a shirt is on sale for 40% off, and you have an additional 20% off coupon. true or false: the shirt will ultimately be 60% off the original price​

Answer:

The answer to your question is

a) 400

b) The function has two real solutions (-2 and 8)

Step-by-step explanation:

Process

1.- Discriminant = b² - 4ac

                         = 12² -4(-2)(32)

                         = 144 + 256

                        = 400

2.- Solutions (using the general formula)

                   x = [tex]\frac{- b +/- \sqrt{b^{2}- 4ac}}{2a}[/tex]

                   x = [tex]\frac{- 12 +/- \sqrt{400}}{2(-2)}[/tex]

                   x = [tex]\frac{-12 +/- 20}{2(-2)}[/tex]

   x₁ = [tex]\frac{-12 + 20}{2(-2)} = \frac{-12 + 20}{-4} = \frac{-8}{4} = - 2[/tex]

   x₂ = [tex]\frac{- 12 - 20}{- 4} = \frac{-32}{- 4} = 8[/tex]

This function has two real solutions (-2, 8)

Answer: C- The value of the discrimination is 400.

E- The function has 2 real solutions.

Step-by-step explanation: Hope that helped!

Answer:

-$80

Step-by-step explanation:

Assuming that the variation in the number of shirts per day is -1 hundred shirs, the variation in profit with respect to time is given by the derivate of the profit equation:

[tex]p=-40x^2+581x-520\\\frac{dp}{dx}=-80x+581[/tex]

Let x be the number of shirts sold in a day, then x-1 is the number of shirts sold in the following day, the change in profit is:

[tex]p'(x) - p'(x-1)=-80x+581 - (-80(x-1)+581)\\p'(x) - p'(x-1)=-80(x-x+1) = -80[/tex]

The profit is changing by -$80 per day.

Answer:

The length of the circular arc is 8.64 inches

Step-by-step explanation:

Length of circular arc (L) = central angle/360° × 2πr

central angle = pie/4 = 45°, r (radius) = 11 inches

L = 45°/360° × 2 × 3.142 × 11 = 8.64 inches (to two decimal places)

Answer:

Step-by-step explanation:

The formula for determining the length of an arc is expressed as

Length of arc = θ/360 × 2πr

Where

θ represents the central angle.

r represents the radius of the circle.

π is a constant whose value is 3.14

From the information given,

Radius, r = 11 inches

θ = pi/4

2π = 360 degrees

π = 360/2 = 180

Therefore,

θ = 180/4 = 45 degrees

Therefore,

Length of arc = 45/360 × 2 × 3.14 × 11

Length of arc = 8.64 inches rounded up to 2 decimal places

Answer:

The margin of error is 3.97 percentage points.

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence interval [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].

For this problem, we have that:

[tex]n = 600, p = 0.56[/tex]

95% confidence interval

So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.56 - 1.96\sqrt{\frac{0.56*0.44}{600}} = 0.5203[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.56 + 1.96\sqrt{\frac{0.56*0.44}{600}} = 0.5997[/tex]

The margin of error is the upper limit subtracted by the proportion, or the proportion subtracted by the lower limit. They are the same values.

So the margin of error is 0.5997 - 0.56 = 0.56 - 0.5203 = 0.0397 = 3.97 percentage points.

Answer:

=19

Step-by-step explanation:

=6÷3+17

=2+17

=19

The tangent line approximation near x=0 for the function f(x) = \\sqrt{10 + x} is found by first calculating its derivative, then using that derivative to construct the equation of the tangent line at x=0, resulting in the linear approximation y = (1/2)(10)^{-1/2}x + \\sqrt{10}.

Finding the tangent line approximation for a function near a point involves using the function's derivative at that point. For the function f(x) = \\sqrt{10 + x}, the derivative at x = 0, denoted as f'(0), will provide the slope of the tangent. To find this, let's differentiate f(x) using the chain rule. The derivative of f(x) with respect to x is (1/2)(10 + x)^{-1/2}. At x = 0, this simplifies to 1/2(\\sqrt{10}), which is the slope of the tangent line at that point. Hence, the tangent line equation is y - f(0) = f'(0)(x - 0), which simplifies to y = (1/2)(10)^{-1/2}x + \\sqrt{10}. This form equation is the linear approximation of f(x) near x = 0.

Answer:

Step-by-step explanation:

The probability that a randomly chosen number between 1 and 100 is divisible by 3, given that the number has at least one digit equal to 5 is 6/19,

It is defined as the ratio of the number of favorable outcomes to the total number of outcomes, in other words, the probability is the number that shows the happening of the event.

It is given that:

The randomly chosen number between 1 and 100 is divisible by 3

Applying conditional probability:

Let A is the event: the numbers divisible by 3

Let B is the event: At least one digit equal to 5

P(A|B) = n(A∩B)/n(B)

P(A|B) = 6/19

Thus, the probability that a randomly chosen number between 1 and 100 is divisible by 3, given that the number has at least one digit equal to 5 is 6/19.

