Plato math help please

Step-by-step explanation:

(12 + -2 )/2

10/2

5 im pretty sure

Answer:

Suppose the bowl is situated such that the rim of the bowl touches the x axis, and the semicircular cross section of the bowl lies below the x-axis (in (iii) and (iv) quadrant ). Then the equation of the cross section of the bowl would be [tex]x^2+y^2=144[/tex], where y≤ 0,

⇒ [tex]y=-\sqrt{144-x^2}[/tex]

Here, h represents the depth of water,

Thus, by using shell method,

The volume of the disk would be,

[tex]V(h) = \pi \int_{-12}^{-12+h} x^2 dx[/tex]

[tex]= \pi \int_{-12}^{-12+h} (144-y^2) dy[/tex]

[tex]= \pi |144y-\frac{y^3}{3}|_{-12}^{-12+h}[/tex]

[tex]=\pi [ (144(-12+h)-\frac{(-12+h)^3}{3}-144(-12)+\frac{(-12)^3}{3}}][/tex]

[tex]=\pi [ -1728 + 144h - \frac{1}{3}(-1728+h^3+432h-36h^2)+1728-\frac{1728}{3}][/tex]

[tex]=\pi [ 144h - \frac{1}{3}(h^3+432h-36h^2}{3}][/tex]

[tex]=\pi [ 144h - \frac{h^3}{3} - 144h + 12h^2][/tex]

[tex]=\pi ( 12h^2 - \frac{h^3}{3})[/tex]

Special cases :

If h = 0,

[tex]V(0) = 0[/tex]

If h = 12,

[tex]V(12) = \pi ( 1728 - 576) = 1152\pi [/tex]

Answer:    

[tex]x-1[/tex] is a factor of [tex]x^n - 1[/tex]

Step-by-step explanation:

[tex]x-1[/tex] is a factor of [tex]x^n - 1[/tex]

We will prove this with the help of principal of mathematical induction.

For n = 1, [tex]x-1[/tex] is a factor [tex]x-1[/tex], which is true.

Let the given statement be true for n = k that is [tex]x-1[/tex] is a factor of [tex]x^k - 1[/tex].

Thus, [tex]x^k - 1[/tex] can be written equal to  [tex]y(x-1)[/tex], where y is an integer.

Now, we will prove that the given statement is true for n = k+1

[tex]x^{k+1} - 1\\=(x-1)x^k + x^k - 1\\=(x-1)x^k + y(x-1)\\(x-1)(x^k + y)[/tex]

Thus, [tex]x^k - 1[/tex] is divisible by [tex]x-1[/tex].

Hence, by principle of mathematical induction, the given statement is true for all natural numbers,n.

Answer:

The cost to rent a trailer for 2.8 hours is $21.4.

The cost to rent a trailer for 3 hours is $23.

The cost to rent a trailer for 8.5 hours is $67.

Step-by-step explanation:

Let x be the number of hours.

It is given that the charge to rent a trailer is $15 for up to 2 hours plus $8 per additional hour or portion of an hour.

The cost to rent a trailer for x hours is defined as

[tex]C(x)=\begin{cases}15 & \text{ if } x\leq 2 \\ 15+8(x-2) & \text{ if } x>2 \end{cases}[/tex]

For x>2, the cost function is

[tex]C(x)=15+8(x-2)[/tex]

We need to find the cost to rent a trailer for 2.8 hours, 3 hours, and 8.5 hours.

Substitute x=2.8 in the above function.

[tex]C(2.8)=15+8(2.8-2)=15+8(0.8)=21.4[/tex]

The cost to rent a trailer for 2.8 hours is $21.4.

Substitute x=3 in the above function.

[tex]C(3)=15+8(3-2)=15+8(1)=23[/tex]

The cost to rent a trailer for 3 hours is $23.

Substitute x=8.5 in the above function.

[tex]C(8.5)=15+8(8.5-2)=15+8(6.5)=67[/tex]

The cost to rent a trailer for 8.5 hours is $67.

Written all the ordered pairs in the form of (hours, cost).

(2.8,21.4), (3,23) and  (8.5,67)

Plot these points on coordinate plane.

