In an arithmetic​ sequence, the nth term an is given by the formula An=a1+(n−1)d​, where a1is the first term and d is the common difference.​ Similarly, in a geometric​ sequence, the nth term is given by an=a1•rn−1.

Use these formulas to determine the indicated term in the given sequence.

The 19th term of 19​,42​,65​,88​,...

The actual movement of the boat is calculated by adding the vectors representing the velocity of the boat and the water current. The net vector shows both the speed and direction of the actual movement.

The actual movement of the boat is determined by adding vectorially, the velocity of the boat and the velocity of the current. This is because we need to consider both the speed and direction of the boat (traveling due east) and the current (flowing southwest).

Let's assume East as +i direction, and North as +j direction. So, the velocity of the boat is 22i km/hr and the velocity of the current is -10i/√2 -10j/√2 km/hr (as it is moving southwest).

To find the net velocity or actual movement, we add these two vectors.

Resultant velocity = 22i -10i/√2 -10j/√2 // Adding the i-components and the j-components

 = (22 -10/√2)i -10/√2 j km/hr

Therefore, the actual movement of the boat is (22 -10/√2)i -10/√2 j km/hr.

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Answer:

less than

Step-by-step explanation:

When we are performing a two-tailed test of whether μ = 100,  the probability of detecting a shift of the mean to 105 will be ___Less__than___ the probability of detecting a shift of the mean to 110.

this means that

mean of 105 < mean of 110.

Answer:

In the first picture, correct answer is b: the inverse of function q is function r, because they are symmetrical about the line y = x.

In the second picture, correct answer is b: {y| 0<= y < 8}.

Step-by-step explanation:

The inverse function of f ( f⁻¹(x) ) must satisfy that: f(f⁻¹(x)) = f⁻¹(f(x)) = x; it returns every point x, transformed under function f, to its original place. In the graph, this property translates in the following statement: the inverse function of a function f is the reflection over the identity function (y = x).

In the graph shown in the question, the blue graph (r), is the one that corresponds to the reflection under the identity function. Therefore, the correct answer is b.

Regarding the second picture, first we need to understand what the range means. The range of a function corresponds to the set of all resulting values of the dependent variable. Since the values taken by the dependent variable span from 0 to 8 (including the 0 but not including the 8), then the answer is b: {y| 0<= y < 8}.

Answer:

[tex]x^3-x^2.[/tex]

Step-by-step explanation:

The word "difference" represents a subtraction. Then the algebraic expression will be of the form a-b.

Now, the difference is between the cube of a number and the square of the number, then let's call the number x. The square of the number is raise to two. Then the square of the number is [tex]x^2[/tex].

The cube of the number is raise to three. Then the cube of the number is [tex]x^3[/tex].

So, the difference between the cube of a number and the square of the number (we are talking about the same number in the square and the cube) is [tex]x^3-x^2[/tex].

Answer and Explanation:

Given : Sides of right triangle 5,12 and 13.

To find : Show that the area of a right triangle of sides 5, 12 and 13 cannot be a square ?

Solution :

If 5,12 and 13 are sides of a right angle triangle then

13 is the hypotenuse as it is largest side.

then we take perpendicular as 12 and base as 5.

The area of the right angle triangle is

[tex]A=\frac{1}{2}\times b\times h[/tex]

Here, h=12 and b=5

[tex]A=\frac{1}{2}\times 5\times 12[/tex]

[tex]A=5\times 6[/tex]

[tex]A=30[/tex]

The area of the right angle triangle is 30 units.

30 is not a perfect square as [tex]30=2\times 3\times 5[/tex]

There is no square pair formed.

Answer: The ratio of boys to girls would be 3 : 5 .

Step-by-step explanation:

Since we have given that

Number of boys and girls = 32

Fraction of boys are going on a field trip = [tex]\dfrac{2}{3}[/tex]

Fraction of girls are going on a field trip = [tex]\dfrac{3}{4}[/tex]

Number of children left = 9

Let the number of boys be 'b'.

Let the number of girls be 'g'.