Learn more about the probability here:

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The probability that a single randomly selected salary is at least $134,000 is approximately 0.5120 or 51.20%.

To find the probability that a single randomly selected salary is at least $134,000, we need to calculate the z-score and use the standard normal distribution.

1. Calculate the z-score:

z = (x - μ) / σ

where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.

In this case, x = $134,000, μ = $133,000, and σ = $31,000.

z = (134000 - 133000) / 31000

z = 1 / 31000

z ≈ 0.0323

2. Find the probability associated with the z-score:

We can use a z-table or a calculator to find the probability.

From the z-table, we find that the probability corresponding to a z-score of 0.0323 is approximately 0.5120.

Therefore, the probability that a single randomly selected salary is at least $134,000 is 0.5120 or 51.20%.

Answer:

P = 4s

Step-by-step explanation:

The perimeter of a geometric shape is simply the sum of all its sides length. Since the shape in question is a square, which means that all of the four sides have the same length 's', the perimeter can be expressed by:

[tex]P = s+s+s+s\\P=4s[/tex]

For any value of 's', the formula above expresses the perimeter 'P' as a function of 's'

Answer: individual

Step-by-step explanation:

An individual is a person or object that is a member of the population being studied. A population is defined as a group of individuals with a common characteristic living and interbreeding within a given area, in statistics, population is a collection of individuals to be studied. Individuals can also be referred to as the objects/person described by a set of data. For example: when studying the height of students in a school, the students attending that school are individuals.

Answer:  0.1

Step-by-step explanation:

WE know that the total probability in an experiments = 1.

i.e. Sum of the probabilities of occurring each event is 1.

i.e. If there are n outcomes in any experiment., then the total probability will be:

[tex]P(E_1)+P(E_2)+P(E_3)+...........+P(E_n)=1[/tex]

Given : An experiment consists of four outcomes ,with P(E1) = 0.2, P(E2) = 0.3, and P(E3) = 0.4.

Then , [tex]P(E_1)+P(E_2)+P(E_3)+P(E_4)=1[/tex]

Substitute corresponding values , we get

[tex]0.2+0.3+0.4+P(E_4)=1[/tex]

[tex]0.9+P(E_4)=1[/tex]

[tex]P(E_4)=1-0.9=0.1[/tex]

Hence , the probability of outcome [tex]E_4[/tex] is 0.1.

The probability of event E4 in the given experiment is 0.1, because the sum of probabilities of all outcomes should equal 1.

The problem falls under the subject of

Probability Theory

within Mathematics. In probability, the total probability of all possible outcomes is always 1. So, for the given problem where you have four events E1, E2, E3, E4, the total probability P(E1)+P(E2)+P(E3)+P(E4) should equal 1. Given that P(E1) = 0.2, P(E2) = 0.3 and P(E3) = 0.4, we can find P(E4) using the equation

1 - P(E1) - P(E2) - P(E3) = P(E4)

. By substituting the given values into this equation, we find that P(E4) = 1 - 0.2 - 0.3 - 0.4 =

0.1

. Therefore, the probability of outcome E4 is 0.1.

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Answer:

A.

sigma > 22500

B.

Null hypothesis:sigma = 22500

Alternative hypothesis:sigma > 22500

Step-by-step explanation:

A.

The claim states that the standard deviation of high school teachers income  is more than 22,500. This can be represented in the symbolic form as sigma > 22500.

B.

The null hypothesis and alternative hypothesis for the given scenario can be written as

Null hypothesis: Standard deviation of income of high school teachers is 22,500.

The standard deviation is represented as sigma.Symbolically it can be written as

Null hypothesis: sigma = 22500

Alternative hypothesis: Standard deviation of income of high school teachers is more than 22,500.

Symbolically it can be written as

Alternative hypothesis: sigma > 22500

Answer:

yes

Step-by-step explanation:

the rate of change is constant

Answer:

20

Step-by-step explanation:

You could find 5/⅓

by using 5 × 3

Knowing that:

i. Any number multiplied by 1, gives the number itself.

ii. Dividing any number by itself gives 1.

You would agree with me that

i. (5×3)/(5×3) = 1

ii. Writing 5/⅓ as 5/⅓ × 1 doesn't change the value.