Final answer:

To find the cost to rent a trailer for 2.8 hours, we consider the flat fee of $15 for the first 2 hours and add the additional cost of $8 for the partial hour beyond 2 hours, resulting in a total cost of $23.

Explanation:

The cost to rent a trailer for a given number of hours is determined by a flat fee of $15 for the first 2 hours and an additional cost of $8 for each extra hour or partial hour. For 2.8 hours, since this exceeds the initial 2-hour period, we calculate the cost as follows:

Therefore, the cost to rent a trailer for 2.8 hours is $23.

Answer:

1. The college (Arts and Science, Business, etc.) you are enrolled in

Nominal

2. The number of students in a statistics course  Ratio

3. The age of each of your classmates  Ratio

4. Your hometown  Nominal

Step-by-step explanation:

Nominal, ordinal, interval, or ratio data are the four fundamental levels of measurement scales that are used to capture data.

Nominal, are used for labeling variables, without any quantitative value.

Ordinal, the order of the values is what is significant, but the differences between each one is not really known.

Interval, we know both, the order and the exact differences between the values

Ratio, they have the order, the exact value between units, and have an absolute zero

Answer:

There are 48 children.

Step-by-step explanation:

Given :There are 3 times as many boys as girls.

            There are 24 more boys than girls,

To Find : how many children are there?

Solution:

Let the number of girls be x

Now we are given that there are 3 times as many boys as girls.

So, no. of boys = 3x

Now we are given that there are 24 more boys than girls.

So, [tex]3x-x=24[/tex]

[tex]2x=24[/tex]

[tex]x=12[/tex]

So, no. of girls = 12

No. of boys = 3x = 3(12) = 36

Now the total no. of children = 12+36 = 48

Hence there are 48 children.

Answer: $957646.07

Step-by-step explanation:

The formula we use to find the accumulated amount of the annuity is given by :-

[tex]FV=m(\frac{(1+\frac{r}{n})^{nt})-1}{\frac{r}{n}})[/tex]

, where m is the annuity payment deposit, r is annual interest rate , t is time in years and n is number of periods.

Given : m= $2000 ; n= 12   [∵12 in a  year] ;   t= 20 years ;   r= 0.063

Now substitute all these value in the formula , we get

[tex]FV=(2000)(\frac{(1+\frac{0.063}{12})^{12\times20})-1}{\frac{0.063}{12}})[/tex]

i.e. [tex]FV=(2000)(\frac{(1+0.00525)^{240})-1}{0.00525})[/tex]

i.e. [tex]FV=(2000)(\frac{(3.51382093497)-1}{0.00525})[/tex]

i.e. [tex]FV=(2000)(\frac{2.51382093497}{0.00525})[/tex]

i.e. [tex]FV=(2000)(478.823035232)[/tex]

i.e. [tex]FV=957646.070464\approx957646.07\ \ \ \text{ [Rounded to the nearest cent]}[/tex]

Hence, the accumulated amount of the annuity= $957646.07

The future value or accumulated amount of an ordinary annuity is calculated using the formula where P is the periodic payment, r is the interest rate per period, n is the number of compounding periods per year, and t is the time in years. Given P = $2000, r = 6.3%, n = 12 and t =  20 years, substituting these values into the formula gives the accumulated amount

To find the future value or accumulated amount of an ordinary annuity, we use the formula: FV = P * (((1 + r)^nt - 1) / r), where P is the periodic payment, r is the interest rate per period, n is the number of compounding periods per year, and t is the time in years.

In the given problem, P = $2000, r = 6.3% or 0.063 (in decimal), n = 12 (since the payments are monthly), and t =  20 years.

Substituting these into the formula, FV = $2000 * (((1 + 0.063 /12)^(12*20) - 1) / (0.063/12)).

Calculating the equation, we'll get the accumulated amount to the nearest cent.

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Answer: 0.339

Step-by-step explanation:

Given : Tour players Harry, Ron, Harmione and Ginny are playing a card game.

. A deck of 52 cards are dealt out equally.