According to question, it becomes ,

[tex]b+g=32------------(1)\\\\32-9=\dfrac{2}{3}b+\dfrac{3}{4}g\\\\23=\dfrac{2b}{3}+\dfrac{3g}{4}-------------(2)[/tex]

From eq(1), we get that g = 32-b

So, it becomes,

[tex]\dfrac{2}{3}b+\dfrac{3}{4}(32-b)=23\\\\\dfrac{2}{3}b+24-\dfrac{3}{4}b=23\\\\\dfrac{2}{3}b-\dfrac{3}{4}b=23-24=-1\\\\\dfrac{8b-9b}{12}=-1\\\\\dfrac{-b}{12}=-1\\\\b=-1\times -12\\\\b=12[/tex]

so, number of girls would be 32 - b = 32 - 12 = 20

So, Ratio of boys to girls in class would be 12 : 20 = 3 : 5.

Therefore, the ratio of boys to girls would be 3 : 5 .

Step-by-step explanation:

Say [tex]a[/tex] is an element of [tex]G[/tex] which might have more than 1 inverse. Let's call them [tex]b[/tex], and [tex]c[/tex]. So that [tex]a[/tex] has apparently two inverses, [tex]b[/tex] and [tex]c[/tex].

This means that [tex]a*b = e[/tex] and that [tex]a*c=e[/tex](where [tex]e[/tex] is the identity element of the group, and * is the operation of the group)

But so we could merge those two equations into a single one, getting

[tex]a*b=a*c[/tex]

And operating both sides by b by the left, we'd get:

[tex]b*(a*b)=b*(a*c)[/tex]

Now, remember the operation on any group is associative, meaning we can rearrange the parenthesis to our liking, gettting then:

[tex](b*a)*b=(b*a)*c[/tex]

And since b is the inverse of a, [tex]b*a=e[/tex], and so:

[tex](e)*b=(e)*c[/tex]

[tex]b=c[/tex] (since e is the identity of the group)

So turns out that b and c, which we thought might be two different inverses of a, HAVE to be the same element. Therefore every element of a group has a unique inverse.

Answer:

Step-by-step explanation:

Given that an experimenter is studying the effects of temperture, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and five different catalysts are under consideration.

a) Experimental runs possible if use of single temperature, pressure and catalyst is there = no of temperatures x no of pressures x no of catalysts

= [tex]3*4*5 = 60[/tex]

b) Here pressure and temperature have no choice as lowest is selected.

no of methods = no of catalysts x 1 x1

= 5

Answer:

4-digit numbers with distinct digits and greater than 4500: 2800 numbers

5-digit numbers with distinct digits: 27216 numbers.

Step-by-step explanation:

If we represent a 4 number digit by ABCD, we have 9 posibilities for A (1,2,3,4,5,6,7,8 and 9, all but 0).

If every digit has to be different, we have 9 posibilities for B: ten digits (0,1,2,3,4,5,6,7,8 and 9 minus the one already used in A).

Int he same way, we have 8 posibilities for C and 7 for D.

Considering all 4-digits numbers, we have 9*9*8*7 = 4536 numers with distinct digits.

To know how many of these numbers are greater than 4500, we can substracte first the numbers that are smaller than 4000: A can take 3 digits (1,2 and 3) and B, C and D the same as before.

3*9*8*7 = 1512 numbers smaller than 4000

Then we can substrat the ones that are between 4000 and 4500

1*4*8*7 = 224 numbers between 4000 and 4500

So, if we substract from the total the numbers that are smaller than 4500 we have the results:

4-digit numbers with distinct digits greater than 4500 = 4536-(1512+224) = 2800

For 5-digit numbers, we can call the number ABCDE.

For A we have 9 digits possible (all but 0).

For B, we also have 9 posibilities (all digits but the one used in A).

For C, we have 8 digits (all 10 but the ones used in A and B).

For D, we have 7 digits.

For E, we have 6 digits.

Multiplying the possible combinations, we have:

9*9*8*7*6 =  27,216 5-digit numbers with distinct digits.

Two thousand five hundred twenty numbers have distinct digits and are more significant than 4500. There are 15120 5-digit odd numbers and 15120 5-digit even numbers with different numerals.

In Mathematics, to calculate how many numbers have distinct digits and are more significant than 4500, consider that any number greater than 4500 and less than 10000 is a 4-digit number. The thousands place can be filled by any number from 5 to 9, giving five options. Any ten digits can fill the hundreds place minus the one used in the thousands, giving nine options. The tens place, similarly, has eight votes. Similarly, the one's site has seven options, as the digit in that place cannot duplicate any of the prior digits. So, the answer is 5*9*8*7 = 2520 distinct numbers.