Then I can write 5/⅓ as

5/⅓ × (5×3)/(5×3) = 1

This can become

[5×(5×3)] / [(⅓) × (5×3)]

= 75/(15/3)

= 75/5

= 15

In a similar way,

12/ 3/5

= [12/ (3/5)] × [(5×3)/(5×3)]

= 12×(5×3) / (3/5)×(5×3)

= (12×5×3) / [(3×5×3)/5]

= 180 / (45/5)

= 180 / 9

= 20

Answer:

a) [tex]P(X=12)=(12C12)(0.2)^{12} (1-0.2)^{12-12}=4.096x10^{-9}[/tex]

b) [tex]P(X \geq 1) =1-P(X<1)= 1-P(X=0)=1-0.0687=0.9313 [/tex]

c) [tex] E(X) = np = 12*0.2= 2.4[/tex]

d) [tex] Sd(X) = \sqrt{np(1-p)}=\sqrt{12*0.2*(1-0.2)}=1.386[/tex]

e) [tex] P(X<2.4) =P(X\leq2) =P(X=0) +P(X=1)+P(X=2)= 0.558[/tex]

f)  [tex] P(X\leq 10) =1.1x10^{-9}[/tex]

Step-by-step explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Solution to the problem

Let X the random variable of interest, on this case we now that:

[tex]X \sim Binom(n=12, p=0.2)[/tex]

The probability mass function for the Binomial distribution is given as:

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]

Where (nCx) means combinatory and it's given by this formula:

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]

Part a

For this case we want to find this probability:

[tex]P(X=12)=(12C12)(0.2)^{12} (1-0.2)^{12-12}=4.096x10^{-9}[/tex]

Part b

[tex]P(X \geq 1) =1-P(X<1)= 1-P(X=0) [/tex]

[tex]P(X=0)=(12C0)(0.2)^{0} (1-0.2)^{12-0}=0.0687[/tex]

[tex]P(X \geq 1) =1-P(X<1)= 1-P(X=0)=1-0.0687=0.9313 [/tex]

Part c

The expected value is given by:

[tex] E(X) = np = 12*0.2= 2.4[/tex]

Part d

The standard deviation is given by:

[tex] Sd(X) = \sqrt{np(1-p)}=\sqrt{12*0.2*(1-0.2)}=1.386[/tex]

Part e

If we want the probability that the number classified as deceptive would be lower than the mean we want:

[tex] P(X<2.4) =P(X\leq2) =P(X=0) +P(X=1)+P(X=2)[/tex]

[tex]P(X=0)=(12C0)(0.2)^{0} (1-0.2)^{12-0}=0.0687[/tex]

[tex]P(X=1)=(12C1)(0.2)^{1} (1-0.2)^{12-1}=0.2062[/tex]

[tex]P(X=2)=(12C2)(0.2)^{2} (1-0.2)^{12-2}=0.2835[/tex]

[tex] P(X<2.4) =P(X\leq2) =P(X=0) +P(X=1)+P(X=2)= 0.558[/tex]

Part f

For this case our random variable would be:

[tex]X \sim Binom(n=200, p=0.2)[/tex]

And we want this probability:

[tex] P(X\leq 10) = P(X=0)+P(X=1)+ .......+P(X=10)[/tex]

And we can use the following excel code to find the answer:

"=BINOM.DIST(10;200;0.2;TRUE)"

And we got: [tex] P(X\leq 10) =1.1x10^{-9}[/tex]

Answer:

6 is a factor of n

Step-by-step explanation:

2 is a factor of n and 3 is a factor of n means

n = 2×3×k

  = 6×k

then  n = 6×k

then 6 is a factor of n

Final answer:

If both 2 and 3 are factors of a number n, then 6 must also be a factor of n, because the product of unique prime factors is always a factor of that number.

Explanation:

The product of unique prime factors of a number will be a factor of that number. Since 2 and 3 are prime factors and both are factors of n, their product (which is 6) must also be a factor of n.

For example, consider the number 12. 12 is divisible by 2 and 12 is divisible by 3, and indeed, 12 is divisible by 6 as well. This holds true for any number n that has 2 and 3 as factors. Thus, we can conclude that 6 is a factor of n if both 2 and 3 are factors of n.

Answer:

1) The solution of the system is

[tex]\left\begin{array}{ccc}x_1&=&5\\x_2&=&8\\x_3&=&-13\end{array}\right[/tex]

2) The solution of the system is

[tex]\left\begin{array}{ccc}x_1&=&2\\x_2&=&-7\\x_3&=&-1\end{array}\right[/tex]

Step-by-step explanation:

1) To solve the system of equations

[tex]\left\begin{array}{ccccccc}&3x_2&-5x_3&=&89\\6x_1&&+x_3&=&17\\x_1&-x_2&+8x_3&=&-107\end{array}\right[/tex]

using the row reduction method you must:

Step 1: Write the augmented matrix of the system

[tex]\left[ \begin{array}{ccc|c} 0 & 3 & -5 & 89 \\\\ 6 & 0 & 1 & 17 \\\\ 1 & -1 & 8 & -107 \end{array} \right][/tex]

Step 2: Swap rows 1 and 2

[tex]\left[ \begin{array}{ccc|c} 6 & 0 & 1 & 17 \\\\ 0 & 3 & -5 & 89 \\\\ 1 & -1 & 8 & -107 \end{array} \right][/tex]

Step 3:  [tex]\left(R_1=\frac{R_1}{6}\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 3 & -5 & 89 \\\\ 1 & -1 & 8 & -107 \end{array} \right][/tex]

