A vehicle factory manufactures cars. The unit cost C (the cost in dollars to make each car) depends on the number of cars made. If x cars are made, then the unit cost is given by the function C(x) = x^2 - 400x + 45,377 . What is the minimum unit cost?
Do not round your answer.

Answer:

The answer is - a) 800

Step-by-step explanation:

In a periodic review system, we calculate the quantity of an item, a company has on hand at specified and fixed interval of time.

Given is : The target inventory level is 1000 units and the on-hand inventory level is 200 units.

So, the quantity will be =[tex]1000-200=800[/tex] units.

The answer is option A.

Rolle's theorem works for a function [tex]f(x)[/tex] over an interval [tex][a,b][/tex] if:

This is our case: [tex]f(x)[/tex] is a polynomial, so it is continuous and differentiable everywhere, and thus in particular it is continuous and differentiable over [0,4].

Also, we have

[tex]f(0)=7=f(4)[/tex]

So, we're guaranteed that there exists at least one point [tex]c\in(a,b)[/tex] such that [tex]f'(c)=0[/tex].

Let's compute the derivative:

[tex]f'(x)=3x^2-2x-12[/tex]

And we have

[tex]f'(x)=0 \iff x= \dfrac{1\pm\sqrt{37}}{3}[/tex]

In particular, we have

[tex]\dfrac{1+\sqrt{37}}{3}\approx 2.36[/tex]

so this is the point that satisfies Rolle's theorem.

The number C that satisfies the conclusion of Rolle's Theorem on the interval [0, 4] is: [tex]\[ c = \frac{1 + \sqrt{37}}{3} \][/tex]

To verify that the function [tex]\( f(x) = x^3 - x^2 - 12x + 7 \)[/tex] satisfies the three hypotheses of Rolle's Theorem on the interval [0, 4] and then to find all numbers c that satisfy the conclusion of Rolle's Theorem, follow these steps:

1. The function f is continuous on the closed interval [a, b]:

  - [tex]\( f(x) = x^3 - x^2 - 12x + 7 \)[/tex] is a polynomial, and polynomials are continuous everywhere.

  - Therefore, f is continuous on [0, 4].

2. The function f is differentiable on the open interval (a, b):

  - Again, [tex]\( f(x) = x^3 - x^2 - 12x + 7 \)[/tex] is a polynomial, and polynomials are differentiable everywhere.

  - Therefore, f is differentiable on (0, 4).

3. f(a) = f(b) :

  - Calculate [tex]\( f(0) \)[/tex] and f(4):

 [tex]\[ f(0) = 0^3 - 0^2 - 12 \cdot 0 + 7 = 7 \] \[ f(4) = 4^3 - 4^2 - 12 \cdot 4 + 7 = 64 - 16 - 48 + 7 = 7 \][/tex]

  - Therefore,  f(0) = f(4) = 7 .

Since all three hypotheses are satisfied, by Rolle's Theorem, there exists at least one number c in (0, 4) such that  f'(c) = 0 .

Finding c

1. Compute the derivative of f:

[tex]\[ f(x) = x^3 - x^2 - 12x + 7 \] \[ f'(x) = 3x^2 - 2x - 12 \][/tex]

2. Set the derivative equal to zero and solve for x:

[tex]\[ f'(x) = 3x^2 - 2x - 12 = 0 \][/tex]

  Solve the quadratic equation using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where  a = 3,  b = -2 , and c = -12 :

[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-12)}}{2 \cdot 3} \] \[ x = \frac{2 \pm \sqrt{4 + 144}}{6} \] \[ x = \frac{2 \pm \sqrt{148}}{6} \] \[ x = \frac{2 \pm 2\sqrt{37}}{6} \] \[ x = \frac{1 \pm \sqrt{37}}{3} \][/tex]

3. Check which solutions are in the interval (0, 4):

  - For [tex]\( x = \frac{1 + \sqrt{37}}{3} \)[/tex]:

 [tex]\[ \frac{1 + \sqrt{37}}{3} \approx \frac{1 + 6.08}{3} \approx \frac{7.08}{3} \approx 2.36 \][/tex]

  - For [tex]\( x = \frac{1 - \sqrt{37}}{3} \)[/tex]:

[tex]\[ \frac{1 - \sqrt{37}}{3} \approx \frac{1 - 6.08}{3} \approx \frac{-5.08}{3} \approx -1.69 \][/tex]

    - This solution is not in the interval (0, 4).