For the second part of your query, the 5-digit odd numbers with non-repeating digits, the tens place must be filled by five possible odd digits (1, 3, 5, 7, 9), and the first place can be filled by any number from 1 to 9, giving nine options. The other three areas have 8, 7, and 6 votes, respectively, leading to 9*8*7*6*5 = 15120 distinct numbers.

As for the 5-digit even numbers with non-repeating digits, the tens place can be filled by five possible even digits (0, 2, 4, 6, 8), and the first place can be filled by any number from 1 to 9, giving nine options. The other three areas have 8, 7, and 6 votes, respectively. So there are 9*8*7*6*5 = 15120 five-digit, distinct digit, even numbers.

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I assume the first equation is supposed to be

[tex]5x-3y+2z=13[/tex]

and not

[tex]5x-3x+2x=4x=13[/tex]

As an augmented matrix, this system is given by

[tex]\left[\begin{array}{ccc|c}5&-3&2&13\\2&-1&-3&1\\4&-2&4&12\end{array}\right][/tex]

Multiply through row 3 by 1/2:

[tex]\left[\begin{array}{ccc|c}5&-3&2&13\\2&-1&-3&1\\2&-1&2&6\end{array}\right][/tex]

Add -1(row 2) to row 3:

[tex]\left[\begin{array}{ccc|c}5&-3&2&13\\2&-1&-3&1\\0&0&5&5\end{array}\right][/tex]

Multiply through row 3 by 1/5:

[tex]\left[\begin{array}{ccc|c}5&-3&2&13\\2&-1&-3&1\\0&0&1&1\end{array}\right][/tex]

Add -2(row 3) to row 1, and add 3(row 3) to row 2:

[tex]\left[\begin{array}{ccc|c}5&-3&0&11\\2&-1&0&4\\0&0&1&1\end{array}\right][/tex]

Add -3(row 2) to row 1:

[tex]\left[\begin{array}{ccc|c}-1&0&0&-1\\2&-1&0&4\\0&0&1&1\end{array}\right][/tex]

Multiply through row 1 by -1:

[tex]\left[\begin{array}{ccc|c}1&0&0&1\\2&-1&0&4\\0&0&1&1\end{array}\right][/tex]

Add -2(row 1) to row 2:

[tex]\left[\begin{array}{ccc|c}1&0&0&1\\0&-1&0&2\\0&0&1&1\end{array}\right][/tex]

Multipy through row 2 by -1:

[tex]\left[\begin{array}{ccc|c}1&0&0&1\\0&1&0&-2\\0&0&1&1\end{array}\right][/tex]

The solution to the system is then

[tex]\boxed{x=1,y=-2,z=1}[/tex]

Answer:

The average rate of change of the population on the intervals [ 5 , 10 ] , [ 5 , 9 ] , [ 5 , 8 ] , [ 5 , 7 ] , and [ 5 , 6 ]  are 734.504, 733.06, 731.62, 730.185 and 728.75 respectively.

Step-by-step explanation:

The given function is

[tex]P(t)=181843(1.04)^{(\frac{t}{10})}[/tex]

where, P(t) is population after t years.

At t=5,

[tex]P(5)=181843(1.04)^{(\frac{5}{10})}=185444.20[/tex]

At t=6,

[tex]P(6)=181843(1.04)^{(\frac{6}{10})}=186172.95[/tex]

At t=7,

[tex]P(7)=181843(1.04)^{(\frac{7}{10})}=186904.57[/tex]

At t=8,

[tex]P(8)=181843(1.04)^{(\frac{8}{10})}=187639.06[/tex]

At t=9,

[tex]P(9)=181843(1.04)^{(\frac{9}{10})}=188376.44[/tex]

At t=10,

[tex]P(10)=181843(1.04)^{(\frac{10}{10})}=189116.72[/tex]

The rate of change of P(t) on the interval [tex][x_1,x_2][/tex] is

[tex]m=\frac{P(x_2)-P(x_1)}{x_2-x_1}[/tex]

Using the above formula, the average rate of change of the population on the intervals [ 5 , 10 ] is

[tex]m=\frac{P(10)-P(5)}{10-5}=\frac{189116.72-185444.20}{5}=734.504[/tex]