Step 4: [tex]\left(R_3=R_3-R_1\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 3 & -5 & 89 \\\\ 0 & -1 & \frac{47}{6} & - \frac{659}{6} \end{array} \right][/tex]

Step 5: [tex]\left(R_2=\frac{R_2}{3}\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 1 & - \frac{5}{3} & \frac{89}{3} \\\\ 0 & -1 & \frac{47}{6} & - \frac{659}{6} \end{array} \right][/tex]

Step 6: [tex]\left(R_3=R_3+R_2\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 1 & - \frac{5}{3} & \frac{89}{3} \\\\ 0 & 0 & \frac{37}{6} & - \frac{481}{6} \end{array} \right][/tex]

Step 7: [tex]\left(R_3=\left(\frac{6}{37}\right)R_3\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 1 & - \frac{5}{3} & \frac{89}{3} \\\\ 0 & 0 & 1 & -13 \end{array} \right][/tex]

Step 8: [tex]\left(R_1=R_1-\left(\frac{1}{6}\right)R_3\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 5 \\\\ 0 & 1 & - \frac{5}{3} & \frac{89}{3} \\\\ 0 & 0 & 1 & -13 \end{array} \right][/tex]

Step 9: [tex]\left(R_2=R_2+\left(\frac{5}{3}\right)R_3\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 5 \\\\ 0 & 1 & 0 & 8 \\\\ 0 & 0 & 1 & -13 \end{array} \right][/tex]

Step 10: Rewrite the system using the row reduced matrix:

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 5 \\\\ 0 & 1 & 0 & 8 \\\\ 0 & 0 & 1 & -13 \end{array} \right] \rightarrow \left\begin{array}{ccc}x_1&=&5\\x_2&=&8\\x_3&=&-13\end{array}\right[/tex]

2) To solve the system of equations

[tex]\left\begin{array}{ccccccc}4x_1&-x_2&+3x_3&=&12\\2x_1&&+9x_3&=&-5\\x_1&+4x_2&+6x_3&=&-32\end{array}\right[/tex]

using the row reduction method you must:

Step 1:

[tex]\left[ \begin{array}{ccc|c} 4 & -1 & 3 & 12 \\\\ 2 & 0 & 9 & -5 \\\\ 1 & 4 & 6 & -32 \end{array} \right][/tex]

Step 2: [tex]\left(R_1=\frac{R_1}{4}\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & - \frac{1}{4} & \frac{3}{4} & 3 \\\\ 2 & 0 & 9 & -5 \\\\ 1 & 4 & 6 & -32 \end{array} \right][/tex]

Step 3: [tex]\left(R_2=R_2-\left(2\right)R_1\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & - \frac{1}{4} & \frac{3}{4} & 3 \\\\ 0 & \frac{1}{2} & \frac{15}{2} & -11 \\\\ 1 & 4 & 6 & -32 \end{array} \right][/tex]

Step 4: [tex]\left(R_3=R_3-R_1\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & - \frac{1}{4} & \frac{3}{4} & 3 \\\\ 0 & \frac{1}{2} & \frac{15}{2} & -11 \\\\ 0 & \frac{17}{4} & \frac{21}{4} & -35 \end{array} \right][/tex]

Step 5: [tex]\left(R_2=\left(2\right)R_2\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & - \frac{1}{4} & \frac{3}{4} & 3 \\\\ 0 & 1 & 15 & -22 \\\\ 0 & \frac{17}{4} & \frac{21}{4} & -35 \end{array} \right][/tex]

Step 6: [tex]\left(R_1=R_1+\left(\frac{1}{4}\right)R_2\right)[/tex]

[tex]\left[ \begin{array}{cccc} 1 & 0 & \frac{9}{2} & - \frac{5}{2} \\\\ 0 & 1 & 15 & -22 \\\\ 0 & \frac{17}{4} & \frac{21}{4} & -35 \end{array} \right][/tex]

Step 7: [tex]\left(R_3=R_3-\left(\frac{17}{4}\right)R_2\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{9}{2} & - \frac{5}{2} \\\\ 0 & 1 & 15 & -22 \\\\ 0 & 0 & - \frac{117}{2} & \frac{117}{2} \end{array} \right][/tex]

Step 8: [tex]\left(R_3=\left(- \frac{2}{117}\right)R_3\right)[/tex]

[tex]\left[ \begin{array}{cccc} 1 & 0 & \frac{9}{2} & - \frac{5}{2} \\\\ 0 & 1 & 15 & -22 \\\\ 0 & 0 & 1 & -1 \end{array} \right][/tex]

Step 9: [tex]\left(R_1=R_1-\left(\frac{9}{2}\right)R_3\right)[/tex]

[tex]\left[ \begin{array}{cccc} 1 & 0 & 0 & 2 \\\\ 0 & 1 & 15 & -22 \\\\ 0 & 0 & 1 & -1 \end{array} \right][/tex]