Answer:Given below

Step-by-step explanation:

A function is said to be odd if

[tex]F\left ( x\right )=F\left ( -x\right )[/tex]

(a)[tex]F\left ( x\right )=x^3+3x[/tex]

[tex]F\left ( -x\right )=-x^3-3x=-\left ( x^3+3x\right )[/tex]

odd function

(b)[tex]F\left ( x\right )=4sin2x[/tex]

[tex]F\left ( -x\right )=-4sin2x[/tex]

odd function

(c)[tex]F\left ( x\right )=x^2+|x|[/tex]

[tex]F\left ( -x\right )=\left ( -x^2\right )+|-x|=x^2+|x|[/tex]

even function

(d)[tex]F\left ( x\right )=e^x[/tex]

[tex]F\left ( -x\right )=e^{-x}[/tex]

neither odd nor even

(e)[tex]F\left ( x\right )=\frac{1}{x}[/tex]

[tex]F\left ( -x\right )=-\frac{1}{x}[/tex]

odd

(f)[tex]F\left ( x\right )=\frac{1}{2}\left ( e^x+e^{-x}\right )[/tex]

[tex]F\left ( -x\right )=\frac{1}{2}\left ( e^{-x}+e^{x}\right )[/tex]

even function

(g)[tex]F\left ( x\right )=xcosx(h)[/tex]

[tex]F\left ( -x\right )=-xcosx(h)[/tex]

odd function

(h)[tex]F\left ( x\right )=\frac{1}{2}\left ( e^x-e^{-x}\right )[/tex]

[tex]F\left ( -x\right )=\frac{1}{2}\left ( e^{-x}+e^{x}\right )[/tex]

odd function

Answer:

if x=-1 then its is NOT in the domain of h.

Step-by-step explanation:

Domain is the set of values for which the function is defined.

we are given the function

h(x) = x + 1 / x^2 + 2x + 1

h(x) = x+1 /x^2+x+x+1

h(x) = x+1/x(x+1)+1(x+1)

h(x) = x+1/(x+1)(x+1)

h(x) = x+1/(x+1)^2

So, the function h(x) is defined when x ≠ -1

Its is not defined when x=-1

So, if x=-1 then its is NOT in the domain of h.

Answer: [tex]x=-1[/tex]

Step-by-step explanation:

Given the function h(x):

[tex]h(x)=\frac{x+1}{ x^2 + 2x + 1}[/tex]

The values that are not in the domain of this function are those values that  make the denominator equal to zero.

Then, to find them, you can make the denominator equal to zero and solve for "x":

[tex]x^2 + 2x + 1=0\\\\(x+1)(x+1)=0\\\\(x+1)^2=0\\\\x=-1[/tex]

Answer:

The total amount to be repaid is equal to $949.66

Step-by-step explanation:

Simple interest is a type of interest which is usually applied on short term loans, where when a payment is made towards this kind of interest the payment first goes towards monthly interest and then the remainder is reverted towards the principal.

FORMULA FOR CALCULATING SIMPLE INTEREST =

[tex]\frac{PRINCIPAL \times RATE OF INTEREST \times TIME PERIOD}{100}[/tex]

Here principal = $922

         interest rate = 6%

         time period = 6 months (when made per annum it will be 6/12)

[tex]\frac{\$ 922 \times 6 \times 1}{100\times 2}[/tex]

SIMPLE INTEREST IS EQUAL TO $27.66

The total amount that is to be repaid is equal to

   PRINCIPAL + SIMPLE INTEREST

= $922 + $27.66

= $949.66

Answer:[tex]\theta =\arctan \mu [/tex]

Step-by-step explanation:

we know force sin component would oppose the weight  of object thus normal reaction will not be W rather it would be

[tex]N=W-Fsin\theta [/tex]

therefore force cos component will balance the friction force

F[tex]cos\theta[/tex] =[tex]\left ( \mu N\right )[/tex]

F[tex]cos\theta[/tex] =[tex]\left ( \mu \left ( W-Fsin\theta \right )\right )[/tex]

F=[tex]\frac{\mu W}{cos\theta +\mu sin\theta}[/tex]

F will be smallest when [tex]cos\theta +\mu sin\theta[/tex] will be maximum

and it will be maximum when we differentiate it to get

[tex]\theta =\arctan \mu[/tex]

The magnitude of the force, F, is varies with the angle the rope makes

with the plane according to the given equations.

F will be smallest when [tex]\underline{\theta \ is \ arctan (\mu)}[/tex].

Reason:

The given parameters are;

Angle the rope makes with the plane = θ

The magnitude of the force is, [tex]F = \dfrac{ \mu \cdot W}{\mu \cdot sin(\theta) +cos(\theta) }[/tex]

The value of θ for which the value of F is smallest.

