Determine whether the geometric series is convergent or divergent. 6 + 5 + 25/6 + 125/36 + ...
If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)

To find the net worth of the Crabby Apple restaurant at the beginning of January before it lost $2500, we add back the loss to the net worth at the end of the month, resulting in an initial net worth of $2900.

The student is asking about calculating the net worth of a restaurant before a financial loss. To find the net worth at the beginning of the month, we would add the loss sustained during that month back to the net worth at the end of the month. Since the restaurant lost $2500 in January, and had a net worth of $400 at the end of January, we can calculate its net worth at the beginning of January by the following calculation:

Net Worth at Beginning of Month = Net Worth at End of Month + Loss During Month

Net Worth at Beginning of Month = $400 + $2500

Net Worth at Beginning of Month = $2900

Therefore, the net worth of the Crabby Apple restaurant at the beginning of January was $2900.

The net worth of the Crabby Apple restaurant at the beginning of January was $2900.

To determine the net worth of the Crabby Apple restaurant at the beginning of January, we need to account for the loss incurred during the month. Given that the net worth at the end of January is $400 and the restaurant lost $2500 during the month, we can set up the following equation:

[tex]\[\text{Net worth at the beginning of January} =[/tex][tex]\text{Net worth at the end of January} + \text{Loss during January}\][/tex]

Substitute the given values into the equation:

[tex]\[\text{Net worth at the beginning of January} = \$400 + \$2500\][/tex]

[tex]\[\text{Net worth at the beginning of January} = \$2900\][/tex]

Answer:

Step-by-step explanation:

Let us calculate std dev and std error of two samples

Sample     N     Mean    Std dev   Std error

  1              65     52.8      9.9         1.22279

  2             111      54.1       11.09      1.0526

Assuming equal variances

df =111+65-2=174

Pooled std deviation = combined std deviation = 10.6677

Pooled std error = 10.6677/sqrt 174 = 0.808

difference in means =52.8-54.1 = -1.3

Margin of error = 0.5923

( t critical 1.6660)

Confidence interval

=(-4.4923, 1.8923)

Since 0 lies in this interval we can accept null hypothesis that 10 year old boys and girls have the same average height

Answer:

Faster worker takes 4 hours and slower worker takes 12 hours.

Step-by-step explanation:

Let x be the time ( in hours ) taken by faster worker,

So, according to the question,

Time taken by slower worker = (x+8) hours,

Thus, the one day work of faster worker = [tex]\frac{1}{x}[/tex]

Also, the one day work of slower worker = [tex]\frac{1}{x+8}[/tex]

So, the total one day work when they work together = [tex]\frac{1}{x}+\frac{1}{x+8}[/tex]

Given,

They take 3 hours in working together,

So, their combined one day work = [tex]\frac{1}{3}[/tex]

[tex]\implies \frac{1}{x}+\frac{1}{x+8}=\frac{1}{3}[/tex]

[tex]\frac{x+8+x}{x^2+8x}=\frac{1}{3}[/tex]  ( Adding fractions )

[tex]3(2x+8)=x^2+8x[/tex]    ( Cross multiplication )

[tex]6x+24=x^2+8x[/tex]       ( Distributive property )

[tex]x^2+2x-24=0[/tex]          ( Subtraction property of equality )

By quadratic formula,

[tex]x=\frac{-2\pm \sqrt{100}}{2}[/tex]

[tex]x=\frac{-2\pm 10}{2}[/tex]

[tex]\implies x=4\text{ or }x=-6[/tex]

Since, hours can not negative,

Hence, time taken by faster worker = x hours = 4 hours,

And, the time taken by slower worker = x + 8 = 12 hours.

To solve this work rate problem, we set up an equation with combined work rates and find that the faster worker takes 3 hours alone, while the slower worker takes 11 hours alone.

The question states that two secretaries can stuff envelopes together in 3 hours. The slower worker takes 8 hours more than the faster worker to complete the job alone. To find how long it takes each secretary to complete the job alone, we can set up an equation using the reciprocal of their work rates.

Let x be the time it takes for the faster worker to stuff the envelopes alone. Then, the slower worker will take x + 8 hours. The work rate of the faster worker is 1/x and the slower worker's rate is 1/(x + 8). Working together, their combined work rate is 1/3 per hour (since they complete the task in 3 hours).

The combined work rate equation will be:

1/x + 1/(x + 8) = 1/3

To solve this equation:

As a result, the faster worker takes 3 hours to complete the job alone, and the slower worker takes 11 hours to complete the job alone.

Answer:

a) bijective

b) neither surjective or injective

c) injective

Step-by-step explanation:

Since we are looking from real numbers to real numbers, we want the following things

1) We want every real number y to get it. (surjective)

2) We want every y that gets hit to be hit only once. (injective).

---

If we have both things then the function is bijective.

a) f(x)=5x+4 this is a line with a positive slope.