Step 10: [tex]\left(R_2=R_2-\left(15\right)R_3\right)[/tex]

[tex]\left[ \begin{array}{cccc} 1 & 0 & 0 & 2 \\\\ 0 & 1 & 0 & -7 \\\\ 0 & 0 & 1 & -1 \end{array} \right][/tex]

Step 11:

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 2 \\\\ 0 & 1 & 0 & -7 \\\\ 0 & 0 & 1 & -1 \end{array} \right]\rightarrow \left\begin{array}{ccc}x_1&=&2\\x_2&=&-7\\x_3&=&-1\end{array}\right[/tex]

Answer:

F(x=11)= (-31)

Step-by-step explanation:

for the function

F(x) = x² - 13*x + 22/x-11 , for x ≠ 11

then in order to define F(x=11) so  F is continuous (see Note below) . By definition of continuity of a function:

F(x) is continuous in x=11 if lim F(x)=F(a) when x→a

then

when x→a , lim x² - 13x + 22/x-11 = lim 11² - 13*11 + 22/11 -11 = -3*11 + 2 = -31 = F(x=11)

then

F(x=11)= (-31)

Note:

F is not continuous in all x since

when x→0⁺ ,  lim (0⁺) ² - 13*0⁺  + 22/0⁺ -11 = (+∞)

when x→0⁻,  lim (0⁻) ² - 13*0⁻  + 22/0⁻ -11 = (-∞)

then

limit F(x) , when x→0 does not exist since the limit from the left and from the right do not converge → since the limit does not exist , the function is not continuous  in x=0

Answer:

a. 24ways

b.256ways

Step-by-step explanation:

the letters IOWA contains for letters, since we are to arrange without repeating any letter, we permutate the letters.

For permutation of n object in r ways is expressed as

P(n,r)=n!/(n-r)!

hence for n=4 and r=4, we have P(4,4)=4!/(4-4)!

P(4,4)=4!/(0)!

P(4,4)=4*3*2*1=24ways

b. To arrange the letters such that each letter can be repeated, we can arrange the letter I in four ways, letter O can be arrange in four ways, letter W can be arranged in four ways and letter A can be arranged in four ways ..

Hence we arrive at

4*4*4*4=256ways

[tex]x''+4x'+53x=15[/tex]

has characteristic equation

[tex]r^2+4r+53=0[/tex]

with roots at [tex]r=-2\pm7i[/tex]. Then the characteristic solution is

[tex]x_c=C_1e^{(-2+7i)t}+C_2e^{(-2-7i)t}=e^{-2t}\left(C_1\cos(7t)+C_2\sin(7t)\right)[/tex]

For the particular solution, consider the ansatz [tex]x_p=a_0[/tex], whose first and second derivatives vanish. Substitute [tex]x_p[/tex] and its derivatives into the equation:

[tex]53a_0=15\implies a_0=\dfrac{15}{53}[/tex]

Then the general solution to the equation is

[tex]x=e^{-2t}\left(C_1\cos(7t)+C_2\sin(7t)\right)+\dfrac{15}{53}[/tex]

With [tex]x(0)=8[/tex], we have

[tex]8=C_1+\dfrac{15}{53}\implies C_1=\dfrac{409}{53}[/tex]

and with [tex]x'(0)=-19[/tex],

[tex]-19=-2C_1+7C_2\implies C_2=-\dfrac{27}{53}[/tex]

Then the particular solution to the equation is

[tex]\boxed{x(t)=\dfrac1{53}e^{-2t}(409cos(7t)-27\sin(7t)+15)}[/tex]

The given equation is a second order linear differential equation. However, there seems to be an inconsistency with the constant right-hand side. In a well-formulated equation, one would solve this by using characteristic roots and trivial solutions, finding the general solution, then applying initial conditions.

The given equation is a second order linear differential equation. Given the initial conditions, including the fact that x(0) = 8 and x˙ = −19, one must adjust for these when solving the differential equation. By utilizing the characteristic equation for determining the roots, the solution and consequent constants are then determined.

However, please note that without some form of driving force (right-hand side function), this is a simple harmonic oscillator equation. Since the right hand side function (15 in this case) is constant, there's an inconsistency in the problem. In this context, for a correct form, it should have a time-dependent function on the right side. If we assume that an inconsistency has occurred and the right side is zero, a full solution could be given.

In a correct equation scenario, one would be able to solve the initial value problem with characteristic roots and trivial solutions, find the particular solution, then a general solution and apply initial value conditions to find specific constants.

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x² + y² + z² - 10x - 4y - 14z + 74 = 0

Step-by-step explanation:

The general equation of a sphere is (x-a)² + (y-b)² + (z-c)² = r²

Where x, y, and z are the coordinates of points on the surface of the sphere.

a, b, and c represents the center of the sphere

r is the radius of the sphere. Note that the radius is always the same for all points on the sphere,

In this equation, the radius r is the largest radius that stays in the octant.