The average rate of change of the population on the intervals [ 5 , 9 ] is

[tex]m=\frac{P(9)-P(5)}{9-5}=\frac{188376.44-185444.20}{4}=733.06[/tex]

The average rate of change of the population on the intervals [ 5 , 8 ] is

[tex]m=\frac{P(8)-P(5)}{8-5}=\frac{187639.06-185444.20}{3}=731.62[/tex]

The average rate of change of the population on the intervals [ 5 , 7 ] is

[tex]m=\frac{P(7)-P(5)}{7-5}=\frac{186904.57-185444.20}{2}=730.185[/tex]

The average rate of change of the population on the intervals [ 5 , 6 ] is

[tex]m=\frac{P(6)-P(5)}{6-5}=\frac{186172.95-185444.20}{1}=728.75[/tex]

Therefore the average rate of change of the population on the intervals [ 5 , 10 ] , [ 5 , 9 ] , [ 5 , 8 ] , [ 5 , 7 ] , and [ 5 , 6 ]  are 734.504, 733.06, 731.62, 730.185 and 728.75 respectively.

Answer:

The correct option is B) The study is an observational study because the survey subjects were not given any treatment.

Step-by-step explanation:

Consider the provided information.

In a study sponsored by a​ company, 12,543 people were asked what contributes most to their happiness, and 87% of the respondents said that it was their job.

Observational study is the study in which observer only observe the subjects, and measure variables of interest without allocating treatments to subjects.

Experiment study is the study in which the researchers are applying treatments to experimental units in the research, then the effect of the treatments on the experimental units is observed.

Now consider the provided statement.

The research is based on information that nobody manipulates any experimental variables.

Hence, the study is an observational study where no treatment is given.

Thus, the correct option is B) The study is an observational study because the survey subjects were not given any treatment.

Answer:  Amount after 5 years become $5937.60.

Step-by-step explanation:

Since we have given that

Principal amount = $5000

Time period = 5 years

Rate of interest for 3 years = 6%

Rate of interest for 2 years = 8%

so, Amount becomes

[tex]Amount=5000(1+\dfrac{6}{1000})^3(1+\dfrac{8}{100})^2\\\\Amount=5000(1+0.006)^3(1+0.08)^2\\\\Amount=5000(1.006)^3(1.08)^2\\\\Amount=\$5937.60[/tex]

Hence, Amount after 5 years become $5937.60.

Answer:

Marginal revenue = R'(Q) = -0.6 Q + 221

Average revenue = -0.3 Q + 221

Step-by-step explanation:

As per the question,

Functions associated with the demand function P= -0.3 Q + 221, where Q is the demand.

Now,

As we know that the,

Marginal revenue is the derivative of the revenue function, R(x), which is equals the number of items sold,

Therefore,

R(Q) = Q × ( -0.3Q + 221) = -0.3 Q² + 221 Q

∴ Marginal revenue = R'(Q) = -0.6 Q + 221

Now,

Average revenue (AR) is defined as the ratio of the total revenue by the number of units sold that is revenue per unit of output sold.

[tex]Average\ revenue\ = \frac{Total\ revenue}{number\ of\ units\ sold}[/tex]

Where Total Revenue (TR) equals quantity of output multiplied by price per unit.

TR = Price (P) × Total output (Q) = (-0.3Q + 221) × Q = -0.3 Q² + 221 Q

[tex]Average\ revenue\ = \frac{TR}{Q}[/tex]

[tex]Average\ revenue\ = \frac{-0.3Q^{2}+221Q}{Q}[/tex]

∴ Average revenue = -0.3Q + 221

Answer:

There is a 0.64% probability that the costumer has filed a claim.

Step-by-step explanation:

Probability:

What you want to happen is the desired outcome.

Everything that can happen iis the total outcomes.

The probability is the division of the number of possible outcomes by the number of total outcomes.

Our problem has these following probabilities:

-78% that a costumer is a good risk.

-20% that a costumer is a medium risk.

-2% that a costumer is a poor risk.

Also:

- 0.5% of a good risk costumer filling an accident claim

- 1% of a medium risk costumer filling an accident claim.

-2.5% of a poor risk costumer filling an accident claim.

The question is:

What is the probability that the customer has filed a claim?