Solution;

When, F is smallest, we have;

[tex]\dfrac{dF}{d \theta} = \dfrac{d}{d\theta} \left(\dfrac{ \mu \cdot W}{\mu \cdot sin(\theta) +cos(\theta) } \right) = \dfrac{-\mu \cdot W \cdot (\mu \cdot cos(\theta) -sin(\theta))}{\left( \mu \cdot sin(\theta) +cos(\theta) \right)^2} = 0[/tex]

Therefore;

-μ·W·(μ·cos(θ) - sin(θ))

μ·cos(θ) = sin(θ)

By symmetric property, we have;

sin(θ) = μ·cos(θ)

[tex]\mathbf{\dfrac{sin(\theta)}{cos(\theta)} = tan (\theta) = \mu}[/tex]

Which gives;

θ = arctan(μ)

Therefore;

F, will be smallest when [tex]\underline{\theta = arctan (\mu)}[/tex].

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Question; The given equation of the magnitude of the force in relation to the angle the rope makes with the plane, θ, is presented as follows;

[tex]F = \dfrac{ \mu \cdot W}{\mu \cdot sin(\theta) +cos(\theta) }[/tex]

Answer:

Step-by-step explanation: manej

Final answer:

In algebra, a tenth of a number is algebraically represented by multiplying the number by [tex]10^{-1}[/tex], which is equivalent to dividing the number by 10. This application of negative exponents simplifies expressions, especially in scientific notation, making it easier to work with large and small quantities.

Explanation:

In algebra, when we refer to a tenth of a number, we are usually dealing with fractions or exponential notation. A tenth of a number can be represented algebraically as the number divided by 10, which is the same as multiplying the number by [tex]10^{-1}[/tex]. This is because negative exponents indicate the reciprocal of a number; in other words, 10-1 equals 1/10 or 0.1.

This concept relates to the powers of ten and how each power of 10 affects the size of a number. For instance, 102 is 100, and 101 is 10, which is ten times smaller than 100. Conversely, 100 is 1, which is ten times smaller than 10, and thus, logically, [tex]10^{-1}[/tex] is 0.1, which is ten times smaller still. In expressing measurements in scientific work, especially for very small numbers, we frequently use this exponential form.

Thus, a tenth of an algebraic expression would mean multiplying the expression by [tex]10^{-1}[/tex] or dividing the expression by 10. This process is a form of simplification and re-scaling of numbers that are commonly used in scientific notation, which includes both positive and negative exponents. By understanding these principles, one can efficiently work with both large and small quantities in scientific and mathematical contexts.

Answer: There are 17576000 ways to generate different license plates.

Step-by-step explanation:

Since we have given that

Numbers are given = 0 to 9 = 10 numbers

Number of letters = 26

We need to generate the license plate numbers.

Since there are repetition allowed.

We would use "Fundamental theorem of counting".

So, the number of different license numbers may be created as given as

[tex]26\times 26\times 26\times 10\times 10\times 10\\\\=26^3\times 10^3\\\\=17576\times 1000\\\\=17576000[/tex]

Hence, there are 17576000 ways to generate different license plates.

Answer: There is 100 ounces of 35% salt solution and 100 ounces of water.

Step-by-step explanation:

Since we have given that

Percent of salt in a solution = 35%

Percent of salt in a mixture = 20%

Number of ounces of a mixture = 175 ounces

We need to find the number of ounces of water and salt as well as .

We would use "Mixture and Allegation":

      Salt                       Water

      35%                           0%

                       20%

---------------------------------------------------------------

20% - 0%          :                35% - 20%

   20%              :                     15%

   4                    :                      3

So, Ratio of salt and water in the mixture is 4 : 3.

So, Number of ounces of salt in the mixture is given  by

[tex]\dfrac{4}{7}\times 175\\\\=100\ ounces[/tex]

Number of ounces of water in the mixture is given by

[tex]\dfrac{3}{7}\times 175\\\\=75\ ounces[/tex]

Hence, there is 100 ounces of 35% salt solution and 100 ounces of water.

Final answer:

To make a 175-ounce mixture with 20% salt, the scientist should mix 75 ounces of water with 100 ounces of the 35% salt solution.

Explanation:

The student is asking for help with a typical mixture problem in algebra that involves determining the amounts of two different concentrations in order to create a mixture with a desired concentration. To solve this, we can set up two equations, one based on the total volume of the mixture and one based on the total amount of salt.

Let x be the amount of water (0% salt) and y be the amount of the 35% salt solution. The total volume should be 175 ounces, so we have:

Equation 1: x + y = 175

The total amount of salt in the solution must be 20% of 175 ounces, which is 35 ounces. So for the salt amount, we have:

Equation 2: 0.35y = 35

Solving Equation 2 gives us y = 100 ounces for the 35% solution. Substituting y in Equation 1, we get x = 75 ounces for the water. Therefore, the scientist should mix 75 ounces of water with 100 ounces of the 35% salt solution to obtain 175 ounces of a 20% salt mixture.