That means it is increasing left to right.  Every increasing or even decreasing line is going to hit every real number y.  This is a bijection.

b) g(x)=2x^2-2 is a parabola. Parabola functions always have y's that get hit more than once and not all y's get hit because the parabola is either open up or down starting from the vertex.  This function is neither injective or surjective.

c) h(x)=1+(2/x)  x is not 0.

1+(2/x) is never 1 because 2/x is never 0 for any x. This means the real number y=1 will never be hit and is therefore not surjective. This function is injective because every 1 that is hit is only hit once. If you want use the horizontal line test to see this.

Answer:

The probability is 0.1444.

Step-by-step explanation:

Let X be the event that college business students cheated on final exams,

Since, the probability that student cheats on exam, p = 75 % = 0.75,

So, the probability that student does not cheat on exam, q = 1 - p = 0.25,

The binomial distribution formula,

[tex]P(x)=^nC_r p^r q^{n-r}[/tex]

Where, [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]

The probability that exactly 30 of the students monitored cheat on the final exam out of 40 students is,

[tex]P(X=30)=^{40}C_{30} 0.75^{30} 0.25^{40-30}[/tex]

[tex]=847660528\times 0.75^{30} \times 0.25^{10}[/tex]

[tex]=0.144364346356[/tex]

[tex]\approx 0.1444[/tex]

The correct probability that exactly 30 of the 40 students monitored cheat on the final exam is 0.0256.

To solve this problem, we can use the binomial probability formula, which is given by:

[tex]\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \][/tex]

 Given:

- ( n = 40 ) (the number of students monitored),

- ( k = 30 ) (the number of students who cheat),

- ( p = 0.75 ) (the probability of a student cheating on the final exam).

We can plug these values into the binomial probability formula:

[tex]\[ P(X = 30) = \binom{40}{30} \cdot (0.75)^{30} \cdot (1-0.75)^{40-30} \][/tex]

First, calculate the binomial coefficient:

[tex]\[ \binom{40}{30} = \frac{40!}{30!(40-30)!} \][/tex]

Next, calculate the powers of p and  (1-p) :

[tex]\[ (0.75)^{30} \] \[ (0.25)^{10} \][/tex]

Now, multiply these values together:

[tex]\[ P(X = 30) = \binom{40}{30} \cdot (0.75)^{30} \cdot (0.25)^{10} \][/tex]

Using a calculator or a software tool, we can compute the exact value of the probability:

[tex]\[ P(X = 30) \approx 0.0256 \][/tex]

Therefore, the probability that exactly 30 of the 40 students monitored cheat on the final exam is approximately 0.0256 when rounded to four decimal places.

Answer:

m<1 = 45°  

m<2 = 76°

m<3 = 80°

Step-by-step explanation:

Points to remember

1). Vertically opposite angles are equal

2). Sum of angles of a triangles is 180°

To find the measures of given angles

From the figure we get,

m<1 = 45°   [Vertically opposite angles]

By using angle sum property <2 + 59 + 45 = 180

m<2 = 180 - 104

m<2 = 76°

Also m<1 + m<3 + 55 = 180

m<3 = 180 - (m<1 + 55)

 = 180 - (45 + 55)

 = 180 - 100

m<3 = 80°

Answer:

125 mL/h

Step-by-step explanation:

Intravenous pumps are used when there is a need of constant pumping of fluids into the body. Injecting every hour with the required volume causes damage.

Volume of 0.45 saline = 1000 mL

Time to be infused = 8 hours

The Intravenous pump should be programmed in the following way

[tex]\frac{Volume}{time}=\frac{1000}{8}\\\Rightarrow flow\ rate=125\ mL/hr[/tex]

Hence, the IV pump should be programmed for 125 mL/hr.

To approximate the volume with 8 boxes, we have to split up the interval of integration for each variable into 2 subintervals, [0, 1] and [1, 2]. Each box will have midpoint [tex]m_{i,j,k}[/tex] that is one of all the possible 3-tuples with coordinates either 1/2 or 3/2. That is, we're sampling [tex]f(x,y,z)=\cos(xyz)[/tex] at the 8 points,

(1/2, 1/2, 1/2)

(1/2, 1/2, 3/2)

(1/2, 3/2, 1/2)

(3/2, 1/2, 1/2)

(1/2, 3/2, 3/2)

(3/2, 1/2, 3/2)

(3/2, 3/2, 1/2)

(3/2, 3/2, 3/2)

which are captured by the sequence

[tex]m_{i,j,k}=\left(\dfrac{2i-1}2,\dfrac{2j-1}2,\dfrac{2k-1}2\right)[/tex]

with each of [tex]i,j,k[/tex] being either 1 or 2.