Then, the number of card each person has = [tex]\dfrac{52}{4}=13[/tex]

If Harmione and Ginny have a total of 8 spades among them, then the total cards the total spades left = 13-8=5

Now, the number of ways to get 3 of 5 spades : [tex]^5C_3=\dfrac{5!}{3!2!}=10[/tex]

Number of ways to draw remaining 10 cards :  [tex]^{21}C_{10}=\dfrac{21!}{10!11!}=352716[/tex]

Also, the total cards Harmione and Ginny have = 13+13=26

Then the total cards left = 26

The number of ways to get 13 cards for Harry :

[tex]^{26}C_{13}=\dfrac{26!}{13!(26-13)!}\\\\=\dfrac{26!}{13!13!}=10400600[/tex]

Now, the probability that Harry has 3 of the remaining 5 spades :_

[tex]\dfrac{^5C_3\times ^{21}C_{10}}{^{26}C_{13}}\\\\=\dfrac{10\times352716}{10400600}\\\\=0.339130434783\approx0.339[/tex]

Hence, the probability that Harry has 3 of the remaining 5 spades= 0.339 (approx)

Answer: The value of x is 3.

Step-by-step explanation:

Since we have given that

10,11, 12 and x

Average of above numbers = 9

As we know that

Average is given by

[tex]\dfrac{\text{Sum of observation}}{\text{Number of observation}}\\\\\\\dfrac{10+11+12+x}{4}=9\\\\10+11+12+x=9\times 4\\\\33+x=36\\\\x=36-33\\\\x=3[/tex]

Hence, the value of x is 3.

Answer:

1.69224581396 Kg/L

Step-by-step explanation:

We are given the measure of the block as 4.45 cm × 3.35 cm × 6.15 cm.

Volume of block = 4.45 cm × 3.35 cm × 6.15 cm = 91.681125 cm cube = 91.681125 × 0.001 L = 0.091681125 L

We did the above step to convert the volume of block into Liter.

Mass of block is given as 155.147 gram = 155.147 × 0.001 kg = 0.155147 kg

We converted the mass of block into kilograms because we need density in Kg/L.

Density is defined as mass per unit volume

Density = [tex]\frac{Mass}{Volume}[/tex]

             = [tex]\frac{0.155147 }{0.091681125}[/tex]]

             = 1.69224581396 Kg/L

The density is found to be approximately 1.688 kg/L.

To find the density of the block, we need to use the density formula:

Density = Mass / Volume

The given dimensions of the block are:

First, calculate the volume:

Volume = Length × Width × Height

Volume = 4.45 cm × 3.35 cm × 6.15 cm

Volume ≈ 91.88925 cubic centimeters (cm)

Next, convert mass to kilograms and volume to liters:

Finally, calculate the density in kg/L:

Density = Mass / Volume

Density ≈ 0.155147 kg / 0.09188925 L

Density ≈ 1.688 kg/L

Thus, the density of the block is approximately 1.688 kg/L.

Answer:

$2191.12

Step-by-step explanation:

We are asked to find the value of a bond after 10 years, if you invest $1000 in a savings bond that pays 4% interest, compounded semi-annually.

[tex]FV=C_0\times (1+r)^n[/tex], where,

[tex]C_0=\text{Initial amount}[/tex],

r = Rate of return in decimal form.

n = Number of periods.

Since interest is compounded semi-annually, so 'n' will be 2 times 10 that is 20.

[tex]4\%=\frac{4}{100}=0.04[/tex]

[tex]FV=\$1,000\times (1+0.04)^{20}[/tex]

[tex]FV=\$1,000\times (1.04)^{20}[/tex]

[tex]FV=\$1,000\times 2.1911231430334194[/tex]

[tex]FV=\$2191.1231430334194[/tex]

[tex]FV\approx \$2191.12[/tex]

Therefore, the bond would be $2191.12 worth in 10 years.

Answer:   29

Step-by-step explanation:

Let S denotes the total number of people surveyed, A denotes the event of having dog , B denotes the event of having cats and C denotes the event of having fish.

Given :     n(S)=50  ;n(A)=11  ;  n(B) =13  and  n(C)=6

Also, n(A∩B)=5 ;  n(A∩C) = 2 and n(B∩C)=3 and n(A∩B∩C)=1

We know that,

[tex]n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A \cap B)-n(A \cap C)-n(B \cap C)-n(A \cap B\cap C)\\\\=11+13+6-5-2-3+1=21[/tex]

Now, the number of people owned none of these pets :-

[tex]n(S)-n(A\cup B\cup C)\\\\=50-21=29[/tex]

Hence, the number of people owned none of these pets =29

Answer:

0.692.