Answer:

The amount invested at 4.5% was [tex]\$3,000[/tex]

The amount invested at 5% was [tex]\$6,000[/tex]

Step-by-step explanation:

we know that

The simple interest formula is equal to

[tex]I=P(rt)[/tex]

where

I is the Final Interest Value

P is the Principal amount of money to be invested

r is the rate of interest  

t is Number of Time Periods

Let

x -----> the amount invested at 4.5%

9,000-x -----> the amount invested at 5%

in this problem we have

[tex]t=1\ year\\ P1=\$x\\ P2=\$(9,000-x)\\I=\$435\\r1=0.045\\r2=0.05[/tex]

substitute

[tex]435=x(0.045*1)+(9,000-x)(0.05*1)[/tex]

[tex]435=0.045x+450-0.05x[/tex]

[tex]0.05x-0.045x=450-435[/tex]

[tex]0.005x=15[/tex]

[tex]x=\$3,000[/tex]

so

[tex]9,000-x=\$6,000[/tex]

therefore

The amount invested at 4.5% was [tex]\$3,000[/tex]

The amount invested at 5% was [tex]\$6,000[/tex]

Answer:

There are  ways for quickly multiply out a binomial that's being raised by an exponent. Like

(a + b)0 = 1

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = (a + b)(a + b)2 = (a + b)(a2 + 2ab + b2) = a3 + 3a2b + 3ab2 + b3

and so on and so on

but there was this mathematician named Blaise Pascal and he found a numerical pattern, called Pascal's Triangle, for quickly expanding a binomial like the ones from earlier. It looks like this

1                           1      

2                   1     2     1

3                1     3     3   1

4           1     4     6     4     1

5       1     5     10     10     5     1

Pascal's Triangle gives us the coefficients for an expanded binomial of the form (a + b)n, where n is the row of the triangle.

Hope this helps!

The Binomial Theorem uses Pascal's triangle for expanding binomials raised to positive integer powers. Pascal's triangle provides the coefficients for each term in the binomial expansion, simplifying the expansion process.

The Binomial Theorem uses Pascal's triangle to expand binomials that are raised to positive integer power. Pascal's triangle is a triangular array of binomial coefficients. Each line of the triangle represents the coefficients of the terms of a binomial expansion. Let's take for instance binomial expansion of (a + b)n = an+nan-1b+…from Pascal's triangle, the coefficients are 1,n, and so on. The role of Pascal's triangle in this case is pivotal in knowing the coefficients of each term in the binomial expansion and thus facilitates the expansion process.

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Answer:  Yes, the given set of vectors is a linearly independent subset of R³.

Step-by-step explanation:  We are given to show that the following set of three vectors is a linearly independent subset of R³ :

B = {(1, 1, 0), (1, 0, 1), (0, 1, 1)} .

Since the given set contains three vectors which is equal to the dimension of R³, so it is a subset of R³.

To check the linear independence, we will find the determinant formed by theses three vectors as rows.

If the value of the determinant is non zero, then the set of vectors is linearly independent. Otherwise, it is dependent.

The value of the determinant can be found as follows :

[tex]D\\\\\\=\begin{vmatrix}1 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 1\end{vmatrix}\\\\\\=1(0\times1-1\times1)+1(1\times0-1\times1)+0(1\times1-0\times0)\\\\=1\times(-1)+1\times(1)+0\\\\=-1-1\\\\=-2\neq0.[/tex]

Since the determinant is not equal to 0, so the given set of vectors is a linearly independent subset of R³.

Thus, the given set is a linearly independent subset of R³.

Answer:

The correct option is 4.

Step-by-step explanation:

It is given that the rate of recipt of income from the sales of vases from 1988 to 1993 can be approximated by

[tex]R(t)=\frac{100}{(t+0.87)^2}[/tex]

billion dollars per year, where t is time in years since January 1988.

We need to estimate the total change in income from January 1988 to January 1993.

[tex]I=\int_{0}^{5}R(t)dt[/tex]

[tex]I=\int_{0}^{5}\frac{100}{(t+0.87)^2}dt[/tex]

[tex]I=100\int_{0}^{5}\frac{1}{(t+0.87)^2}dt[/tex]

On integration we get

[tex]I=-100[\frac{1}{(t+0.87)}]_{0}^{5}[/tex]

[tex]I=-100(\frac{1}{5+0.87}-\frac{1}{0+0.87})[/tex]

[tex]I=-100(-0.979)[/tex]

[tex]I=97.9[/tex]

[tex]I\approx 98[/tex]

The total change in income from January 1988 to January 1993 is $98. Therefore the correct option is 4.