Then the integral of [tex]f(x,y,z)[/tex] over [tex]B[/tex] is approximated by the Riemann sum,

[tex]\displaystyle\iiint_B\cos(xyz)\,\mathrm dV\approx\sum_{i=1}^2\sum_{j=1}^2\sum_{k=1}^2\cos m_{i,j,k}\left(\frac{2-0}2\right)^2[/tex]

[tex]=\displaystyle\sum_{i=1}^2\sum_{j=1}^2\sum_{k=1}^2\cos\frac{(2i-1)(2j-1)(2k-1)}8[/tex]

[tex]=\cos\dfrac18+3\cos\dfrac38+3\cos\dfrac98+\cos\dfrac{27}8\approx\boxed{4.104}[/tex]

(compare to the actual value of about 4.159)

The Midpoint Rule for triple integrals approximates the integral of a function by taking the function's value at the midpoint of sub-boxes into which the domain is divided. In this problem, B is divided into eight 1x1x1 sub-boxes, and cos(xyz) is evaluated at each sub-box's center.

The Midpoint Rule for triple integrals is a method for approximating the value of a triple integral over a box B by dividing the box into smaller sub-boxes and evaluating the function at the midpoint of each sub-box.

For the given integral of cos(xyz) over the box B defined by the ranges 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, and 0 ≤ z ≤ 2, we are asked to divide B into eight sub-boxes of equal size.

This involves calculating the midpoints for each sub-box (xi, yj, zk) and then evaluating the function at these points.

The size of each sub-box would be 1 x 1 x 1 since the original box dimensions are 2 x 2 x 2.

The midpoints for the sub-boxes will be at (0.5, 0.5, 0.5), (0.5, 0.5, 1.5), (0.5, 1.5, 0.5), (0.5, 1.5, 1.5), (1.5, 0.5, 0.5), (1.5, 0.5, 1.5), (1.5, 1.5, 0.5), and (1.5, 1.5, 1.5).

The volume of each sub-box is 1, so the Riemann sum becomes the sum of cos(xyz) evaluated at these midpoints times the volume of each sub-box.

After calculating the Riemann sum, it provides an approximate value for the triple integral.

The actual calculation would require substituting each midpoint into the function cos(xyz) to get the respective value for each sub-box, then summing these values and finally multiplying by the volume of a single sub-box to get an approximation for the entire integral.

Answer:

No, he should have set the sum of ∠AED and ∠DEC equal to 180°, rather then setting ∠AED and ∠DEC equal to each other.

Step-by-step explanation:

He was using the measurements of m∠AED & m∠CED, which are supplementary angles, not vertical angles (therefore making them, when combined, equal to 180°).

If they were vertical angles (at the case of m∠AED & m∠BEC, or the other pair), then yes, they will be congruent. But in this case, they are not, so you don't solve it like they are vertical angles.

~

Answer:

B is the answer.

Step-by-step explanation:

Yes, he solved for the correct answer.

No, he should have set the sum of ∠AED and ∠DEC equal to 180°, rather than setting ∠AED and ∠DEC equal to each other.

No, he should have added 8 to both sides rather than subtracting 8 from both sides.

No, he should have multiplied both sides by 16 rather than dividing both sides by 16.

Answer:

55.5556 hours.

Step-by-step explanation:

Let's solve the problem.

The amount of dopamine rate applied to a person is based on the formula: 10mcg/kg/min. Such relation can be express as follows:

(10mcg/kg/min)=

(10mcg/kg)*(1/min)

Now by multiplying by the weight (12 kg) of the 2 years old person, we have:

(10mcg/kg)*(1/min)*(12kg)=

(10mcg*12kg/kg)*(1/min)=

(120mcg)*(1/min)=

120mcg/min, which is the rate of dopamine infusion, which can be express as:

(120mcg/min)*(60min/1hour)=

(120mcg*60min)/(1hour*1min)=

7200mcg/hour=

1hour/7200mcg, which means that for each hour, 7200mcg dopamine are infused.  

Because the D5W product has 400 mg of dopamine, then we need to convert 400 mg to X mcg of dopamine in order to use the previous obtained rate. This means:

Because 1mcg=0.001mg then:

(400mg)*(1mcg/0.001mg)=

(400mg*1mcg)/(0.001mg)=

400000mcg, which is the amount of dopamine in D5W.

Now, using the amount of dopamine in D5W and the applied rate we have:

(rate)*(total amount of dopamine)=hours of infusion

(1hour/7200mcg)*(400000mcg)=hours of infusion

(1hour*400000mcg)/(7200mcg)=hours of infusion

(55.5556 hours) =hours of infusion

In conclusion, the infusion will last 55.5556 hours.

Answer:

The two lengths are 3.4303 and 9.8606 centimeters

Step-by-step explanation:

The metal bar is divided into two pieces, so we are going to call X the length of the first piece and Y the length of the second piece.