Step-by-step explanation:

This is a Binomial Probability of Distribution  with P(success) = 0.67.  Prob  success  >= 20) , 31 trials.

From  Binomial Tables we see that the required probability  = 0.692.

Answer:

(B) 18.

Step-by-step explanation:

We are asked to find the perimeter of a quadrilateral with vertices at (-1, 4), (7,4), (7,5), and (-1. 5).

First of all, we will draw vertices of quadrilateral on coordinate plane and connect the vertices as shown in the attached photo.

We can see that our quadrilateral is a parallelogram, whose parallel sides are equal.

[tex]\text{Perimeter of quadrilateral}=8+1+8+1[/tex]

[tex]\text{Perimeter of quadrilateral}=16+2[/tex]

[tex]\text{Perimeter of quadrilateral}=18[/tex]

Therefore, the perimeter of the given quadrilateral is 18 units.

Answer:

D. A, B, and C

Step-by-step explanation:

Option (A) is true because when two straight lines intersect to each other we get two pair of vertically opposite angles and the angles opposite to each other is always equal.

Option (B) is also correct as If the sum of two angles is equal to 180°, then they are supplementary to each other.

Option (C) is also correct as If the sum of the two angles is equal to 90°, then they are Complementary to each other.

Hence, Option (D) is correct.

Answer:

$767.49

Step-by-step explanation:

given,

Amount of money = $16,500

quarterly rate = 3.6/4 = 0.9 %

times = 6 × 4 = 24 quarters.                            

[tex]A =\dfrac{P(r(1+r)^n)}{(1+r)^n-1}\\\\A =\dfrac{16500\times(0.009(1+0.009)^{24})}{(1+0.009)^{24}-1}\\A = \$ 767.49[/tex]          

hence, the payment to amortize the dept  will be equal to $767.49  .

Answer: a) 15:26, and b) 15:11.

Step-by-step explanation:

Since we have given that

Number of graded tests = 15

Number of total tests = 26

Number of ungraded tests is given by

[tex]26-15\\\\=11[/tex]

a) Proportion of all exams has Dr. Fitxgerald graded is given by

15:26.

b) Ratio of graded to ungraded tests is given by 15:11

Hence, a) 15:26, and b) 15:11.

(a) The proportion of all exams graded by Dr. Fitzgerald is [tex]\(\frac{15}{26}\)[/tex].

(b) The ratio of graded to ungraded tests is [tex]\(\frac{15}{26 - 15}\) or \(\frac{15}{11}\)[/tex].

(a) To find the proportion of exams graded by Dr. Fitzgerald, we divide the number of exams graded by the total number of exams. This gives us the fraction:

[tex]\[ \text{Proportion graded} = \frac{\text{Number of exams graded}}{\text{Total number of exams}} = \frac{15}{26} \][/tex]

This fraction represents the part of the whole set of exams that has been graded.

(b) To find the ratio of graded to ungraded tests, we take the number of exams that have been graded and divide it by the number of exams that have not been graded. The number of ungraded exams is the total number of exams minus the number of graded exams:

[tex]\[ \text{Number of ungraded exams} = \text{Total number of exams} - \text{Number of exams graded} = 26 - 15 = 11 \][/tex]

Now, we can find the ratio:

[tex]\[ \text{Ratio of graded to ungraded tests} = \frac{\text{Number of exams graded}}{\text{Number of exams ungraded}} = \frac{15}{11} \][/tex]

This ratio tells us how many times greater the number of graded exams is compared to the number of ungraded exams.

Answer:

108 one-bedroom units

72 two-bedroom units

36 three-bedroom units

Step-by-step explanation:

Let x, y, z the number of one-bedroom, two-bedroom and three-bedroom units respectively. Then  

1) x+y+z = 216

2)     y+z = x

3)         x = 3z

Multiplying equation 1) by -1 and adding it to 2), we get

-x = x-216 so, x = 216/2 = 108

x = 108

Replacing this value in 3) we get

z = 108/3 = 36

z = 36

Replacing now in 2)

y+36 = 108, y = 108-36 and

y = 72

In the planned apartment complex, there will be 0 one-bedroom units, 216 two-bedroom townhouses, and 0 three-bedroom townhouses.