Answer:

[tex]\frac{1}{35}[/tex]

Step-by-step explanation:

On a single roll of a pair of dice. When a pair of dice are rolled the possible outcomes are as follows:

(1,1)         (1,2)          (1,3)  (1,4)  (1,5)  (1,6)

(2,1)  (2,2)  (2,3)  (2,4)  (2,5)  (2,6)

(3,1)  (3,2)  (3,3)  (3,4)  (3,5)  (3,6)

(4,1)  (4,2)  (4,3)  (4,4)  (4,5)  (4,6)

(5,1)  (5,2)  (5,3)  (5,4)  (5,5)  (5,6)

(6,1)  (6,2)  (6,3)  (6,4)  (6,5)  (6,6)

The number of outcomes that gives us 12 are (6,6). There is only one outcome that gives us sum 12.

Total outcomes = 36

Odd against favor = [tex]\frac{non \ favorable\ outcomes}{favorable \ outcomes}[/tex]

Number of outcomes of getting sum 12 is 1

Number of outcomes of not getting sum 12 is 36-1= 35

odds against rolling a sum of 12= [tex]\frac{1}{35}[/tex]

Final answer:

A detailed explanation of the odds against rolling a sum of 12 on a pair of dice.

Explanation:

On a single roll of a pair of dice, the odds against rolling a sum of 12 are:

There is only one way to roll a 12, which is by getting a 6 on each die.

The probability of rolling a 6 on one die is 1/6 or approximately 0.166.

The probability of rolling a 12 on both dice is (1/6) * (1/6) = 1/36, which is about 2.8%.

ANSWER

[tex]x = \frac{3}{2} \: or \: x = - 3[/tex]

EXPLANATION

We want to find the roots of the parabola with equation:

[tex]2 {x}^{2} + 5x - 9 = 2x[/tex]

We need to write this in the standard quadratic equation form.

We group all terms on the left to get:

[tex]2 {x}^{2} + 5x - 2x - 9 = 0[/tex]

We simplify to get:

[tex]2 {x}^{2} +3x- 9 = 0[/tex]

We now compare to:

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

[tex] \implies \: a = 2 , \: \: b = 3 \: \: and \: c=- 9[/tex]

[tex] \implies ac = 2 \times - 9 = - 18[/tex]

The factors of -18 that sums up to 3 are -3, 6.

We split the middle term with these factors to get:

[tex]2 {x}^{2} +6x - 3x- 9 = 0[/tex]

Factor by grouping:

[tex]2x(x + 3) -3(x + 3) = 0[/tex]

Factor again to obtain:

[tex](2x - 3)(x + 3) = 0[/tex]

Apply the zero product principle to get:

[tex]2x - 3 = 0 \: or \: x + 3 = 0[/tex]

[tex] \implies \: x = \frac{3}{2} \: or \: x = - 3[/tex]

Answer:

y(x) = 1 - x

Step-by-step explanation:

Given the two parametric equations:

[tex]  x(t)=cos^{2}(6t)  [/tex]  ---(1)

[tex] sin^{2}(6t) [/tex] ----(2)

We can add eq (1) and eq (2) and consider the trigonometric identity:

[tex]  cos^{2}(6t)+sin^(6t) = 1  [/tex]

so,

[tex]   x+y=1  [/tex]

in other way we  can express this like:

[tex] y(x)=1-x [tex].

Answer:

The Hindu-Arabic numeral form of the given expanded numeral is 842.

Step-by-step explanation:

The given expanded numeral is

[tex](8\times 10^2)+(4\times 10)+(2\times 1)[/tex]

We need to express the given expanded numeral as a Hindu-Arabic numeral.

According to Hindu-Arabic numeral the given expanded numeral is written as

[tex](8\times 10^2)+(4\times 10)+(2\times 1)=(8\times 100)+(4\times 10)+(2\times 1)[/tex]

On simplification we get,

[tex](8\times 10^2)+(4\times 10)+(2\times 1)=(800)+(40)+(2)[/tex]

[tex](8\times 10^2)+(4\times 10)+(2\times 1)=842[/tex]

Therefore the Hindu-Arabic numeral form of the given expanded numeral is 842.

Answer:

2.8

Step-by-step explanation:

The weighted average is found by dividing the total number of points by the total number of credits.

GPA = (4×3 + 3×4 + 3×4 + 3×4 + 2×3 + 1×2) / (3 + 4 + 4 + 4 + 3 + 2)

GPA = 56 / 20

GPA = 2.8

The GPA of the student will be 2.8.

Algebra is the study of mathematical symbols, and the rule is the manipulation of those symbols.

One common system for computing a grade point average​ (GPA) assigns 4 points to an​ A, 3 points to a​ B, 2 points to a​ C, 1 point to a​ D, and 0 points to an F.