From the phrase: one piece is 3 centimeters longer than twice the length of the other, we can separate and rewrite as:

One piece - is - 3 centimeters - longer than - twice the length of the other

X                   =               3                      +               2         *       Y

So, X=3+2Y is our first equation

From the phrase: the sum of the squares of the two lengths is 109 square centimeters, we can rewrite as:

The sum of the squares of the two lengths - is - 109 square centimeters

               [tex]X^{2} +Y^{2}[/tex]                                            =      109

So, X^2+Y^2=109 is our second equation

Replacing the first equation on the second question we get:

[tex]X^{2} +Y^{2} =109\\(3+2Y)^{2} +Y^{2} =109[/tex]

[tex]9+(2*3*2Y)+4Y^{2} +Y^{2} -109=0[/tex]

[tex]5Y^{2} +12Y-100=0[/tex]

Solving this equation we find two solutions:

Y=3.4303 and Y= -5.8303

Since the question is talking about the length there is no sense use Y=-5.8303, then our first length is 3.4303

So replacing this value on the first equation we get:

X= 3 + 2*Y

X= 3 + 2*3.4303

X= 9.8606

Finally the two length are 3.4303 and 9.8606 centimeters

Answer:

C. If I lower my grades then I work alone

Step-by-step explanation:

In logic, the contrapositive of a conditional statement, [tex]P\to Q[/tex] is [tex]\neg Q\to \neg P[/tex].

In other words, the contrapositive of "If P then Q" is "If not Q, then not P".

The given conditional statement is:

"If I form a study group then I raise my grades."

We can break this conditional statement into the following propositional statements:

P : I form a study group

¬P : I work alone

Q : I raise my grades

¬Q : I lower my grades

The contrapositive of  "If I form a study group then I raise my grades." becomes:

" If I lower my grades then I work alone"

The correct choice is C

To find the contrapositive of the given conditional statement "If I form a study group then I raise my grades," we have to negate both the hypothesis (the "if" or P part) and the conclusion (the "then" or Q part) and also reverse their order.
The original statement is in the form: If P then Q.
P: I form a study group.
Q: I raise my grades.
The contrapositive is: If not Q then not P.
Not Q: I do not raise my grades.
Not P: I do not form a study group.
So, the contrapositive of the original statement would be: "If I do not raise my grades then I do not form a study group."
Now let's match this to one of the choices given:
A. If I work alone then I lower my grades. (This is not the contrapositive; it introduces new concepts that weren't part of the original statement.)
B. If I raise my grades then I form a study group. (This is the converse, not the contrapositive.)
C. If I lower my grades then I work alone. (This is the contrapositive: "lower my grades" is equivalent to "do not raise my grades," and "work alone" is equivalent to "do not form a study group.")
D. If I form a study group then I lower my grades. (This is not the contrapositive; it's the inverse of the original statement.)
E. If I form a study group then I raise my grades. (This is the original statement.)
F. If I raise my grades then I work alone. (This is not the contrapositive; it introduces the concept of working alone.)
The correct choice that represents the contrapositive of the original statement is C. If I lower my grades then I work alone.

Answer:

Step-by-step explanation:

Let p : It is raining

Truth value of this statement is True.

q : I will give you a ride home after class

Truth value of the statement q is True.

So for the statement p→q (If it is raining, then I will give you a ride home)

truth value will be True.

Therefore, when statement p is True and q is False, p→q will be False.

Answer Step-by-step explanation:

Given statement if a-b is even and b-c is even then a-c is even .

Let p: a-b and b-c are even

q: a-c is even.

Converse: If a-c is even then a-b and b-c are both even.

Inverse:If a-b and b-c  are not both even then a-c is not even.

If a= Even number

b= Even number

c=Even number

If a-c  is even then a-b and b-c are both even..Hence, the converse statement is true.

If a=Odd number

b=Odd number

c= Odd number

If a-c is even then a-b and b-c are both even number .Hence, the converse  statement is true.

If a=Even number

b= Even number

c= Odd number

a-b  and b-c are both odd not even number but a-c is even number

a=8,b=6 c=3

a-b=8-6=2

b-c=6-3=3

a-c=8-3=5

If a-c is odd then a-b even but b-c is odd .Hence , the converse statement is false.But the inverse statement is true.

If a= Odd number

b=Even number

c= Even number

If a-b is odd and b-c is even then a-c is odd not even . Hence, the inverse statement is true.

If a= Odd number

b=Eve number

c=Odd number

a=9,b=6,c=5

a-b=9-6=3

b-c=6-5=1

a-c=9-5=4

Here, a-b and b-c are not both even but a-c is even .Hence, the inverse statement is false.

Answer:

  1a. y = 5x +3

  1b. y = -1/7x + 2

  1c. y = 2x +3

  2a. y = -1/2x +3; slope = -1/2, y-intercept = +3

  2b. y = 3/2x +6; slope = 3/2, y-intercept = +6

Step-by-step explanation:

Apparently the "y-intercept form" referred to is the one more commonly called "slope-intercept form." That form is ...

  y = mx + b . . . . . . . . m is the slope; b is the y-intercept

You arrive at this form by solving each equation for y. You do that the same way you solve any equation for any variable: undo what is done to the variable of interest.