In the given question, (5,2,7) is the center of the sphere.

Therefore, substitute this into the general equation to get:

(x-5)²+(y-2)²+(z-7)² = r² ---------------------------------------------(i)

To find the radius r, we have to look at the distance from the center coordinate to each bounding planes xy-plane, xz-plane, and yz-plane.

The distance from the center to the xy-plane is the center of the z coordinate which is 7. The distance from the center to the xz-plane is the center of the y coordinate which is 2. The distance from the center to the yz-plane is the center of the x coordinate which is 5.

Therefore, to determine the radius contained in the first octant, we need to choose the smallest distance so as not to cross into a second octant. That will also be the largest possible radius for it not to cross into a different octant.

The smallest distance therefore is 2. So we substitute r = 2 into equation (i) above to get:

(x-5)²+(y-2)²+(z-7)² = 2²

(x-5)²+(y-2)²+(z-7)² = 4

Therefore (x-5)²+(y-2)²+(z-7)²-4 = 0  ----------------------------------------(ii)

A monic formula is a formula where the highest power of its single variable has a coefficient of 1.

Therefore, we expand equation (ii) in form of a monic formula to get

x² + y² + z² - 10x - 4y - 14z + 74 = 0

The highest power of x², y², and z² is 1

Answer:

The probability that at most 3 attempts are required to produce a craft item with satisfactory quality is 0.9375

Step-by-step explanation:

Let E be a random variable denoting the event that an artist makes a craft item with satisfactory level.

Then the random variable E follows a Geometric distribution.

A Geometric distribution is defined as the number of failures (k) before the first success.

The probability function of Geometric distribution is:

[tex]P(X=k)=(1-p)^{k}p[/tex], p = Probability of success and k = 0, 1, 2, 3...

The probability of success is, p = 0.5 and the number of failures is, k = 3.

Compute the probability of at most 3 attempts before the first success is:

[tex]P(X\leq 3) =P(X=3)+P(X = 2)+P(X=1) +P(X = 0)\\=[(1-0.5)^{0}*0.5]+[(1-0.5)^{1}*0.5]+[(1-0.5)^{2}*0.5]+[(1-0.5)^{3}*0.5]\\=0.9375[/tex]

Therefore, the probability that at most 3 attempts are required to produce a craft item with satisfactory quality is 0.9375.

Answer:

11)y = [tex]\frac{3}{2} e^{x} - \frac{1}{2} e^{-x}[/tex]

12)y = [tex]\frac{e^{2} }{1+e^{2} } (e^{x} - e^{-x} )[/tex]

13)y = [tex]5e^{-(x+1)}[/tex]

14)y = 0

Step-by-step explanation:

Given data:

[tex]y=c_{1} e^{x} +c_{2} e^{-x}[/tex]

y''-y=0

The equation is

[tex]m^{r}[/tex]-1 = 0

(m-1)(m+1) = 0

if  above equation is zero then either

m - 1 = 0 or  m + 1 = 0

m = 1        ,    m  = - 1

11)

y(0) = 1 , y'(0) = 2

[tex]y'=c_{1} e^{x} -c_{2} e^{-x}[/tex]

[tex]c_{1}[/tex] +  [tex]c_{2}[/tex] = 1   (y(0) = 1) (1)

[tex]c_{1}[/tex] -  [tex]c_{2}[/tex] = 2   (y'(0) = 2)  (2)

adding 1 & 2

2[tex]c_{1}[/tex] = 3

[tex]c_{1}[/tex] = 3/2

3/2 +  [tex]c_{2}[/tex] = 1

[tex]c_{2}[/tex]  = 1 -  3/2

[tex]c_{2}[/tex] = - 1/2

y = [tex]\frac{3}{2} e^{x} - \frac{1}{2} e^{-x}[/tex]

12)

y(0) = 1 , y'(0) = e

[tex]c_{1}[/tex] +  [tex]c_{2}[/tex] = 0 (y(0) = 1) (3)

[tex]c_{1}[/tex] = - [tex]c_{2}[/tex]

[tex]e=c_{1} e -c_{2} e^{-1}[/tex]   (y'(0) = 2)  (4)

[tex]e=c_{1} e -\frac{c_{2} }{e} }[/tex]

[tex]e =\frac{c_{1} e^{2} -c_{2} }{e} }[/tex]

[tex]e^{2} ={c_{1} e^{2} -c_{2} }[/tex]

replace [tex]c_{2}[/tex] = [tex]c_{1}[/tex] by equation 3

[tex]e^{2} ={c_{1} e^{2} -c_{1} }[/tex]

taking common [tex]c_{1}[/tex]

[tex]e^{2} =c_{1} ({e^{2} -1 })[/tex]

[tex]\frac{e^{2} }{({e^{2} -1 })} =c_{1}[/tex]

[tex]-\frac{e^{2} }{({e^{2} -1 })} =c_{2}[/tex]