Let x be the number of one-bedroom units. Since the number of two- and three-bedroom townhouses equals the number of one-bedroom units, let y be the number of two-bedroom townhouses and z be the number of three-bedroom townhouses. We know that x + y + z = 216. Additionally, x = 3z because the number of one-bedroom units will be 3 times the number of three-bedroom townhouses. Substituting x = 3z into the first equation gives 3z + y + z = 216. Simplifying this equation, we get 4z + y = 216.

Now, we can solve this system of equations to find the values of x, y, and z. Subtracting y from both sides of the equation 4z + y = 216 gives 4z = 216 - y. Let's call this equation (1). Substituting x = 3z and y = 216 - 4z into the equation x + y + z = 216 gives 3z + (216 - 4z) + z = 216. Simplifying this equation, we get 4z + 216 = 216. Subtracting 216 from both sides of the equation gives 4z = 0. Let's call this equation (2).

Since equation (1) and equation (2) both have 4z on the left side, we can equate the right sides of the equations. This gives 216 - y = 0. Solving for y, we find y = 216. Plugging this value of y into equation (1), we get 4z = 216 - 216, which simplifies to 4z = 0. Solving for z, we find z = 0. Finally, plugging the value of z into the equation x = 3z, we get x = 3(0), which simplifies to x = 0.

Therefore, there are 0 one-bedroom units, 216 two-bedroom townhouses, and 0 three-bedroom townhouses in the complex.

Answer:

6233500

Step-by-step explanation:

We are given that CEO of a company that sells car stereos has determined the profit x number of stereos.

The profit of selling x number of stereos is given by

[tex]P(x)=-0.04x^2=100x-16500[/tex]

We have to find the value of profit when the company selling 12500 stereos.

Substitute the value of x=12500

Then, we get

[tex]P(12500)=-.04(12500)^2+1000(12500)-16500[/tex]

[tex]P(12500)=-6250000+12500000-16500=-6266500+12500000[/tex]

[tex]P(12500)=6233500[/tex]

Hence, the company should expect profit 6233500 from selling 12500 stereos.

Answer:

The chances it will rain two days in a row is 12%

Step-by-step explanation:

The forecast calls for a 30% chance of snow today

So, chance of snowfall today = 30% = 0.3

A 40% chance of snow tomorrow.

So, chance of snowfall tomorrow= 40% = 0.4

The chances it will rain two days in a row = [tex]0.4  \times 0.3[/tex]

                                                                    = [tex]0.12[/tex]  

So, percent  it will rain two days in a row = [tex]0.12 \times 100 = 12\%[/tex]

Hence the chances it will rain two days in a row is 12%

Constraints are simply the subjects of an objective function.

The point of intersection is:  [tex]\mathbf{(x_1,y_1) = (9.54,-4.19)}[/tex]

The constraints are given as:

[tex]\mathbf{9x_1 + 7x_2 \ge 57}[/tex]

[tex]\mathbf{4x_1 + 6x_2 \ge 13}[/tex]

Express [tex]\mathbf{4x_1 + 6x_2 \ge 13}[/tex] as an equation

[tex]\mathbf{4x_1 + 6x_2= 13}[/tex]

Subtract 6x2 from both sides

[tex]\mathbf{4x_1 = 13 - 6x_2}[/tex]

Divide through by 4

[tex]\mathbf{x_1 = \frac{1}{4}(13 - 6x_2)}[/tex]

Substitute [tex]\mathbf{x_1 = \frac{1}{4}(13 - 6x_2)} \\[/tex] in [tex]\mathbf{9x_1 + 7x_2 \ge 57}[/tex]

[tex]\mathbf{9 \times \frac{1}{4}(13 - 6x_2) + 7x_2 \ge 57}[/tex]

Open brackets

[tex]\mathbf{29.25 - 13.5x_2 + 7x_2 \ge 57}[/tex]