In these general form equations, the y-variable term has a value multiplying y and some other terms added to that. First of all, you subtract the added terms (from both sides of the equation). This transforms ...

  ax +by +c = 0

into

  by = -ax -c

Next, you divide by the constant that is multiplying y. Of course all terms on both sides of the equation are divided by that, so you now have ...

  y = (-a/b)x -c/b

The slope is -a/b, and the y-intercept is -c/b.

___

1a. 5x -y +3 = 0

  -y = -5x -3 . . . . . subtract non-y terms

  y = 5x +3 . . . . . . divide by -1

__

1b. x +7y -14 = 0

  7y = -x +14 . . . . . subtract non-y terms

  y = -1/7x +2 . . . . .divide by 2

__

1c. 6x -3y +9 = 0

  -3y = -6x -9 . . . . subtract terms not containing y

  y = 2x +3 . . . . . . divide by -3

__

2a. x +2y -6 = 0

  2y = -x +6 . . . . . subtract non-y terms

  y = -1/2x +3 . . . . divide by 2 -- this is graphed as the red line below

__

2b. 3x -2y +12 = 0

  -2y = -3x -12 . . . . subtract non-y terms

  y = 3/2x +6 . . . . . divide by -2 -- this is graphed as the blue line below

_____

Comment on the 2nd problem

Of course, the y-intercept (constant term in slope-intercept form) is the point on the y-axis where the line crosses. The slope tells you the ratio of "rise" to "run". That is, a slope of -1/2 means the line drops one unit for each 2 units it goes to the right. A slope of 3/2 means the line increases (rises) by 3 units for each 2 units it goes to the right.

Answer:

so principal amount is $7862.07

Step-by-step explanation:

Given data

rate = 7*1/4 % = 29/4 %

interest = $190

time = 4 months = 4/12 year

to find out

principal

solution

we know the simple interest formula  i.e.

interest = ( principal × rate  × time ) /100             ..................1

now put all value rate time and interest in equation 1 we get interest here

interest = ( principal × rate  × time ) /100  

190 =  ( principal × 29/4  × 4/12 ) /100  

principal = 190 × 12  ×100 /   29

principal = 7862.068966

so principal amount is $7862.07

Answer:

A. It is false that the bus has an engine and the people have money.

Step-by-step explanation:

Given statement is -

The bus does not have an engine or the people do not have money.

The De Morgan's laws states that the complement of the union of two given sets is the intersection of their complements.

Also, the complement of the intersection of these two sets is the union of their complements.

negative (p ∨ q)⇔ p ∧ negative q

So, here option A is correct. It is false that the bus has an engine and the people have money.

Final answer:

Using De Morgan's laws, the equivalent statement for 'The bus does not have an engine or the people do not have money' is 'It is not the case that the bus has an engine and the people have money,' which corresponds to answer choice A.

Explanation:

The question asks to use De Morgan's laws to write an equivalent statement for the sentence: "The bus does not have an engine or the people do not have money." Let's denote 'the bus has an engine' by A and 'the people have money' by B. The original statement can be written in logical form as ¬A ∨ ¬B, which using De Morgan's laws, translates to ¬(A ∧ B). This means that the equivalent statement is: "It is not the case that the bus has an engine and the people have money." Therefore, the correct answer is A.

Answer with explanation:

⇒(2 x y )d x+(x-6 y) d y=0

P= 2 x y

Q=x-6 y

[tex]P_{y}=2 x\\\\Q_{x}=1[/tex]

So this Differential Equation is exact.

To solve this, we will first evaluate,[tex]\varphi (x,y)[/tex].

[tex]\varphi_{x}=P\\\\\varphi_{y}=Q\\\\\varphi=\int P d x\\\\= \int 2 x y  dx\\\\\varphi=x^2 y\\\\\varphi(x,y)=x^2y+k(y)------(1)[/tex]

Differentiating with respect to , y

[tex]\varphi'(x,y)=x^2+k'(y)=Q=x-6 y\\\\\rightarrow x-6 y-x^2=k'(y)\\\\ k(y)=\int (x-6 y -x^2) dy\\\\k(y)=x y-3 y^2-x^2 y+f\\\\\varphi(x,y)=x^2y+x y-3 y^2-x^2 y+f\\\\\varphi(x,y)=x y-3 y^2+f[/tex]

Substituting the value of , k(y) in equation 1.

This is required Solution of exact differential equation.

Answer:

Largest possible rank =6

Minimum possible rank=0

Step-by-step explanation:

We are given that A is a matrix of order [tex] 6\times 9 [/tex]

We have to find the maximum possible rank and minimum possible rank of A.

In given matrix we have 6 rows and 9 columns.

Rank: Rank is defined as the number of non zero rows or columns of matrix and any row or column is not a linear  combination of other two or more rows or columns

Rank of matrix of order [tex] m\times n [/tex]

Where m< n

Then largest possible rank is m

We are given a matrix of order[tex]6\times 9[/tex] where 6< 9

Therefore, the largest possible rank is 6.