∴ y = [tex]\frac{e^{2} }{1+e^{2} } (e^{x} - e^{-x} )[/tex]

13)

y(-1) = 5 , y'(-1) = -5

[tex]c_{1}[/tex][tex]e^{-1}[/tex] +  [tex]c_{2}[/tex][tex]e^{1}[/tex] = 5   (y(-1) = 5 ) (5)

[tex]c_{1}[/tex][tex]e^{-1}[/tex] -  [tex]c_{2}[/tex][tex]e^{1}[/tex] = -5    (y'(-1) = -5)  (6)

Adding 5&6

2[tex]c_{1}[/tex] [tex]e^{-1}[/tex] = 0

[tex]c_{1}[/tex] = 0

[tex]c_{2}[/tex][tex]e^{1}[/tex] = 5 - [tex]c_{1}[/tex][tex]e^{-1}[/tex]

[tex]c_{2}[/tex][tex]e^{1}[/tex] = 5 - 0

[tex]c_{2}[/tex]= 5/e

y = [tex]5e^{-1} e^{-x}[/tex]

y = [tex]5e^{-(x+1)}[/tex]

14)

y(0) = 0 , y'(0) = 0

[tex]c_{1}[/tex] +  [tex]c_{2}[/tex] =  0 (y(0) = 0) (7)

[tex]c_{1}[/tex] -  [tex]c_{2}[/tex] = 0   (y'(0) = 0)  (8)

Adding 7 & 8

2[tex]c_{1}[/tex] = 0

[tex]c_{2}[/tex] =

y = 0

Answer:

8.99 days elapsed. Option (C) is correct

Step-by-step explanation:

The distribution  has density function k/t+1 for a constant k and t between 0 and 6.5 . Since the distribution is symmetrical in 6.5, the area it forms between 0 and 6.5 should be 1/2, thus

[tex]\frac{1}{2} = \int\limits_0^{6.5} \frac{k}{t+1} \, dt = k *(ln(t+1) \, |_0^{6.5}) = k * (ln(7.5)-ln(1)) = k*ln(7.5)[/tex]

Hence k = 1/(2ln(7.5)), approx 1/4.

We need to find the percentil 0.6, since the integral of the random variable is 1/2 over the first half, we need to find t such that the integral of the random variable between o and 6.5 + t is 0.6. This is equivalent to find t such that the integral between 6.5 and 6.5+t is 0.1. Due to the  over 6.5, this t should satisfy that the integral between 6.5-t and 6.5 is also 0.1. Lets compute the integral and find t

[tex]\int\limits^{6.5}_{6.5-t} {\frac{k}{t+1}} \, dx = \frac{1}{2ln(7.5)}*(ln(t+1) \, |_{6.5-t}^{6.5} \, ) = \frac{1}{2ln(7.5)} * (ln(7.5)-ln(7.5-t)) = \\\frac{1}{2} - \frac{ln(7.5-t)}{2ln(7.5)} = 0.1[/tex]

Therefore,

[tex]\frac{ln(7.5-t)}{2ln(7.5)} = 0.4\\\\ln(7.5-t) = 0.8*ln(7.5)\\\\7.5-t = e^{0.8*ln(7.5)}\\\\t = 7.5-e^{0.8*ln(7.5)} = 2.49[/tex]

As a result, the student sould have registered 2.49 days after the day 6.5, thus it should have registeredd at day 8.99. Option (C) is correct.

Answer:  The proofs are given below.

Step-by-step explanation:  We are given to prove that the following statements are tautologies using truth table :

(a) ¬r ∨ (¬r → p)                              b. ¬(p → q) → ¬q

We know that a statement is a TAUTOLOGY is its value is always TRUE.

(a) The truth table is as follows :

r                 p                 ¬r                       ¬r→p                     ¬r ∨ (¬r → p)

T                T                   F                         T                                T

T                F                   F                         T                                T

F                T                   T                         T                                T

F                F                   T                         F                                T  

So, the statement (a) is a  tautology.

(b) The truth table is as follows :

p                 q                 ¬q                       p→q             ¬(p→q)          ¬(p→q)→q

T                T                   F                         T                      F                    T

T                F                   T                         F                      T                    T

F                T                   F                         T                      F                    T

F                F                   T                         T                       F                   T

So, the statement (B) is a  tautology.              

Hence proved.

Final answer:

To prove that a statement is a tautology using propositional logic, we need to show that the statement is true under all possible truth values of its variables. By applying the laws of implication, disjunction, and contradiction, we can prove that the given statements are tautologies.