[tex]P = P_[1} + P_{2} + P_{3}[/tex], in which:

-[tex]P_{1}[/tex] is the probability that a good risk costumer is chosen and files a claim. This probability is: the probability of a good risk costumer being chosen multiplied by the probability that a good risk costumer files a claim. So:

[tex]P_[1} = 0.78*0.005 = 0.0039[/tex]

-[tex]P_{2}[/tex] is the probability that a medium risk costumer is chosen and files a claim. This probability is: the probability of a medium risk costumer being chosen multiplied by the probability that a medium risk costumer files a claim. So:

[tex]P_[2} = 0.20*0.01 = 0.002[/tex]

-[tex]P_{3}[/tex] is the probability that a poor risk costumer is chosen and files a claim. This probability is: the probability of a poor risk costumer being chosen multiplied by the probability that a poor risk costumer files a claim. So:

[tex]P_[3} = 0.02*0.025 = 0.0005[/tex]

[tex]P = P_[1} + P_{2} + P_{3} = 0.0039 + 0.002 + 0.0005 = 0.0064[/tex]

There is a 0.64% probability that the costumer has filed a claim.

The probability that a randomly selected customer has filed a claim is calculated using the total law of probability, which yields a result of 0.0064 or 0.64%, after considering the probabilities of each risk group filing a claim.

To calculate the probability that a randomly chosen customer has filed a claim, we need to use the total law of probability. This involves multiplying the probability of a customer being in each risk group by the probability that a customer in that risk group files a claim, and then summing these products.

For good risks, this probability is 0.78 (the percentage of good risk customers) multiplied by 0.005 (the probability a good risk customer files a claim): 0.78 * 0.005 = 0.0039.

For medium risks, the calculation is 0.20 * 0.01 = 0.0020.

For poor risks, the calculation is 0.02 * 0.025 = 0.0005.

Adding these probabilities together gives us the total probability that a customer has filed a claim: 0.0039 + 0.0020 + 0.0005 = 0.0064, or 0.64% if expressed as a percentage.

Therefore, the probability that a randomly selected customer from the insurance company has filed a claim is 0.0064 or 0.64%, rounded to four decimal places.

Answer:$624

Total sales: $52 x 48 = $2,496

Cost of ingredients:  

$2,496

4

= $624

Step-by-step explanation:

$624

Total sales: $52 x 48 = $2,496

Cost of ingredients:  

$2,496

4

= $624

Answer:

The ingredients will cost $624 for all of the birthday cakes.

Step-by-step explanation:

This week the bakery has orders for 48 birthday cakes. Each cake sells for $52.

1/4 of each cake's selling price is spent on ingredients.

This becomes [tex]\frac{1}{4}\times52= 13[/tex] dollars

Hence, the total cost of ingredients for 48 cakes will be "

[tex]13\times48=624[/tex] dollars

Therefore, the ingredients will cost $624 for all of the birthday cakes.

Answer:

Kay's husband drove at a speed of 50 mph

Step-by-step explanation:

This is a problem of simple motion.

First of all we must calculate how far Kay traveled to her job, and then estimate the speed with which her husband traveled later.

d=vt

v=45 mph

t= 20 minutes/60 min/hour = 0.333 h (to be consistent with the units)

d= 45mph*0.333h= 15 miles

If Kay took 20 minutes to get to work and her husband left home two minutes after her and they both arrived at the same time, it means he took 18 minutes to travel the same distance.

To calculate the speed with which Kate's husband made the tour, we will use the same initial formula and isolate the value of "V"

d=vt; so

v=[tex]\frac{d}{t}[/tex]

d= 15 miles

t= 18 minutes/60 min/hour = 0.30 h  (to be consistent with the units)

v=[tex]\frac{d}{t}=\frac{15 miles}{0.3 h}=50mph[/tex]

Kay's husband drove at a speed of 50 mph

Answer:

$2,850,000

Step-by-step explanation:

Data provided in the question:

Investment amount asked for by the company = $950,000

Exchange of equity = 25%

Now,

Equity exchanged = [tex]\frac{\textup{Amount invested}}{\textup{Post money evaluation}}\times100[/tex]

or

Post money evaluation = [tex]\frac{\textup{950,000}}{\textup{25}}\times100[/tex]

or

Post money evaluation = $3,800,000

Therefore,

Implied valuation = Post money evaluation - Amount invested

or

Implied valuation = $3,800,000 - $950,000 = $2,850,000

Answer:

I am not entirely sure what you are asking, but I believe the answer is 4*2=8

Step-by-step explanation:

This is because:

8 increased by the product of a number and 4 is at most 20.