The minimum possible rank  is zero because a  given matrix can be zero matrix therefore, the rank of zero matrix is zero.

Largest possible rank =6

Minimum possible rank=0

Answer:

Maximum velocity, v = 50 ft/s

Step-by-step explanation:

Given

1000[tex]\frac{dv}{dt}=5000-100v[/tex]    -----------(1)

Dividing (1) by 1000, we get

[tex]\frac{dv}{dt}=5-\frac{v}{10}[/tex]

[tex]\frac{dv}{dt}+\frac{v}{10}=5[/tex]     -----------------(2)

Now we can solve the above equation using method of integrating factors

[tex]u(t)=e^{\int \frac{1}{10}dt}[/tex]

[tex]u(t)=e^{\frac{1}{10}t}[/tex]

Now multiplying each side of (2) by integrating factor,

[tex]e^{\frac{1}{10}t}(\frac{dv}{dt})+\frac{v}{10}e^{\frac{1}{10}t}=5e^{\frac{1}{10}t}[/tex]

Combining the LHS into one differential we get,

[tex]\frac{d}{dt}\left ( e^{\frac{1}{10}t}v \right ) = \int 5e^{\frac{1}{10}t}.dt[/tex]

[tex]e^{\frac{1}{10}t}v = 50e^{\frac{1}{10}t}[/tex] + c

v(t)=50+ce

Appltying the initial condition v(0)=0, we get

[tex]0=50+ce^{-\frac{1}{10}(0)}[/tex]

0=50+c

c=-50

So the particular solution is

[tex]v(t)=50-50e^{-\frac{1}{10}t}[/tex]

[tex]v(t)=50\left (1-e^{-\frac{1}{10}t}\right)[/tex]

Therefore, the maximum velocity is 50 ft/s

The forces balance out at 50 ft/s, which is the maximum velocity the boat can achieve.

To determine the maximum velocity of the boat, we need to consider the forces acting on it. The equation given is:

1000  [tex]\frac{dv}{dt}[/tex] = 5000 - 100v

At maximum velocity, the acceleration of the boat will be zero, which means  [tex]\frac{dv}{dt}[/tex] = 0. Therefore, setting the left-hand side of the equation to zero, we get:

0 = 5000 - 100v

Solving for v, we have:

Thus, the maximum velocity of the boat is 50 ft/s. This is where the force provided by the motor equals the resistive force from the water.

Answer:

f(x) = -logx + 4.

Step-by-step explanation:

y = -log x  approaches  the point (0, 1)  but this approaches (0, 5)

Looks like the graph of - log x + 4.

Answer:

[tex]y = 5-\sqrt{x}[/tex]

Step-by-step explanation:

Note that the graph shown is a transformation of the parent function

[tex]y = \sqrt{x}[/tex]

The main function cuts the x-axis at x = 0 and has no negative values of y. In addition, the main function is always growing.

Note that the function shown is decreasing and intercepts the y-axis at y = 5.

Therefore, the graph of the main function has been reflected on the x-axis and displaced 5 units upwards.

If we do y = f(x) then to reflect the main function on the  x-axis and move it upwards 5 units we make the following transformation

[tex]y = -f(x) + 5[/tex].

Then the graphed function is:

[tex]y = 5-\sqrt{x}[/tex]

Answer:

The method for selecting a stratified sample is to order a population in some way and then select members of the population at regular intervals. - FALSE.

This can be re written as :The method for selecting a systematic sample is to order a population in some way and then select members of the population at regular intervals.

Explanation for stratified sampling:

In stratified sampling, the members of a population are divided into two or more strata with similar characteristics and then a random sample is selected from each strata. This way ensures that members of each group within a population will be sampled.

Explanation for cluster sampling:

The method for selecting a cluster sample is to order a population in some way and then select members of the population at regular intervals.

In cluster sampling, the population is divided into​​ clusters, and all of the members of one or more​ clusters are selected.

Final answer:

The statement provided is false as it describes the method of systematic sampling instead of stratified sampling. Stratified sampling requires dividing the population into strata and randomly selecting individuals from each stratum, while systematic sampling selects members at regular intervals.

Explanation:

The statement is false. The correct method for selecting a stratified sample is to divide the population into groups known as strata, and then use simple random sampling to identify a proportionate number of individuals from each stratum. The statement describes the method for selecting a systematic sample, not a stratified sample. A systematic sample is obtained by ordering a population and then selecting members at regular intervals, not by stratifying them.

On the other hand, a cluster sample involves dividing the population into clusters (groups), often geographically defined, and then randomly selecting some of these clusters. All the members from these selected clusters are included in the sample. Hence, the method of selecting individuals at regular intervals is not used in cluster sampling either.

