 + 37 + 55 + 79 + 87 = 191 + 67 + 37 + 55 + 79 + 87

45 + 58 + 29 + 59 + 67 + 37 + 55 + 79 + 87 = 258 + 37 + 55 + 79 + 87

45 + 58 + 29 + 59 + 67 + 37 + 55 + 79 + 87 = 295 + 55 + 79 + 87

45 + 58 + 29 + 59 + 67 + 37 + 55 + 79 + 87 = 350 + 79 + 87

45 + 58 + 29 + 59 + 67 + 37 + 55 + 79 + 87 = 429+ 87

45 + 58 + 29 + 59 + 67 + 37 + 55 + 79 + 87 = 516

HEnce, the solution is 516

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The answers to Q11 to 14 with the figure representing angles with protractor are as follows:

11.Acute angle= JEH & measure=25°

12.Obtuse angle=JEK & measure=75°

13.One right angle=DEJ

14.One straight angle=DEF

What is protractor ?

With the aid of the scale on it, it serves as a tool for measuring angles. Protractors often feature two sets of numbers that face each other. A protractor is a tool having a semicircular shape and markings from 0° to 180° on both its inner and outer scales. Angles come in many different shapes and sizes in geometry, including obtuse (>90), acute (90), right (90), reflex (180–270), complete (360), and straight (>90).(180).

11)Acute angles:

HEJ=25

JEK=50

KEF=40

HED=65

12)Obtuse angles:

KED=140

JEK=75

13)right angles:

JEF

JED

14)straight lines

DEF

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Answer:

I'm sorry, but there seems to be an error in the formula you provided. The correct formula for finding the volume (V) of a rectangular prism is V = l x w x h, where l, w, and h are the length, width, and height of the prism, respectively.

We are given the graph of

y = h(x)

If the graph of y = h(x-3) is drawn on the same set of axes as the graph above, then the two graphs intersect at one point. What is that point?

We begin by identifying two points on the graph of h(x).

These are;

P1(-2,3), P2(1,3)

next, we are told that the graph of y = h(x - 3) is drawn on the same axis, which means the graph of the function has been shifted 3 units to the right.

Therefore, the points identified above will now be;

P 1 Rightarrow (- 2 + 3, 3)

P_{1}' = (1, 3)

P 2 Rightarrow (1 + 3, 3)

P_{2}' = (4, 3)

In summary,

The point is

1) The opposite face of the black circle is, Plus +.

2) The volume of the sliced cylinder is 96π cubic cm.

We have to given that;

Three different views of identical cubes are shown at right.

Now, From first and third dice,

Star is just opposite to symbol ~.

Hence, From second and third dice,

The opposite face of the black circle is, +

2) For the volume of the sliced cylinder, we need to first find the volume of the full cylinder and then subtract the volume of the smaller cylinder that's left after the slice is made.

Hence, The volume of the full cylinder is given by:

V = πr²h

Where r is the radius of the base and h is the height of the full cylinder.

Using the given values, we can find:

V = π(4²)(18)

V = 288π cubic cm

Now, let's find the volume of the smaller cylinder. The shorter height of 12 cm tells us that the slice removed 6 cm from the top of the cylinder. So the height of the smaller cylinder is,

18 - 6 = 12 cm.

The radius remains the same at 4 cm.

Thus, the volume of the smaller cylinder is:

V = π(4²)(12)

V = 192π cubic cm

Finally, we can find the volume of the sliced cylinder by subtracting the volume of the smaller cylinder from the volume of the full cylinder;

⇒ V = (288π) - (192π)

⇒ V = 96π cubic cm

Therefore, the volume of the sliced cylinder is 96π cubic cm.

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Answer:

The probability that a randomly chosen college student does not take statistics is 0.76.

Step-by-step explanation:

The probability of an event happening and the probability of its complement (the event not happening) must add up to 1. Therefore, if the probability that a randomly chosen college student takes statistics is 0.24, then the probability that they do not take statistics is:

1 - 0.24 = 0.76

So the probability that a randomly chosen college student does not take statistics is 0.76 or 76%.

The similar triangle is solved and the width of the river is R = 33.88 m

Given data ,

Let the base of the larger triangle be = 59 m

Let the base of the smaller triangle be = 35 m

Now , height of smaller triangle = 20.1 m

And , width of the river = height of the larger triangle

From the similar triangles , we get

59 / 35 = R / 20.1

Multiply by 20.1 on both sides , we get

R = 33.88 m

Hence , the width of river is 33.88 m

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An equation and solution that describe the situation is: B. The equation 1/8(x + 16) = 76 represents the situation, where x is the food bill.