8      +                           p                               +    4    = 20

We are trying to find p.

8 +4= 12

20 - 12= 8

So I believe the answer is 8.

(If the answer is wrong, plz tell me in the comments)

Answer:

3

Step-by-step explanation:

eight increased by the product of a number and 4 is at most 20

these implies

8 + a x4 = 20

a is the number whose product with 4 is increased by eight to give at most 20

8 + a x4 = 20

8 + 4a = 20

subtract 8 from both sides

4a = 20 -8

4a= 12

divide both sides by 4

a = 12/4 = 3

the number is 3

Answer:

x = 10

Step-by-step explanation:

8 + 12 = 2x

8 + 12 = 20

20 = 2x

----   -----

2      2

10 = x

x = 10

Hey!

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

Solution:

8 + 12 = 2x

~Divide 2 to both sides

20/2 = 2x/2

~Simplify

10 = x

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

Answer:

x = 10

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

Hope This Helped! Good Luck!

Answer:

The solution for this differential equation is [tex]y=\sqrt{-2cos(x)+6}[/tex]

Step-by-step explanation:

This differential equation [tex]\frac{dy}{dx}=\frac{sin(x)}{y}[/tex] is a separable First-Order ordinary differential equation.

We know this because a first-order differential equation is separable if and only if it can be written as

[tex]\frac{dy}{dx}=f(x)g(y)[/tex] where f and g are known functions.

And we have

[tex]\frac{dy}{dx}=\frac{sin(x)}{y}\\ \frac{dy}{dx}=sin(x)\frac{1}{y}[/tex]

To solve this differential equation we need to integrate both sides

[tex]y\cdot dy=sin(x)\cdot dx\\ \int\limits {y\cdot dy}= \int\limits {sin(x)\cdot dx}[/tex]

[tex]\int\limits {y\cdot dy}=\frac{y^{2} }{2} + C[/tex]

[tex]\int\limits {sin(x) \cdot dx}=-cos(x) + C[/tex]

[tex]\frac{y^{2} }{2} + C=-cos(x) + C[/tex]

We can make a new constant of integration [tex]C_{1}[/tex]

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

We need to isolate y

[tex]\frac{y^{2} }{2}=-cos(x) + C_{1}\\y^2=-2cos(x)+2*C_{1}\\\mathrm{For\:}y^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}y=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}\\y=\sqrt{-2cos(x)+c_{1} } \\y=-\sqrt{-2cos(x)+c_{1} }[/tex]

We have the initial conditions y(0)=2 so we can find the value of the constant of integration for [tex]y=\sqrt{-2cos(x)+c_{1} } [/tex]

[tex]2=\sqrt{-2\cos \left(0\right)+c_1}\\2= \sqrt{-2+c_1} \\c_1=6[/tex]

For [tex]y=-\sqrt{-2cos(x)+c_{1} } [/tex] there is not solution for [tex]c_{1}[/tex] in the domain of real numbers.

The solution for this differential equation is [tex]y=\sqrt{-2cos(x)+6}[/tex]

Answer:

214

Step-by-step explanation:

The playing field is 53.33 yards wide and 120 yards long you would need to find the area so multiply 53.33 by 120 yards. That equals 6399.6 , the thickness they are applying is 1.2 millimeters. You would divide the area, 6399.6 by 1.2 which would equal 5333. Divide that by 25 gallons and it equals 213.32, you would need to purchase 214, rounded.

Answer:

The number of containers to purchase is   [tex]N_V= 67.85[/tex]

Step-by-step explanation:

From the question we are told that

        The playing field  width is [tex]w_f = 53.33 \ yard = 53.33*0.9144 = 48.76m[/tex]

        The playing field length is [tex]l_f = 120 \ yards = 120 * 0.9144 = 109.728m[/tex]

The volume of one container is [tex]V= 25 \ gallon = 25 * 0.00378541 = 0.094625m^3[/tex]

        The thickness of the painting is  [tex]t = 1.2 \ mm = 1.2 * 0.001 = 0.0012m[/tex]

The area of the playing field is [tex]A = 48.76 * 109.728[/tex]

                                         [tex]=5350.337m^2[/tex]

The number of container of paint needed [tex]N_V[/tex] [tex]= \frac{area \ of \ playing \ field(A) * thickness \ of \ paint \ application(t) }{volume\ single \ container(V)}[/tex]

=>    [tex]N_V = \frac{5350.337 * 0.0012}{0.094625}[/tex]

             [tex]N_V= 67.85[/tex]

Answer:

The amount of nuclear fuel required is 1.24 kg.