[tex]\mathbf{29.25-6.5x_2 \ge 57}[/tex]

Collect like terms

[tex]\mathbf{-6.5x_2 \ge 57 - 29.25}[/tex]

[tex]\mathbf{-6.5x_2 \ge 27.25}[/tex]

Divide both sides by -6.5

[tex]\mathbf{x_2 \ge -4.19}[/tex]

Substitute -4.19 for x2 in [tex]\mathbf{4x_1 + 6x_2 \ge 13}[/tex]

[tex]\mathbf{4x_1 + 6 \times -4.19 \ge 13}[/tex]

[tex]\mathbf{4x_1 - 25.14 \ge 13}[/tex]

Add 25.14 to both sides

[tex]\mathbf{4x_1 \ge 38.14}[/tex]

Divide both sides by 4

[tex]\mathbf{x_1 \ge 9.54}[/tex]

Hence, the values are:

[tex]\mathbf{(x_1,y_1) = (9.54,-4.19)}[/tex]

Read more about inequalities at:

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The solution of the problem involves finding the values of x1 and x2 which satisfy both inequalities when plotted on a graph. This can be done by simplifying  the equations and comparing them.

To solve this problem, we need to find where the two inequalities intersect. This means that we need to find the values of x1 and x2 which satisfy both inequalities.

Let's start with the first inequality '9x1 + 7x2 ≥ 57'. This means that the sum of 9 times x1 and 7 times x2 should be greater than or equal to 57. You can simplify this inequality by dividing the entire expression by the smallest coefficient which is 9, getting 'x1 + (7/9)x2 ≥ 57/9'.

Similarly, simplifying the second inequality '4x1 + 6x2 ≥ 13' by dividing by the smallest coefficient which is 4, we get 'x1 + (3/2)x2 ≥ 13/4'.

By comparing these two simplified inequalities, you should be able to identify the values of x1 and x2 where both inequalities are satisfied.

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Answer:

(E) None of these above are true.

Step-by-step explanation:

Married = 74% or 0.74

College graduates = 42% or 0.42

pr(married | college graduates) = 0.56

(A) These events are pairwise disjoint. This is false. Pairwise disjoint are also known as mutually exclusive events. Here we can see that both events are occurring at same time.

(B) These events are independent events. This is also false.

(C) These events are both independent and pairwise disjoint. False

(D) A worker is either married or a college graduate always. False

Here Probability(A or B) shall be 1

= Pr(A) + Pr(B) - Pr( A and B) = 0.74 + 0.42 - 0.56 * 0.42 = 0.9248

This is not equal to 1.

(E) None of these above are true. This is true.

Answer: c.  $4668.30

Explanation:

Given:

Sales = $513000

Sales made by an individual = 14% of $513000

Sales made by an individual = [tex]\frac{14}{100}\times 513000[/tex]

Sales made by an individual = $71280

Commission made on this sales = 6.5% of  $71280

Commission made on this sales = [tex]\frac{6.5}{100}\times 71280[/tex]

Commission made on this sales = $4668.30

The digital scale should display 3.0 when weighing 3 lbs of ham. This is because 3 pounds exactly can be displayed as the decimal 3.0 after converting the number into a fraction, 3/1, and dividing the numerator by the denominator.

When Daniel wants to buy 3 lb of ham, the digital scale at the grocery store should display the decimal 3.0. This is because 3 pounds exactly translates to 3.0 in decimal terms.

The process of converting a number like 3 into a fraction would begin by writing it as 3/1 (as any number can be written over 1).

To convert that into decimal form, you would divide the top number (numerator) by the bottom number (denominator), so 3 ÷ 1 = 3.0.

Thus, the digital scale should read 3.0 when Daniel weighs out his 3 lbs of ham.

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Answer:

No, Jay is not correct.

Step-by-step explanation:

Quotient of powers property:

For any non-zero number a and any integer x and y:

[tex]\frac{a^x}{a^y}=a^{x-y}[/tex]

According to by the quotient of powers property

[tex]\frac{0^5}{0^2}=0^{5-2}\Rightarrow 0^3=0[/tex]

We need to check whether Jay is correct or not.