Then the GPA of a student who gets an A in a 3​-credit ​course, a B in each of three 4​-credit ​courses, a C in a 3​-credit ​course, and a D in a 2​-credit ​course will be

GPA = (4×3 + 3×4 + 3×4 + 3×4 + 2×3 + 1×2) / (3 + 4 + 4 + 4 + 3 + 2)

GPA = 56 / 20

GPA = 2.8

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Answer:

Step-by-step explanation:

We know that X-Y = X∩Y'

Using it ,we get

A ∩(B∩C') which can be written as (A∩B)∩C' or (A∩B) - C

And right hand side is

(A∩B)-(B∩C) =B ∩(A-C) = B∩(A∩C') = A∩B∩C'

Since both left and right side both leads to same expression A∩B∩C'

Therefore both are equal.

Answer:  

The probability that the student will get at least 4 of the questions​ right is 0.0823044.

Step-by-step explanation:

For each question we have 3 choices. So,total choices will be :

[tex]3\times3\times3\times3\times3\times3=729[/tex]

Getting 4 correct means, 4 corrects and two wrongs

Now, as there are 3 answer choices, out of which only one will be correct, so 2/3 is the probability if a question is answered wrong.

And 1/3 is the probability if a question is answered correctly.

Hence, we can consider this probability :

[tex]P=(2/3)*(2/3)*(1/3)*(1/3)*(1/3)*(1/3)[/tex] = 4/729

=> P = 0.00548696

We can select any combination of 2 from 6 for being wrong, so we will multiply P by (6,2)=6!/(2!*4!) = 15

So the answer is P*15 =[tex]0.00548696*15=0.0823044[/tex]

The probability that the student will get at least 4 of the questions​ right is 0.0823044.

With 3 choices per question, the probability of getting at least 4 out of 6 questions correct is approximately 0.0823044

1: Total Choices

Each question has 3 possible answers.

So, the total choices for 6 questions would be 3 raised to the power of 6 (3^6).

2: Probability of Getting 4 Correct and 2 Wrong

Getting 4 correct and 2 wrong means selecting 4 correct answers out of 6 questions.

The probability of a question being answered correctly is 1/3, and the probability of being answered incorrectly is 2/3.

So, the probability of getting 4 correct and 2 wrong is calculated using combinations (6 choose 4) multiplied by (1/3)^4 multiplied by (2/3)^2.

3: Calculate Probability

(6 choose 4) is the number of ways to choose 4 correct answers out of 6 questions, which is 15.

The probability (P) is then calculated as 15 multiplied by (1/3)^4 multiplied by (2/3)^2.

4: Multiply by Number of Combinations

Since there are 15 ways to choose 4 correct answers out of 6 questions, multiply the probability by 15.

So, the probability that the student will get at least 4 of the questions right is approximately 0.0823044.

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Answer: 0.357

Step-by-step explanation:

Given : The number of Hatfield  = 6

The number of McCoys = 2

The number of companies = 8

The number of construction jobs -3

Now, the required probability is given by :-

[tex]\dfrac{^6C_3\times^2C_0}{^8C_3}\\\\=\dfrac{\dfrac{6!}{3!(6-3)!}}{\dfrac{8!}{3!(8-3)!}}=0.357142857143\approx0.357[/tex]

Hence, the probability that all 3 jobs go to Hatfields =0.357

The statement is given by:

∀ x ,  [tex]x^4>x[/tex]

This statement is false

Since, if we consider,

[tex]x=\dfrac{1}{2}[/tex]

then we have:

[tex]x^4=(\dfrac{1}{2})^4\\\\i.e.\\\\x^4=\dfrac{1}{2^4}\\\\i.e.\\\\x^4=\dfrac{1}{16}[/tex]

Also, we know that:

[tex]\dfrac{1}{16}<\dfrac{1}{2}[/tex]

( Since, two number with same numerator; the number with greater denominator is smaller than the number with the smaller denominator )

Hence, we get:

[tex]x^4<x[/tex]

when [tex]x=\dfrac{1}{2}[/tex]

Hence, the result :

[tex]x^4>x[/tex] is not true for all x belonging to real numbers.

Hence, the given statement is a FALSE statement.

Final answer:

The statement (∀x)(x⁴ > x) is false, as it does not hold true for all real numbers. For instance, when x is a negative number like -1, the inequality x⁴ > x is false.

Explanation:

False

Explanation:

Given statement: (∀x)(x⁴ > x)

This statement asserts that for all real numbers x, x⁴ will be greater than x. However, this statement is false because it doesn't hold for all real numbers. For instance, when x is a negative number such as -1, (-1)4 is greater than -1, which means the inequality x⁴ > x is false.