Step-by-step explanation:

From the principle of mass energy equivalence we know that energy generated by mass 'm' in an nuclear plant is

[tex]E=m\cdot c^2[/tex]

where

'c' is the speed of light in free space

Since the power plant operates at 1200 MW thus the total energy produced in 1 year equals

[tex]E=1200\times 10^6\times 3600\times 24\times 365=3.8\times 10^{16}Joules[/tex]

Thus using the energy produced in the energy equivalence we get

[tex]3.8\times 10^{16}=mass\times (3\times 10^{8})^2\\\\\therefore mass=\frac{3.8\times 10^{16}}{9\times 10^{16}}=0.422kg[/tex]

Now since the efficiency of conversion is 34% thus the fuel required equals

[tex]mass_{required}=\frac{0.422}{0.34}=1.24kg[/tex]

Answer:

$1895.64

Step-by-step explanation:

Given:

Principle for the first year = $100

rate of interest = 6% compounded monthly

thus,

rate per month, r = \frac{\etxtup{6}}{\textup{12}}= 0.5% = 0.005

Total time = 10 year\

Now,

for the first year

number of months, n = 12

Amount at the end of first year = Principle × ( 1 + r )ⁿ

on substituting the values, we get

Amount at the end of first year = 100 × ( 1 + 0.005 )¹²

or

Amount at the end of first year = $106.17

Therefore,

The principle amount for the consecutive years will be

= $1000 + Amount at the end of first year

=  $1000 + $106.17 = $1106.17

Thus, for the rest 9 years

n = 9 × 12 = 108

Principle = $1106.17

Final amount after the end of 10th year = Principle × ( 1 + r )ⁿ

or

Final amount after the end of 10th year = $1106.17 × ( 1 + 0.005 )¹⁰⁸

or

Final amount after the end of 10th year = $1895.64

Answer: 532

Step-by-step explanation:

Let x be the number of students and y be the number of non-students.

Then,  by considering the given information, we have

[tex]x+y=817-----(1)\\\\2x+3y=1919-----------(2)[/tex]

Multiply 2 on the both sides of equation (1), we get

[tex]2x+2y=1634--------(3)[/tex]

Subtract (3) from (2), we get

[tex]y=285[/tex]

Put the value of y in (1), we get

[tex]x+285=817\\\\\Rightarrow\ x=817-285=532[/tex]

Hence, 532 student tickets were sold .

[tex]\left[\begin{array}{ccc}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{array}\right]=\left[\begin{array}{ccc}-a_{1}^{2}-b_{1}^{2}\\-a_{2}^{2}-b_{2}^{2}\\-a_{3}^{2}-b_{3}^{2}\end{array}\right][/tex]

In the question,

We have to find out the circumcentre of the circle passing through the triangle with the vertices (a₁, b₁), (a₂, b₂) and (a₃, c₃).

So,

The circle is passing through these points the equation of the circle is given by,

[tex]x^{2}+y^{2}+ax+by+c=0[/tex]

On putting the points in the circle we get,

[tex]x^{2}+y^{2}+ax+by+c=0\\(a_{1})^{2}+(b_{1})^{2}+a(a_{1})+b(b_{1})+c=0\\and,\\(a_{2})^{2}+(b_{2})^{2}+a(a_{2})+b(b_{2})+c=0\\and,\\(a_{3})^{3}+(b_{3})^{3}+a(a_{3})+b(b_{3})+c=0\\[/tex]

So,

[tex](a_{1})^{2}+(b_{1})^{2}+a(a_{1})+b(b_{1})+c=0\\a(a_{1})+b(b_{1})+c=-(a_{1})^{2}-(b_{1})^{2}\,.........(1)\\and,\\a(a_{2})+b(b_{2})+c=-(a_{2})^{2}-(b_{2})^{2}\,.........(2)\\and,\\a(a_{3})+b(b_{3})+c=-(a_{3})^{3}-(b_{3})^{3}\,.........(3)\\[/tex]

On solving these equation using, Matrix method we can get the required equation of the circle,

[tex]\left[\begin{array}{ccc}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{array}\right]=\left[\begin{array}{ccc}-a_{1}^{2}-b_{1}^{2}\\-a_{2}^{2}-b_{2}^{2}\\-a_{3}^{2}-b_{3}^{2}\end{array}\right][/tex]

This is the required answer.