Answer:

total return is -34.4828 %  and The annual return is - 8.1094 %

Step-by-step explanation:

given data

principal amount of share = $2900

selling amount pay = $1900

time period (t) = 5 years

to find out

total return and The annual return

solution

we know interest formula

amount = principal [tex](1 + r/100)^{t}[/tex]

put all these value amount principal, time n we get rate

1900 = 2900 [tex](1 + r/100)^{5}[/tex]

[tex](1 + r/100)^{5}[/tex]  = 1900/2900

1 + r/100 = 0.9189

r = −8.1094

The annual return is - 8.1094 %

and total return is ((selling amount pay - principal amount ) /  principal  amount )  × 100

put these value and we get total return

total return  = ((1900 - 2900) / 2900 )  × 100

total return  =  -0.344828  × 100

total return  =  -34.4828 %

The area is given by the integral,

[tex]\displaystyle\int_{-1/2}^{1/2}(1-2x^2-|x|)\,\mathrm dx[/tex]

The integrand is even, so we can simplify the integral somewhat as

[tex]\displaystyle2\int_0^{1/2}(1-2x^2-|x|)\,\mathrm dx[/tex]

When [tex]x\ge0[/tex], we have [tex]|x|=x[/tex], so this is also the same as

[tex]\displaystyle2\int_0^{1/2}(1-2x^2-x)\,\mathrm dx[/tex]

which has a value of

[tex]2\left(x-\dfrac23x^3-\dfrac12x^2\right)\bigg|_0^{1/2}=2\left(\dfrac12-\dfrac1{12}-\dfrac18\right)=\boxed{\dfrac7{12}}[/tex]

so that A = 7.

Answer:

(x+3)^2 + (y-5)^2 = 4^2

or

(x+3)^2 + (y-5)^2 = 16

Step-by-step explanation:

The equation of a circle can be written in the form

(x-h)^2 + (y-k)^2 = r^2

where (h,k) is the center and r is the radius

(x--3)^2 + (y-5)^2 = 4^2

(x+3)^2 + (y-5)^2 = 4^2

or

(x+3)^2 + (y-5)^2 = 16

Answer with explanation:

Given → A, B ⊆R, are non-empty sets.

Also, A ⊆ B.

It is also given that , B has least upper bound.

To Prove:→A has a least upper bound,

and → lub A ≤ lub B.

Proof:

    A and B are non empty sets.

Also, A ⊆ B

it means there are some elements in B which is not in A and there are some elements in set B which may be greater or smaller than or equal to set A.

The meaning of least upper bound is the the smallest member of the set from which all members are greater.

As, A⊆B

So, there are two possibilities ,either

→ Least upper bound set A = least upper bound set B------(1)

→ Least upper bound set A < least upper bound set B-------(2)

Combining (1) and (2)

→→→l u b(A) ≤ l u b (B)

[tex]\vec F(x,y,z)=\dfrac74\cos(xyz)\langle yz,xz,xy\rangle[/tex]

Computing the line integral directly is cumbersome, if not impossible by elementary means. Let's instead try to determine if [tex]\vec F[/tex] is conservative. We look for a scalar function [tex]f(x,y,z)[/tex] such that [tex]\nabla f=\vec F[/tex]. We should have

[tex]\dfrac{\partial f}{\partial x}=yz\cos(xyz)[/tex]

(ignoring the 7/4 for a moment). Integrating both sides wrt [tex]x[/tex] gives

[tex]\displaystyle\int\cos(xyz)yz\,\mathrm dx=\sin(xyz)+g(y,z)[/tex]

Then differentiating wrt [tex]y[/tex] gives

[tex]\dfrac{\partial(\sin(xyz))}{\partial y}=xz\cos(xyz)=xz\cos(xyz)+\dfrac{\partial g}{\partial y}[/tex]

[tex]\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)[/tex]

Differentiating wrt [tex]z[/tex] gives

[tex]\dfrac{\partial(\sin(xyz))}{\partial z}=xy\cos(xyz)=xy\cos(xyz)+\dfrac{\mathrm dh}{\mathrm dz}[/tex]

[tex]\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C[/tex]

So we have (and here we re-introduce the 7/4)

[tex]f(x,y,z)=\dfrac74\sin(xyz)+C[/tex]

and by the fundamental theorem of calculus,

[tex]\displaystyle\int_C\nabla f\cdot\mathrm d\vec r=f(\vec b)-f(\vec a)[/tex]

where [tex]\vec a[/tex] and [tex]\vec b[/tex] are vectors representing the start- and endpoints of [tex]C[/tex]. So

[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\frac74\sin\frac\pi6-\frac74\sin\frac\pi2=\boxed{\frac78}[/tex]

The problem asks to evaluate a line integral of a given function over a line segment. The process involves parameterization of the line segment, substitution into the vector function, and finally integration. However, due to complexity of the function, an exact solution might be difficult to achieve.

The problem asks us to evaluate a line integral, specifically ∫ C → F ⋅ → d r, where the function → F ( x , y , z ) = ⟨ 1.75 y z cos ( x y z ) , 1.75 x z cos ( x y z ) , 1.75 x y cos ( x y z ) ⟩, and C is the line segment joining two points.