Answer:

There are 26 investors which contributed 6,000

And 34 investors which contributed 3,000

Step-by-step explanation:

You need to set up a two equation problem

[tex]\left \{ {{258,000=6,000y+3,000x} \atop {60=y+x}} \right.[/tex]

Now you have to clear "x" or "y" from the second equation:

[tex]y = 60 - x[/tex]

And replace on the first equation:

[tex]258,000 = 6,000 (60 - x) + 3,000x\\258,000 = 360,000 - 6,000x + 3,000x\\3,000x = 360,000 - 258,000\\x = 102,000/3,000\\x = 34[/tex]

And now you use this "x" vale on the second equation

[tex]60 = y + x\\60 = y + 34\\60 - 34 = y\\y = 26[/tex]

There are 26 investors which contributed 6,000

And 34 investors which contributed 3,000

Step-by-step explanation:

If p is the number of physics books and s is the number of sociology books, then:

p + s = 440

s = p - 54

Substituting:

p + (p - 54) = 440

2p - 54 = 440

2p = 494

p = 247

Solving for s:

s = p - 54

s = 247 - 54

s = 193

The store sold 247 physics books and 193 sociology books.

Answer:

The dimensions of the package is [tex]r=\frac{48}{\pi}\ \text{and} \ h=48[/tex].

Step-by-step explanation:

Consider the provided information.

As it is given that, cylindrical package to be sent by a postal service can have a maximum combined length and girth is 144 inches.

Therefore,

144 = 2[tex]\pi[/tex]r + h

144-2[tex]\pi[/tex]r = h

The volume of a cylindrical package can be calculated as:

[tex]V=\pi r^{2}h[/tex]

Substitute the value of h in the above equation.

[tex]V=\pi r^{2}(144-2\pi r)[/tex]

Differentiate the above equation with respect to r.

[tex]\frac{dV}{dr}=2\pi r(144-2\pi r)+\pi r^{2}(-2\pi)[/tex]

[tex]\frac{dV}{dr}=288\pi r-4{\pi}^2 r^{2}-2{\pi}^2 r^{2}[/tex]

[tex]\frac{dV}{dr}=288\pi r-6{\pi}^2 r^{2}[/tex]

[tex]\frac{dV}{dr}=-6\pi r(-48+\pi r)[/tex]

Substitute [tex]\frac{dV}{dr}=0[/tex] in above equation.

[tex]0=-6\pi r(-48+\pi r)[/tex]

Therefore,

[tex]0=-48+\pi r[/tex]

[tex]r=\frac{48}{\pi}[/tex]

Now, substitute the value of r in 144-2[tex]\pi[/tex]r = h.

[tex]144-2\pi\frac{48}{\pi}=h[/tex]

[tex]144-96=h[/tex]

[tex]48=h[/tex]

Therefore the dimensions of the package should be:

[tex]r=\frac{48}{\pi}\ \text{and} \ h=48[/tex]

This is about optimization problems in mathematics.

Dimensions; Height = 48 inches; Radius =  48/π inches

Girth is same as perimeter which is circumference of the circular side.

Thus; Girth = 2πr

2πr + h = 144

h = 144 - 2πr

Volume of cylinder; V = πr²h

Let us put 144 - 2πr for h to get;

V = πr²(144 - 2πr)

V = 144πr² - 2π²r³

Differentiating with respect to r gives;

dV/dr = 288πr - 6π²r²

Thus; 288πr - 6π²r² = 0

Add 6π²r² to both sides to get;

288πr = 6π²r²

Rearranging gives;

288/6 = (π²r²)/πr

48 = πr

r = 48/π inches

h = 144 - 2π(48/π)

h = 144 - 96

h = 48 inches

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Answer:

False

Step-by-step explanation:

Given that a researcher testing the effects of two treatments for anxiety computed a 95% confidence interval for the difference between the mean of treatment 1 and the mean of treatment 2

Also that  confidence interval includes zero.

When confidence interval includes zero, we need not reject the null hypothesis since null hypothesis claims that difference =0

When confidence interval includes 0 it confirms that there is no difference and hence null hypothesis should be accepted.

Final answer:

The statement is false; if a 95% confidence interval for the difference between two treatment means includes zero, it indicates no significant difference, implying the null hypothesis cannot be rejected.

Explanation:

If a researcher testing the effects of two treatments for anxiety computed a 95% confidence interval for the difference between the mean of treatment 1 and the mean of treatment 2, and this confidence interval includes the value of zero, the correct interpretation is false regarding the statement that she should reject the null hypothesis that the two population means are equal. A confidence interval that contains zero indicates that the difference between the two treatments could be zero, suggesting there is no significant difference between the two treatments.

Therefore, there is insufficient evidence to reject the null hypothesis, and it is retained.