The center of the circumscribed circle of the symmetric isosceles triangle is at the origin, and the radius is equal to the length of the triangle's equal sides, denoted as r.

To find the center and the radius of the circle which circumscribes the triangle with vertices at ai, a, and a3, we must first understand the nature of the triangle. Given that the triangle is described to be symmetric with equal sides AB = BC = r, it is an isosceles triangle. The perpendicular bisector of the base a will pass through the midpoint of the base and the opposite vertex, given it is symmetric about this bisector. This will also be the diameter of the circumscribed circle. Consequently, as the triangle is isosceles and symmetric, we can use the properties of similar isosceles triangles to solve for the center and radius of the circumscribed circle.

Since the base a will also be the diameter of the circumscribed circle, and we know from geometry that the diameter is twice the radius (a = 2r), the radius of the circumscribing circle is r. The center of this circle is at the midpoint of the base a in the given symmetric form. Therefore, the center of the circle is at the origin due to the symmetric property and the radius remains r.

Answer:

Everything is verified in the step-by-step explanation.

Step-by-step explanation:

We have the following differential equation:

[tex]x'' + 2x' + x = 0[/tex]

This differential equation has the following characteristic polynomial:

[tex]r^{2} + 2r + 1 = 0[/tex]

This polynomial has two repeated roots of [tex]r = -1[/tex].

Since the roots are repeated, our solution has the following format:

[tex]x(t) = c_{1}e^{-t} + c_{2}te^{-t}[/tex]

This shows that the family of solutions satisfies the equation for all values of the constants. The values of the constants depends on the initial conditions.

Lets solve the system with the initial conditions given in the exercise.

[tex]x(0) = 8[/tex]

[tex]c_{1}e^{0} + c_{2}(0)e^{0} = 8[/tex]

[tex]c_{1} = 8[/tex]

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

[tex]x'(0) = 8[/tex]

[tex]x(t) = c_{1}e^{-t} + c_{2}te^{-t}[/tex]

[tex]x'(t) = -c_{1}e^{-t} + c_{2}e^{-t} - c_{2}te^{-t}[/tex]

[tex]-c_{1}e^{0} + c_{2}e^{0} - c_{2}(0)e^{0} = 8[/tex]

[tex]-c_{1} + c_{2} = 8[/tex]

[tex]c_{2} = 8 + c_{1}[/tex]

[tex]c_{2} = 8 + 8[/tex]

[tex]c_{2} = 16[/tex]

With these initial conditions, we have the following solution

[tex]x(t) = 8e^{-t} + 16te^{-t}[/tex]

Answer:

The amount in the account at the end of 8 years is about $8070.71.

Step-by-step explanation:

Given information:

Principal = $5000

Interest rate = 6% = 0.06 compounded monthly

Time = 8 years

The formula for amount after compound interest is

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

where,

P is principal.

r is rate of interest.

n is number of times interest compounded in a year.

t is time in years.

Substitute P=5000, r=0.06, n=12 and t=8 in the above formula.

[tex]A=5000(1+\frac{0.06}{12})^{(12)(8)}[/tex]

[tex]A=5000(1.005)^{96}[/tex]

[tex]A=5000(1.61414270846)[/tex]

[tex]A=8070.7135423[/tex]

[tex]A\approx 8070.71[/tex]

Therefore the amount in the account at the end of 8 years is about $8070.71.

Answer:

[tex]\lim_{x\rightarrow \infty}C=C[/tex]

Step-by-step explanation:

We are given that f(x)=c=Constant for all x in R

We have to find that why  f(x) not equal to zero when x approaches to zero.

[tex]\lim_{x\rightarrow \infty}f(x)[/tex]

[tex]\lim_{x\rightarrow \infty}C=C[/tex] not equal to zero

We are given that function which is constant for all x in R.

When x approaches then the value of function does not change. it remain same for all x in R because function is constant.

Hence, when x tends to infinity then f(x) is not equal to zero.