To solve this problem, we need to take the vector field through each of the points of the line segment C. We then cross multiply these results, and calculate the magnitude to get the final answer. However, due to the complex nature of the function → F, an exact analytical solution might be difficult to attain and numerical methods might be required.

Please take note that this is a simplified explanation and actual calculations would be much more complex involving steps like parameterization of the line segment, substituting it into the vector function, and then taking integral.

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Step-by-step explanation:

Probability, P = [tex]\frac{no. of favourable outcomes}{Total no. of possible outcomes}[/tex]

Out of 8846 heart pacemaker malfunctions cases, caused by firmware cases are 375

then, no. of cases not caused by firmware are:

8846 - 375 = 8471

Probability for three different pacemakers respectively is given by:

P(1) = [tex]\frac{8471}{8846}[/tex]

we select the next malfunctioned pacemaker from the remaining i.e., out of 8471, excluding the chosen malfunctioned pacemaker

P(2|1) =  [tex]\frac{8470}{8845}[/tex]

Therefore, the events are not independent of each other

Now, if the selection is without replacement, then

P(3|1 & 2) =  [tex]\frac{8469}{8844}[/tex]

Now, by general multiplication rule(as the events are not independent):

P(none of the 3 are caused by malfunction) =

[tex]\frac{8471}{8846}\times\frac{8470}{8845}\times\frac{8469}{8844}[/tex]

= 0.9019

P(none of the 3 are caused by malfunction)  = 90.19%

The probability is high, therefore the whole batch will be accepted.

Probability of an event represents the chances of that event to occur.

Suppose there are n mutually independent events.

The probability of their simultaneously occurrence is given as

[tex]P(A_1 \cap A_2 \cap ... \cap A_n) = P(A_1) \times P(A_2) \times ... \times P(A_n)[/tex]

(This is true only if all those events are mutually independent).

The probability of getting a failure(malfunction) in a randomly selected pacemaker for the given case is

[tex]\dfrac{375}{8846} \approx 0.042[/tex]

The probability of a pacemaker working fine = 1- its failure probability  = 1 - 0.042 = 0.958

Since each of the 3 selected pacemakers are independent for their failure or success of each other, thus, if we have:

[tex]A_1[/tex] = Event of properly working of first pacemaker

[tex]A_2[/tex] = Event of properly working of second pacemaker

[tex]A_3[/tex] = Event of properly working of third pacemaker

Then, as all three are independent for their working from each other, thus,

[tex]P(A_1 \cap A_2 \cap A_3) = P(A_1) \times P(A_2)\times P(A_3) = (0.958)^3 \approx 0.879\\\\\begin{aligned}{P(\text{batch getting selected}) &= P(\text{All three pacemaker working})\\& = P(A_1 \cap A_2 \cap A_3) \\&\approx 0.879\\\end{aligned}[/tex]

Thus, as this probability is high, thus, this procedure is likely to result in the entire batch being accepted.

Thus,

Learn more about multiplication rule of probability here:

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Answer: a) [tex]t=\frac{2.303}{k}\log\frac{a}{a-x}[/tex]

b) 3.15 weeks.

c) 0.11 grams

Step-by-step explanation:

a) radioactive decay follows first order kinetics and thus Expression for rate law for first order kinetics is given by:

[tex]k=\frac{2.303}{t}\log\frac{a}{a-x}[/tex]

where,

k = rate constant  = ?

t = time taken for decomposition = 1 week

a = initial amount of the reactant  = 1 g

a - x = amount left after decay process  = 0.8 g

Now put all the given values in above equation, we get

[tex]k=\frac{2.303}{1}\log\frac{1}{0.8}[/tex]

[tex]k=0.22weeks^{-1}[/tex]

2) To calculate the half life, we use the formula :

[tex]t_{\frac{1}{2}=\frac{0.693}{k}[/tex]

[tex]t_{\frac{1}{2}=\frac{0.693}{0.22}=3.15weeks[/tex]

Thus half life of Thorium 234 is 3.15 weeks.

3) amount of Thorium 234 present after 10 weeks:

[tex]10=\frac{2.303}{0.22}\log\frac{1}{a-x}[/tex]

[tex](a-x)=0.11g[/tex]

Thus amount of Thorium 234 present after 10 weeks is 0.11 grams

Answer:

[tex]36.93\ ft[/tex]

Step-by-step explanation:

step 1

Find the circumference of the circle

The circumference is equal to

[tex]C=\pi D[/tex]

we have

[tex]D=25.2\ ft[/tex]

substitute

[tex]C=25.2\pi\ ft[/tex]

step 2

we know that

The circumference of a circle subtends a central angle of 360 degrees

so

using proportion

Find out the arc length by a central angle of 168 degrees

[tex]\frac{25.2\pi}{360}=\frac{x}{168}\\\\x=25.2\pi*168/360\\\\x=(25.2*3.14)*168/360\\\\x=36.93\ ft[/tex]