In order to write a linear equation to describe this situation, we would a assign variable to the food bill, and then translate the word problem into a linear equation as follows:

Let the variable x represent the total amount of the food bill.

Since this group of 8 friends spent a total of $76, which included the food bill and a tip of $16, and decided to split the bill and tip evenly among themselves, a linear equation that can be used to model this situation is given by;

1/8(x + 16) = 76

(x + 16) = 76(8)

x + 16 = 608

x = 608 - 16

x = $592.

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the area of the trapezoidal base of the first prism is 9 square units, and the area of the trapezoidal base of the second prism is 16 square units.

What is Volume of both the trapezoidal prisms?

volume = area of cross section of object*  height of object (A)

V = (1/2) * h * (b1 + b2) * H

For the first trapezoid prism, we have:

24 = (1/2) * 6 * (b1 + b2) * 8

Simplifying this equation, we get:

3 = (b1 + b2)

For the second trapezoid prism, we have:

24 = (1/2) * 8 * (b1' + b2') * 6

Simplifying this equation, we get:

4 = (b1' + b2')

A = (1/2) * h * (b1 + b2)

For the first trapezoidal prism:

A = (1/2) * 6 * 3

A = 9 square units

For the second trapezoidal prism:

A = (1/2) * 8 * 4

A = 16 square units

Therefore, the area of the trapezoidal base of the first prism is 9 square units, and the area of the trapezoidal base of the second prism is 16 square units.

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Changing the base from 5/7 to 7, the end behavior of the function would be given as follows:

The end behavior of a function is given by the limit of the function is the input x goes to either negative infinity or positive infinity.

The function for this problem is given as follows:

f(x) = 4(5/7)^x.

Changing the base to 7, it would have an absolute value greater than 1, hence the function would be increasing with the end behavior given as follows:

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The length of the diagonal of the cube is x = 3√2 inches

Given data ,

Let the length of the diagonal of the cube be x

Now , the side length of the cube is = 3 inches

From the Pythagoras theorem , we get

x = √s² + s²

On simplifying , we get

x = √ ( 3² + 3² )

x = √18

On simplifying , we get

x = 3√2 inches

Hence , the diagonal is 3√2 inches

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a) The mean number of cases in groups of 17,199 children is 0.187.

b) The probability of 0 or 1 case in a group of 17,199 children is 0.832.

c)  The probability of more than one case in a group of 17,199 children is 0.168.

d) The expected number of cases is 3.214 and the actual number of cases is 4

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain.

For example, if you flip a fair coin, the probability of it landing heads up is 0.5, because there are two equally likely outcomes (heads and tails), and one of them is heads. Similarly, if you roll a fair six-sided die, the probability of rolling a 3 is 1/6, because there are six equally likely outcomes (rolling a 1, 2, 3, 4, 5, or 6), and only one of them is a 3.

a. Assuming that the tumor occurs as usual, the mean number of cases in groups of 17,199 children can be calculated as:

mean = (probability of a single case) x (number of children) = 0.000011 x 17,199 = 0.187

So the mean number of cases in groups of 17,199 children is 0.187.

b. The probability of 0 or 1 case can be found using the Poisson distribution, which models the probability of a certain number of events occurring in a fixed interval of time or space, given the mean rate of occurrence. In this case, the Poisson distribution can be used to find the probability of 0 or 1 case in a group of 17,199 children, given the mean of 0.187.

The probability of 0 or 1 case can be calculated as follows:

P(0 or 1 case) = P(0 case) + P(1 case)

= (e⁽⁻⁰°¹⁸⁷⁾ * 0.187⁰) / 0! + (e⁽⁻⁰°¹⁸⁷⁾ * 0.187¹) / 1!

= 0.832

Therefore, the probability of 0 or 1 case in a group of 17,199 children is 0.832.

c. The probability of more than one case can be found as:

P(more than one case) = 1 - P(0 or 1 case)

= 1 - 0.832

= 0.168

Therefore, the probability of more than one case in a group of 17,199 children is 0.168.

d. To determine whether the cluster of four cases is attributable to random chance, we can compare the actual number of cases with the expected number of cases based on the mean rate of occurrence.

The expected number of cases in a group of 17,199 children can be found by multiplying the mean rate of occurrence (0.187) by the number of groups:

expected number of cases = 0.187 x 17199 = 3.214

Since the expected number of cases is 3.214 and the actual number of cases is 4.

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