Understanding confidence intervals is crucial in hypothesis testing. A 95% confidence interval includes the true mean 95% of the time if the same experiment is repeated under the same conditions. Including zero in this interval suggests that the effect of the treatments could be negligible, meaning we cannot confidently claim there is a difference between the treatments based on the data provided.

Answer:1,2,3

Step-by-step explanation:

F(x)=[tex]x^{3}[/tex]+2[tex]x^2[/tex]-[tex]5x[/tex]-[tex]6[/tex]=0

disintegrating 2[tex]x^2[/tex] to [tex] x^2[/tex] + [tex]x^2[/tex]

[tex]x^{3}[/tex]+[tex]x^2[/tex]+[tex]x^2[/tex]-[tex]5x-6[/tex]=0

[tex]x^2[/tex][tex]\left ( x+1\right )[/tex]+[tex]x^2[/tex]-5x-6=0

[tex]x^2[/tex][tex]\left ( x+1\right )[/tex]+[tex]x^2[/tex]-6x+x-6=0

[tex]x^2[/tex][tex]\left ( x+1\right )[/tex]+[tex]\left (x-6 \right )[/tex][tex]\left ( x+1\right )[/tex]=0

[tex]\left ( x+1\right )[/tex][tex]\left ( x^2+x-6\right )[/tex]=0

[tex]\left ( x+1\right )[/tex][tex]\left ( x^2+3x-2x-6\right )[/tex]=0

[tex]\left ( x+1\right )[/tex][tex]\left ( x+3\right )[/tex][tex]\left ( x-2\right )[/tex]=0

Answer:

B. $271

Step-by-step explanation:

Given,

Present value of the loan, PV = $ 6000,

Annual rate of interest = 8 % = 0.08,

So, the monthly rate of interest, r = [tex]\frac{0.08}{12}[/tex],

Also, time = 2 years,

So, the total number of months, n = 24,

Hence, the monthly payment would be,

[tex]A=\frac{PV(r)}{1-(1+r)^{-n}}[/tex]

[tex]=\frac{6000(\frac{0.08}{12})}{1-(1+\frac{0.08}{12})^{-24}}[/tex]

[tex]=\$271.363748737[/tex]

[tex]\approx \$271[/tex]

Option B is correct.

Answer: The proportion of oil tankers that had spills is [tex]\dfrac{156}{474}[/tex] or 0.329.

Step-by-step explanation:

Since we have given that

Number of tankers is drawn = 474

Number of tankers did not have spills = 318

Number of tankers have spills = 474 - 318 = 156

Proportion of oil tankers that had spills is given by

[tex]\dfrac{Containing\ spill}{Total}=\dfrac{156}{474}=0.329[/tex]

Hence, the proportion of oil tankers that had spills is [tex]\dfrac{156}{474}[/tex] or 0.329.

Answer:  The required solution is

[tex]y=(-2+t)e^{-5t}.[/tex]

Step-by-step explanation:   We are given to solve the following differential equation :

[tex]y^{\prime\prime}+10y^\prime+25y=0,~~~~~~~y(0)=-2,~~y^\prime(0)=11~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

Let us consider that

[tex]y=e^{mt}[/tex] be an auxiliary solution of equation (i).

Then, we have

[tex]y^prime=me^{mt},~~~~~y^{\prime\prime}=m^2e^{mt}.[/tex]

Substituting these values in equation (i), we get

[tex]m^2e^{mt}+10me^{mt}+25e^{mt}=0\\\\\Rightarrow (m^2+10y+25)e^{mt}=0\\\\\Rightarrow m^2+10m+25=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{since }e^{mt}\neq0]\\\\\Rightarrow m^2+2\times m\times5+5^2=0\\\\\Rightarrow (m+5)^2=0\\\\\Rightarrow m=-5,-5.[/tex]

So, the general solution of the given equation is

[tex]y(t)=(A+Bt)e^{-5t}.[/tex]

Differentiating with respect to t, we get

[tex]y^\prime(t)=-5e^{-5t}(A+Bt)+Be^{-5t}.[/tex]

According to the given conditions, we have

[tex]y(0)=-2\\\\\Rightarrow A=-2[/tex]

and

[tex]y^\prime(0)=11\\\\\Rightarrow -5(A+B\times0)+B=11\\\\\Rightarrow -5A+B=11\\\\\Rightarrow (-5)\times(-2)+B=11\\\\\Rightarrow 10+B=11\\\\\Rightarrow B=11-10\\\\\Rightarrow B=1.[/tex]

Thus, the required solution is

[tex]y(t)=(-2+1\times t)e^{-5t}\\\\\Rightarrow y(t)=(-2+t)e^{-5t}.[/tex]