No, Jay is not correct because quotient of powers property is used for non-zero numbers.

[tex]\frac{0^m}{0^n}=\frac{0}{0}=unde fined[/tex]

Therefore, Jay is not correct.

Answer:

[tex]P(x) = x - x^2 + x^3 - x^4+x^5[/tex]

Step-by-step explanation:

Let us first remember how a Taylor polynomial looks like:

Given a differentiable function [tex]f[/tex] then we can find its Taylor series to the [tex]nth[/tex] degree as follows:

[tex]P(x) = f(x_{0}) + f'(x_{0}).(x-x_{0}) + \frac{f''(x_{0})}{2!}.(x-x_{0})^2+.....+\frac{f^n(x_{0})}{n!}.(x-x_{0})^n + R_{n}(x).(x-x_{0})^n[/tex]

Where [tex]R_{n}(x)[/tex] represents the Remainder and [tex]f^n(x)[/tex] is the [tex]nth[/tex] derivative of [tex]f[/tex].

So let us find those derivatives.

[tex]f(x) = \frac{x}{1+x}\\f'(x) = \frac{1}{(1+x)^2}\\f''(x) = \frac{-2}{(1+x)^3}\\f'''(x) = \frac{6}{(1+x)^4}\\f''''(x) = \frac{-24}{(1+x)^5}\\f'''''(x) = \frac{120}{(1+x)^6}[/tex]

The only trick for this derivatives is for the very first one:

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

Then it's only matter of replacing on the Taylor Series and replacing [tex]x_{0}=0[/tex]

Answer: 0.03

Step-by-step explanation:

Total number of napkins: 5 x 525 = 2,625

75/2626 = 0.02857, which rounds to 0.03

The cost per individual napkin, when rounded to the nearest hundredth, is $0.03.

To begin finding the cost per napkin, we first need to find out how many napkins are purchased every 3 months. Since each box contains 525 napkins and you purchase 5 boxes every 3 months, that would be 525 * 5 = 2625 napkins. The cost of these napkins is $75.00.

So, to find the cost per individual napkin, you would divide the total cost by the total number of napkins. That would be 75 / 2625 = $0.028571... When rounded to the nearest hundredth, this becomes $0.03. So, each individual napkin costs $0.03. Therefore, the correct answer is (C) 0.03.

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Answer:

d) $37800

Step-by-step explanation:

Cost of making x items = [tex]C(x)=25x + 300[/tex]

Cost of making [tex]1500[/tex] items = [tex]C(1500)=25(1500) + 300\\C(1500)= 37500 + 300\\C(1500)= 37800[/tex]

Cost of making [tex]1500[/tex] items = $37800

d) $37800 is the correct answer

[tex]2xyy'+y^2-4x^3=0[/tex]

Let [tex]z(x)=y(x)^2[/tex], so that [tex]z'(x)=2y(x)y'(x)[/tex] (which appears in the first term on the left side):

[tex]xz'+z=4x^3[/tex]

This ODE is linear in [tex]z[/tex], and we don't have to find any integrating factor because the left side is already the derivative of a product:

[tex](xz)'=4x^3\implies xz=x^4+C\implies z=\dfrac{x^4+C}x[/tex]

[tex]\implies y(x)=\sqrt{\dfrac{x^4+C}x}[/tex]

With [tex]y(1)=2[/tex], we get

[tex]2=\sqrt{1+C}\implies C=3[/tex]

so the solution is as given in your post.

Answer:

The length of the diagonal is 22.6 ft.

Step-by-step explanation:

To find the length of the diagonal of a square, multiply the length of one side by the square root of 2:

If the length of one side is x, [tex]length = x\sqrt{2}[/tex] as you can see in the image attached.

This fact is a consequence of applying the Pythagoras' Theorem to find the length of the diagonal if we know the side length of the square.

[tex]length^{2}  = x^{2}+x^{2}  \\ length=\sqrt{x^{2}+x^{2}} \\ length=\sqrt{2x^{2} } \\ length=x\sqrt{2}[/tex]

We know that the length of one side is 16 ft so [tex]length = 16\sqrt{2}=22.627[/tex] and round to the nearest tenth is 22.6 ft