Explanation:

To prove that a statement is a tautology using the laws of propositional logic, we need to show that the statement is true under all possible truth value assignments of its variables. Let's consider each statement:

a. ¬r ∨ (¬r → p)

We can use the law of implication, which states that ¬p ∨ q is equivalent to p → q, to rewrite the statement as ¬r ∨ (r → p). By applying the law of disjunction, which states that p ∨ (q ∧ r) is equivalent to (p ∨ q) ∧ (p ∨ r), we can further rewrite the statement as (¬r ∧ r) ∨ (r ∨ p). Using the law of contradiction, which states that p ∧ ¬p is always false, we can simplify the statement to r ∨ p, which is a tautology.

b. ¬(p → q) → ¬q

We can use the law of implication to rewrite the statement as ¬(¬p ∨ q) → ¬q. By applying De Morgan's law, which states that ¬(p ∨ q) is equivalent to ¬p ∧ ¬q, we can simplify the statement to (p ∧ ¬q) → ¬q. Using the law of contradiction, we know that p ∧ ¬p is always false, so the statement simplifies to false → ¬q, which is always true. Therefore, it is a tautology.

Answer:

the explanation is given below.

Step-by-step explanation:

from this, the lowest number is 1 and the highest number is 100

Answer:

Mean = 68.375

Step-by-step explanation

The mean will be:

Mean= sum of scores / number of students

Mean=1641/24

Mean = 68.375

Final answer:

Using the pigeonhole principle, we can prove that among 502 positive integers, at least two will have the same remainder when divided by 1000, implying their difference is divisible by 1000.

Explanation:

The question asks to prove that among 502 positive integers, there are always two integers so that either their sum or their difference is divisible by 1000. This statement can be understood through the pigeonhole principle, which in basic terms means if you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon.

In this case, consider the remainders when these integers are divided by 1000. Since there are only 1000 possible remainders (from 0 to 999), and we have 502 numbers, at least two of them must have the same remainder when divided by 1000, according to the pigeonhole principle.

Let these two numbers be a and b, where without loss of generality, a ≥ b. If a and b have the same remainder when divided by 1000, then a - b is divisible by 1000. Alternatively, if we had a case where the sum is considered, assuming complementary pairs mod 1000, a similar argument involving the pigeonhole principle can conclude that there must be at least one pair whose sum or difference gives a number divisible by 1000, satisfying the initial claim.

Answer:  The required laplace transform of g(t) is [tex]\dfrac{5}{(s+5)^2}.[/tex]

Step-by-step explanation:  We are given to find the laplace transform of the following function :

[tex]g(t)=5te^{-5t}.[/tex]

We know the following formulas for laplace transform :

[tex](i)~L\{t^ne^{at}\}=\dfrac{n!}{(s-a)^{n+1}},\\\\(ii)~L\{cf(t)\}=cL\{f(t)\}.[/tex]

In the given function function, we have

c = 5,  n = 1  and  a = -5.

Therefore, we get

[tex]L\{g(t)\}\\\\=L\{5te^{-5t}\}\\\\=5L\{te^{-5t}\}\\\\\\=5\times\dfrac{1!}{(s-(-5))^{1+1}}\\\\\\=\dfrac{5}{(s+5)^2}.[/tex]

Thus, the required laplace transform of g is [tex]\dfrac{5}{(s+5)^2}.[/tex]

The Laplace transform of the given function is [tex]\frac{5} { (s + 5)^2}[/tex].

The Laplace transform of a function g(t) is defined as:

[tex]L{(g(t))} = \int\limits^{\infty}_0 e^-^s^tg(t) dt[/tex]

We need to find the Laplace transform of [tex]g(t) = 5te^-^5^t[/tex]. To do this, we use the shifting theorem and known transforms.

First, recall the Laplace transform of [tex]nte^-^a^t[/tex] is:

[tex]L(nte^-^a^t)} = \frac{n! } {(s + a)^n^+^1}[/tex]

For our function [tex]g(t)[/tex] :

a = 5

n = 1

Applying the formula:

[tex]L{(5te^-^5^t)} = \frac{(5 * 1! )}{(s + 5)^2}[/tex]

After simplifying, we get:

[tex]L\leftparanthesis(\ 5te^-^5^t)\rightparanthesis\ = \frac{5 }{ (s + 5)^2}[/tex]

Answer:

No, because it is not the tens digit that goes up by 1 in these numbers, it is the unit digit.

Step-by-step explanation:

It is important to know the concepts of units, tenths and cents.

For example

1 = 1 unit

10 = 1*10 + 0 = The tens digit is one the unit digit is 0

21 = 2*10 + 1 = The tens digit is two and the unit digit is 1.

120 = 1*100 + 2*10 + 0 = The cents digit is 1, the tens digit is two and the unit digit is 0.

So

Adding 1 is the same as the unit digit going up by 1.

Adding 10 is the same as the tens digit going up by 1.

Adding 100 is the same as the cents digit going up by 1.

In this problem, we have that:

864,865,866,867,868,869

Each value is the 1 added to the previous value, that is, the unit digit goes up by 1.

Mariel thinks the tens digit goes up by 1 in these numbers. Do you agree?

No, because it is not the tens digit that goes up by 1 in these numbers, it is the unit digit.

Answer:

disagree, its the unit number that goes up by 1

Step-by-step explanation: