Mathematics College

When integrating polar coordinates, when should one use the polar differential element, [tex]rdrd \theta [/tex]
and when should one just use[tex]drd \theta [/tex] ?

For instance, do you use the former or latter if changing to polar variables when solving a surface integral?

Answers

Answer 1
To answer your first question: Whenever you convert from rectangular to polar coordinates, the differential element will *always* change according to

[tex]\mathrm dA=\mathrm dx\,\mathrm dy\implies\mathrm dA=r\,\mathrm dr\,\mathrm d\theta[/tex]

The key concept here is the "Jacobian determinant". More on that in a moment.

To answer your second question: You probably need to get a grasp of what the Jacobian is before you can tackle a surface integral.

It's a structure that basically captures information about all the possible partial derivatives of a multivariate function. So if [tex]\mathbf f(\mathbf x)=(f_1(x_1,\ldots,x_n),\ldots,f_m(x_1,\ldots,x_n))[/tex], then the Jacobian matrix [tex]\mathbf J[/tex] of [tex]\mathbf f[/tex] is defined as

[tex]\mathbf J=\begin{bmatrix}\mathbf f_{x_1}&\cdots&\mathbf f_{x_n}\end{bmatrix}=\begin{bmatrix}{f_1}_{x_1}&\cdots&{f_m}_{x_n}\\\vdots&\ddots&\vdots\\{f_m}_{x_1}&\cdots&{f_m}_{x_n}\end{bmatrix}[/tex]

(it could be useful to remember the order of the entries as having each row make up the gradient of each component [tex]f_i[/tex])

Think about how you employ change of variables when integrating a univariate function:

[tex]\displaystyle\int2xe^{x^2}\,\mathrm dr=\int e^{x^2}\,\mathrm d(x^2)\stackrel{y=x^2}=\int e^y\,\mathrm dy=e^{r^2}+C[/tex]

Not only do you change the variable itself, but you also have to account for the change in the differential element. We have to express the original variable, [tex]x[/tex], in terms of a new variable, [tex]y=y(x)[/tex].

In two dimensions, we would like to express two variables, say [tex]x,y[/tex], each as functions of two new variables; in polar coordinates, we would typically use [tex]r,\theta[/tex] so that [tex]x=x(r,\theta),y=y(r,\theta)[/tex], and

[tex]\begin{cases}x(r,\theta)=r\cos\theta\\y(r,\theta)=r\sin\theta\end{cases}[/tex]

The Jacobian matrix in this scenario is then

[tex]\mathbf J=\begin{bmatrix}x_r&y_\theta\\y_r&y_\theta\end{bmatrix}=\begin{bmatrix}\cos\theta&-r\sin\theta\\\sin\theta&r\cos\theta\end{bmatrix}[/tex]

which by itself doesn't help in integrating a multivariate function, since a matrix isn't scalar. We instead resort to the absolute value of its determinant. We know that the absolute value of the determinant of a square matrix is the [tex]n[/tex]-dimensional volume of the parallelepiped spanned by the matrix's [tex]n[/tex] column vectors.

For the Jacobian, the absolute value of its determinant contains information about how much a set [tex]\mathbf f(S)\subset\mathbb R^m[/tex] - which is the "value" of a set [tex]S\subset\mathbb R^n[/tex] subject to the function [tex]\mathbf f[/tex] - "shrinks" or "expands" in [tex]n[/tex]-dimensional volume.

Here we would have

[tex]\left|\det\mathbf J\right|=\left|\det\begin{bmatrix}\cos\theta&-r\sin\theta\\\sin\theta&r\cos\theta\end{bmatrix}\right|=|r|[/tex]

In polar coordinates, we use the convention that [tex]r\ge0[/tex] so that [tex]|r|=r[/tex]. To summarize, we have to use the Jacobian to get an appropriate account of what happens to the differential element after changing multiple variables simultaneously (converting from one coordinate system to another). This is why

[tex]\mathrm dx\,\mathrm dy=r\,\mathrm dr\,\mathrm d\theta[/tex]

when integrating some two-dimensional region in the [tex]x,y[/tex]-plane.

Surface integrals are a bit more complicated. The integration region is no longer flat, but we can approximate it by breaking it up into little rectangles that are flat, then use the limiting process and add them all up to get the area of the surface. Since each sub-region is two-dimensional, we need to be able to parameterize the entire region using a set of coordinates.

If we want to find the area of [tex]z=f(x,y)[/tex] over a region [tex]\mathcal S[/tex] - a region described by points [tex](x,y,z)[/tex] - by expressing it as the identical region [tex]\mathcal T[/tex] defined by points [tex](u,v)[/tex]. This is done with

[tex]\mathbf f(x,y,z)=\mathbf f(x(u,v),y(u,v),z(u,v))[/tex]

with [tex]u,v[/tex] taking on values as needed to cover all of [tex]\mathcal S[/tex]. The Jacobian for this transformation would be

[tex]\mathbf J=\begin{bmatrix}x_u&x_v\\y_u&y_v\\z_u&z_v\end{bmatrix}[/tex]

but since the matrix isn't square, we can't take a determinant. However, recalling that the magnitude of the cross product of two vectors gives the area of the parallelogram spanned by them, we can take the absolute value of the cross product of the columns of this matrix to find out the areas of each sub-region, then add them. You can think of this result as the equivalent of the Jacobian determinant but for surface integrals. Then the area of this surface would be

[tex]\displaystyle\iint_{\mathcal S}\mathrm dS=\iint_{\mathcal T}\|\mathbf f_u\times\mathbf f_v\|\,\mathrm du\,\mathrm dv[/tex]

The takeaway here is that the procedures for computing the volume integral as opposed to the surface integral are similar but *not* identical. Hopefully you found this helpful.

Related Questions

What is the probability that Eric rolls an even number or draws an ace from a standard deck of cards?

Answers

Evidently the quesiton is incomplete.

It is logic that the actions are roll a die and draw a card.

The two events are indepent, so the combined probability is the sum of the individual probabiliites.

Probability = number of positive events / number of possible events

Probability of rolling an even number = number of faces with even numbers / total number of numbers.

Probability of rolling an even number = 3 / 6 = 1/2

Probability of drawing an ace = number of aces in the deck of cards / number of cards

Probability of drawing an ace = 4 / 52 = 1 / 13

Combined probability = 1/2 + 1/13 = (13 + 2) / 26 = 15 / 26

Answer: option D. 15 / 26

Which of the following represents the zeros of the function
g(x) = x3 - 9x2 + 2x + 48 ?

A. x= -8, x= 2 , and x= -3
B. x = 8, x = -2 , and x = 3
C. x = 6, x = -4 , and x = 9
D. x = -6, x = 4 , and x = -9

Answers

g(x) = x^3 - 9x^2 + 2x + 48 ?

Probe some roots. When you use x = - 2

you will have: (-2)^3 - 9(-2)^2 + 2(-2) + 48 = -8 - 36 - 4 + 48 = 0

So, - 2 is a root

From that you can divide x^3 - 9x^2 + 2x + 48 by x + 2 and you will get

x^2 - 11x + 24

Then you can factor that: (x - 8)(x - 3)

So, the three roots are x = - 2, x = 3 and x = 8, which is the option B.

Answer: option B. x = 8, x = -2 , and x = 3
 

Which equation can be used to find the answer? A playground with four sides has a perimeter of 52 ft. Three of the sides have lengths of 9 ft, 16 ft, and 19 ft. What is the length of the fourth side? A. s – 9 + 16 + 19 = 52 s =26 26 ft B. s + 9 + 16 + 19 = 52 s = 8 8 ft C. s – 52 = 9 + 16 – 19 s = 58 58 ft D. s – 9 – 16 – 19 = 52 s = 96 96 ft

Answers

What we know:
Perimeter=52 ft
Perimeter=s1+s2+s3+s4
s1=9ft
s2=16ft
s3=19ft
s4=s

What we need to find: equation for given information and solution to s4=s

Perimeter=s1+s2+s3+s4
52=9+16+19+s

52=9+16+19+s
52=44+s               like terms added
52-44=44-44+s     additive inverse
8=s

Solution: B. s + 9 + 16 + 19 = 52 s = 8 8 ft

Option B)s + 9 + 16 + 19 = 52. Solving this gives the fourth side as 8 ft.

To find the length of the fourth side, you need to know the equation that correctly represents the given information. The perimeter of the playground is the sum of all four sides. Therefore, the correct equation to find the fourth side (s) is:

B. s + 9 + 16 + 19 = 52

9 + 16 + 19 = 44.

s + 44 = 52.

s = 52 - 44.

s = 8.

Therefore, the length of the fourth side is 8 ft.

So, the correct option is B. s + 9 + 16 + 19 = 52.

What is the probability of selection for any man in a proportionate random sample, where a sample of 100 will be drawn from a population of 1,000 that is 50% male and 50% female?

Answers

Working Principle: Stratified Random Sampling

nx = (Nx/N)*n

where:
    nx = sample size for stratum x
    Nx = population size for stratum x
    N = total population size 
    n = total sample size

Given:

  Nx = 100
  N = 1000
  n = 0.5*(1000) = 500

Required: Probability of Man to be selected

Solution:

nx = (Nx/N)*n
nx = (100/1000)*500 = 50 men

ny = (Nx/N)*n
ny = (100/1000)*500 = 50 women


Probability of Man to be selected = nx/(nx + ny)*100 = 50/(50+50)*100 = 50%

ANSWER: 50%

PLEASE HELP ASAP DUE IN A DAY What is the unit rate for 768.57 m 41.1 h? Enter your answer, as a decimal, in the box.

Answers

The unit rate for 768.57 meters in 41.1 hours is approximately 18.68 meters per hour. This calculation is derived by dividing the distance by the time to find the rate of movement.

To find the unit rate, divide the distance by the time:

[tex]\[ \text{Unit rate} = \frac{\text{Distance}}{\text{Time}} \][/tex]

[tex]\[ \text{Unit rate} = \frac{768.57 \, \text{m}}{41.1 \, \text{h}} \][/tex]

[tex]\[ \text{Unit rate} \approx \frac{768.57}{41.1} \, \text{m/h} \][/tex]

[tex]\[ \text{Unit rate} \approx 18.68 \, \text{m/h} \][/tex]

So, the unit rate is approximately 18.68 meters per hour.

how do i calculate an estimated regression line with the information
n=12, ∑x=66, ∑y=588, ∑xy=2244, ∑x2=396

Answers

The regression line is
y = ax + b
where
a = [(Σy)(Σx²) - (Σx)(Σxy)]/D
b = [n(Σxy) - (Σx)(Σy)]/D
D = n(Σx²) - (Σx)²

Therefore,
D = 12(396) - 66² = 396
a = [588*396 - 66*2244]/396 = 214
b = [12*2244 - 66*588]/396 = -30

Answer:
The regression line is y = 214x - 30

Randy school requires that there are two adult chaperones for every 18 students when the students go on a field trip to the museum if there are 99 students going to the museum how many adult shop around our need it

Answers

Use a proportion.

2 adults to 18 students is as x adults to 99 students.

2/18 = x/99

18x = 2 * 99

18x = 198

x = 11

Answer: 11 adults

The sum of two positive numbers is 12. what two numbers will maximize the product g

Answers

Given two numbers x and y such that:

x + y = 12   ...    (1)


two numbers will maximize the product g

from  equation (1) 

y = 12 - x  

Using this value of y, we represent xy as

xy = f(x)= x(12 - x)

 f(x) = 12x - x^2

Differentiating the above function:

f'(x) = 12 - 2x

Maximum value of f(x) occurs at point for which f'(x) = 0.

Equating f'(x) to 0 we get:

12 - 2x = 0

 2x =  12

> x = 12/2 = 6

Substituting this value of x in equation (2):

y = 12 - 6 = 6

Therefore, value of xy is maximum when:

x = 6 and y = 6

The maximum value of xy = 6*6 = 36

The two numbers that sum up to 12 and maximize the product are 6 and 6.

Let's denote the two numbers by x and y. We know that x + y = 12, and we want to maximize the product P = xy.

First, express y in terms of x,

y = 12 - x

P = x(12 - x)
= 12x - x²

To find the maximum value of P, we can take the derivative of P with respect to x and set it to zero,

dP/dx = 12 - 2x

Set dP/dx to 0: 12 - 2x = 0

x = 6

Since x + y = 12, if x = 6, then y = 12 - 6 = 6.

The two numbers are 6 and 6.
This combination will maximize the product, which is 6 * 6 = 36.

In conclusion, the two numbers that add up to 12 and maximize their product are both 6.

Find the length and width of a rectangle when the width is 4ft. Shorter than the length. The perimeter of the rectangle is greater than 72ft.

Answers

length- 38 ft
width= 34

Johns collection contains US, Indian and British stamos. If the stamps ratios of US to Indian stamps is 5 to 2, and the ratio of Indian stamps to British stamps is 5 to 1, what is the ratio of US to British stamps

Answers

Final answer:

The ratio of US to British stamps is 25 to 2.

Explanation:

To find the ratio of US to British stamps, we need to consider the ratios provided in the question. The ratio of US to Indian stamps is 5 to 2, and the ratio of Indian stamps to British stamps is 5 to 1. We can combine these ratios by multiplying the two ratios together. 5/2 multiplied by 5/1 is equal to 25/2. Therefore, the ratio of US to British stamps is 25 to 2.

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There are 365,493 blue pens in a warehouse and 549384 black pens how many pens are there in all

Answers

365,493 + 549,384 = 914,877
To find the answer you add the numbers together to find the sum. 
365,493+549,384

Your answer would be 366,042.384

I hope this helps!

If an article costs a store $7.45 and if it sells the article for $9.95, what percentage (to the nearest tenth) markup is it using?

Answers

subtract to find the difference:

9.95 - 7.45 = 2.50

 divide 2.50 by cost price to get mark up percentage

2.50 / 7.45= 0.3355

= 33.6 percent to the nearest tenth

Answer:

Markup Percentage is 33.56 %

Step-by-step explanation:

Given: Cost of an article = $ 7.45

           Selling price of the article = $ 9.95.

To find: Markup percentage or profit percentage

Clearly Selling price is greater we have profit in this case.

Profit amount = 9.95 - 7.45 =  $ 2.50

Profit percentage = [tex]\frac{2.50}{7.45}\times100=33.5570469799=33.56[/tex] %

Therefore, Markup Percentage is 33.56 %

What number should I multiply 1 1/4 by to get 7/12

Answers

Let's take our given information and transform it into numbers. We will let x equal the "mystery" number we need to find. Here is our equation:
[tex]1 \frac{1}{4} x= \frac{7}{12} [/tex]
Now, all we need to do is convert the mixed fraction into an improper fraction:
[tex] \frac{5}{4} x= \frac{7}{12}[/tex] 
Now, just multiply the reciprocal of 5/4 with 7/12, giving us:
[tex] x=\frac{4}{5}* \frac{7}{12} [/tex]
Finally, just straight up multiply to get an answer of x = 28/60, which can be simplified down to x = 7/15. Therefore, the number you have to multiply 1 1/4 to get 7/12 is 7/15. Hope this helped!
1 and 1/4 is the same as 5/4 so essentially we are asking what do we multiple 5/4 by to get 7/12. Try to keep things as fractions initially to see if there is a fractional answer with whole numbers.

Let x represent the number:
x . 5/4 = 7/12
i.e. 5x/4 = 7/12
We want to get x on it's own so multiply both sides of equation by 4
5x = 4 . (7/12) = 28/12
now divide both sides by 5 to get x on its own
x = (28/12) / 5
This does not equate to a whole number fraction as 28 and 12 are not divisible by 5. So simply convert to decimal
x = 2.3333 / 5 = 0.46666



A cylindrical shaped vase has the radius of 3cm and a height of 18cm. how much water is needed to fill the vase 3/4 of the way?

Answers

Answer:

Step-by-step explanation:

To fill the entire vase:

pi(3)^2(18)= 508.9 cubic centimeters

To fill vase 3/4 of the way:

3/4= .75

So then you take what it takes to fill the vase (508.9) and multiply that by .75 which equals 381.70 cubic centimeters.

3/4 of the volume of water needed to fill the vase is 162π cm^3.

To find how much water is needed to fill the vase 3/4 of the way, we first need to calculate the volume of the vase and then determine 3/4 of that volume.

The formula to calculate the volume of a cylinder is:

[tex]Volume = \pi * radius^2 * height[/tex]

Given:

Radius (r) = 3 cm

Height (h) = 18 cm

Volume of the entire vase:

[tex]Volume = \pi * (3 cm)^2 * 18 cm\\Volume = \pi* 9 cm^2 * 18 cm\\Volume = 162\pi cm^3[/tex]

Now, to find 3/4 of the volume, multiply the total volume by 3/4:

[tex]3/4 * 162\pi cm^3 = 3 * 54\pi cm^3 = 162\pi cm^3[/tex]

So, 3/4 of the volume of water needed to fill the vase is 162π cm^3.

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I need to solve and show work for 3x over4 divided by 5-1 over 8

Answers

3or4vxe
     4  



Is the answer 
Sounds as tho' you want to evaluate

3x
---
  4

over

5x-1
------
    8

To accomplish this, invert the 2nd fraction and then multiply:


3x     5x-1
---   * ------  = 15x^2-3x   divided by 32.
  4       8

At an interest rate of 8% compounded annually, how long will it take to double the following investments?
$50
$500
$1700

Answers

let's see, if the first one has a Principal of $50, when it doubles the accumulated amount will then be $100,

recall your logarithm rules for an exponential,

[tex]\bf \textit{Logarithm of exponentials}\\\\ log_{{ a}}\left( x^{{ b}} \right)\implies {{ b}}\cdot log_{{ a}}(x)\\\\ -------------------------------\\\\ \qquad \textit{Compound Interest Earned Amount} \\\\ [/tex]

[tex]\bf A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\to &\$100\\ P=\textit{original amount deposited}\to &\$50\\ r=rate\to 8\%\to \frac{8}{100}\to &0.08\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annnually, thus once} \end{array}\to &1\\ t=years \end{cases} \\\\\\ 100=50\left(1+\frac{0.08}{1}\right)^{1\cdot t}\implies 100=50(1.08)^t \\\\\\ \cfrac{100}{50}=1.08^t\implies 2=1.08^t\implies log(2)=log(1.08^t) \\\\\\ [/tex]

[tex]\bf log(2)=t\cdot log(1.08)\implies \cfrac{log(2)}{log(1.08)}=t\implies 9.0065\approx t\\\\ -------------------------------\\\\ [/tex]

now, for the second amount, if the Principal is 500, the accumulated amount is 1000 when doubled,

[tex]\bf \qquad \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\to &\$1000\\ P=\textit{original amount deposited}\to &\$500\\ r=rate\to 8\%\to \frac{8}{100}\to &0.08\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annnually, thus once} \end{array}\to &1\\ t=years \end{cases} \\\\\\ 1000=500\left(1+\frac{0.08}{1}\right)^{1\cdot t}\implies 1000=500(1.08)^t \\\\\\ [/tex]

[tex]\bf \cfrac{1000}{500}=1.08^t\implies 2=1.08^t\implies log(2)=log(1.08^t) \\\\\\ log(2)=t\cdot log(1.08)\implies \cfrac{log(2)}{log(1.08)}=t\implies 9.0065\approx t\\\\ -------------------------------[/tex]

now, for the last, Principal is 1700, amount is then 3400,

[tex]\bf \qquad \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\to &\$3400\\ P=\textit{original amount deposited}\to &\$1700\\ r=rate\to 8\%\to \frac{8}{100}\to &0.08\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annnually, thus once} \end{array}\to &1\\ t=years \end{cases}[/tex]

[tex]\bf 3400=1700\left(1+\frac{0.08}{1}\right)^{1\cdot t}\implies 3400=1700(1.08)^t \\\\\\ \cfrac{3400}{1700}=1.08^t\implies 2=1.08^t\implies log(2)=log(1.08^t) \\\\\\ log(2)=t\cdot log(1.08)\implies \cfrac{log(2)}{log(1.08)}=t\implies 9.0065\approx t[/tex]

Cameron can display his 12 models cars

Answers

I don’t understand your question.

Over the last three evenings, Raina received a total of 83 phone calls at the call center . The first evening, she received 7 more calls than the third evening. The second evening, she received 2 times as many calls as the third evening. How many phone calls did she receive each evening ?

Answers

x = 1st evening, y = 2nd evening, z = 3rd evening

x + y + z = 83
x = z + 7
y = 2z

z + 7 + 2z + z = 83
4z + 7 = 83
4z = 83 - 7
4z = 76
z = 76/4
z = 19 <=== 3rd evening calls

x = z + 7
x = 19 + 7
x = 26 <=== 1st evening calls

y = 2z
y = 2(19)
y = 38 <=== 2nd evening calls

If you put all the flowers together end to end, what would be the total length of all the flowers?

Answers

1/8 + 3(1/4) + 2(1/2) + 5/8 + 3/4 + 7/8 + 1 =
1/8 + 3/4 + 1 + 5/8 + 3/4 + 7/8 + 1 =
13/8 + 6/4 + 2 =
13/8 + 12/8 + 16/8 =
41/8 =
5 1/8 inches <==

The salaries of the president and the Vice President of a certain country total $562,500 a year. If the president makes $237,400 more than the Vice President , find each of their salaries.

Answers

To do this question, you simply add what the Vice President makes to how much more the President makes. $562,500+ $237,400=799,900.

Each class at briarwood elementary collected at least 54 cans of food lduring the food drive.If tbere are 29 clases in the school what was the least numbet of cans collected?

Answers

the answer is 54 cans of food
Let's turn the words into numbers:
For each class → x ≥ 54
Full School → x ≥ 54(29)

If we simplify the second inequality, we get this:
x ≥ 1566

The answer to your query is x ≥ 1566. Hope this helps and have a fantastic day!

Some luggage pieces have wheels and a handle so that the luggage can be pulled along the ground. Suppose the length from the bottom of the bag to the place on the floor perpendicular to the hand on the bag is 14 inches, and the length of the bag with its handle is 19 inches. At which angle made by the bag and the floor would it be comfortable to roll the bag?

Answers

Draw a right triangle with adjacent-leg equal to the length from the bottom to the place on the floor, whose meausre is 14 inches, and the hypotenuse is 19 inches.

The trigonometric ratio cosine, relates the angle with those lengths:

cos(angle) = adjacent-leg / hypotenuse = 14 / 19

=> angle = arc cosine (14/19) = 42.5°

Answer: 42.5°

A student correctly answers 15 of the first 20 questions on an examination.

Answers

I imagine you're being asked to predict the success rate in answering questions AFTER the 20th has been answered.  15/20 represents a success rate of 0.75.  It's reasonable to use 0.75 as the probability of success on the 21st question and beyond.

If 4f(x)+f(5-x)=x2

What is f(x)?

Answers

We are given that [tex]4f(x)+f(5-x)=x^2[/tex].

Substitute x with 5-x, then the above equation becomes:

[tex]4f(5-x)+f(5-(5-x))=(5-x)^2[/tex], that is

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


So, we have the following system of equations:

i) [tex]4f(x)+f(5-x)=x^2[/tex]
ii) [tex]4f(5-x)+f(x)=(5-x)^2[/tex]

multiply the first equation by -4, so that we eliminate f(5-x)'s

i) [tex]-16f(x)-4f(5-x)=-4x^2[/tex]
ii) [tex]4f(5-x)+f(x)=(5-x)^2[/tex]

adding the 2 equations side by side we have:

[tex]-15f(x)=-4x^2+(5-x)^2[/tex]

expanding the binomial, and collecting same terms we have:

[tex]-15f(x)=-4x^2+(25-10x+x^2)[/tex]

[tex]-15f(x)=-3x^2-10x+25[/tex]

dividing by -5:

[tex]3f(x)=\frac{3}{5}x^2+2x-5[/tex]

dividing by 3:

[tex]\displaystyle{f(x)= \frac{1}{5}x^2+ \frac{2}{3}x-\frac{5}{3} [/tex]


Answer: [tex]\displaystyle{f(x)= \frac{1}{5}x^2+ \frac{2}{3}x-\frac{5}{3} [/tex]

The cost CC (in dollars) of making nn watches is represented by C=15n+85C=15n+85. How many watches are made when the cost is $385?

Answers

[tex]\bf \stackrel{cost}{C}=15\stackrel{watches}{n}+85\qquad \boxed{C=385}\qquad \stackrel{cost}{385}=15n+85 \\\\\\ 380=15n\implies \cfrac{380}{15}=n\implies \cfrac{76}{3}=n\implies 25\frac{1}{3}=n[/tex]

so, 25 whole watches and hmmmm just the wristband of another maybe.

Find all values of x (if any) where the tangent line to the graph of the given equation is horizontal. HINT [The tangent line is horizontal when its slope is zero.] (If an answer does not exist, enter DNE. Enter your answers as a comma-separated list.) y = −9x2 − 2x

Answers

By "y = −9x2 − 2x" I assume you meant  y = −9x^2 − 2x (the "^" symbol represents exponentiation).

Let's find the first derivative of y with respect to x:  dy/dx = -18x - 2.  This is equivalent to the slope of the tangent line to the (parabolic) curve.  Now let this derivative (slope) = 0 and solve for the critical value:  -18x - 2 = 0, or
-18x = 2.  Solving for x,   x = -2/18,    or    x = -1/9.

When x = -1/9, y = -9(-1/9)^2 - 2(-1/9).  This simplifies to y = -9/9 + 2/9, or 
y = -7/9.

The only point at which the tangent to the curve is horiz. is (-1/9,-7/9).

The tangent line is the point that touches a graph at a point.

The value of x at the tangent line to the graph of [tex]\mathbf{y=-9x^2 - 2x}[/tex] is [tex]\mathbf{x = -\frac 19 }[/tex]

The function is given as:

[tex]\mathbf{y=-9x^2 - 2x}[/tex]

Differentiate both sides with respect to x

[tex]\mathbf{y' =-18x - 2}[/tex]

Set the above equation to 0, to calculate the value of x

[tex]\mathbf{-18x - 2 = 0}[/tex]

Collect like terms

[tex]\mathbf{-18x = 2 }[/tex]

Divide both sides by -18

[tex]\mathbf{x = -\frac 19 }[/tex]

Hence, the value of x when at tangent line to the graph of [tex]\mathbf{y=-9x^2 - 2x}[/tex] is [tex]\mathbf{x = -\frac 19 }[/tex]

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Suppose five construction companies have the ability to build a factory overseas to produce a manufactured good. The marginal cost of building a factory for each construction company is shown in the table​ below: Producer Marginal Cost Company 1 ​$1,000,000 Company 2 ​$1,250,000 Company 3 ​$1,300,000 Company 4 ​$1,350,000 Company 5 ​$1,500,000 If the market price of an overseas factory is ​$ 1 470 000 ​, what is the surplus for these five​ companies?

Answers

The surplus for the five companies is amounting to $830,000

What is the surplus for the five companies?

Producer Surplus is the difference between the price at which a producer is willing to spend and the price which he is getting from the market for his product.

The minimum price at which a producer is willing to spend is the Marginal Cost of producing that product.Therefore, Producer surplus is Market Price- Marginal CostProducer Marginal Cost Market Price Producer Surpus Company 1 1,000,000 1,470,000 (1,470,000-1,000,000)=470,000

Company 2 1,250,000 1,470,000 (1,470,000-1,250,000)=220,000

Company 3 1,300,000 1,470,000 (1,470,000-1,300,000)=170,000

Company 4 1,350,000 1,470,000 (1,470,000-1,350,000)=120,000

Company 5 1,500,000 1,470,000 (1,470,000-1,500,000)=-30,000

Total producer surplus= 470,000 + 220,000 + 170,000 + -30,000 = 830,000

The surplus for the five companies is amounting to $830,000

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Final answer:

The surplus for Companies 1, 2, and 3 is $470,000, $220,000, and $170,000 respectively, as their marginal costs are less than the market price of $1,470,000. Companies 4 and 5 do not have a surplus since their marginal costs are above the market price. The total surplus is $860,000.

Explanation:

The surplus for the five construction companies based on their marginal costs and the market price of an overseas factory can be calculated as the difference between the market price and the marginal cost for each company. Only the companies that have a marginal cost lower than the market price will experience a surplus. Given that the market price is $1,470,000:

Company 1 has a surplus of $470,000 ($1,470,000 - $1,000,000).

Company 2 has a surplus of $220,000 ($1,470,000 - $1,250,000).

Company 3 has a surplus of $170,000 ($1,470,000 - $1,300,000).

Companies 4 and 5 do not have a surplus because their marginal costs are higher than the market price. The total surplus for the three companies with a surplus is $860,000 ($470,000 + $220,000 + $170,000).

Charlie has 5 times as many stamps as Ryan. They have 1,608 stamps in all. How many more stamps does CHarlie have than Ryan?

Answers

Charlie-5x           Charlie has 268*5=1340
Ryan-x                 1340+268=1608

5x+x=1608          1340-268=1072
6x=1608              
x=268                  Charlie has 1072 more stamps then Ryan

Ryan has 268 stamps







What is the value of r, the part of the job that Marina can complete in 1 hour? 0.1 0.4 0.5 0.6

Answers

so, Katherine can do 0.1 of the job per hour. (if she needs 10 hours for the job, each hour she can do the whole divided by the time, so 1/10, which is 0.1)

if in two hours she and Marina can be done, then Marina will do 1(the whole job)-2/10 (twice as much as she can do in an hour)=0.8 of the job. So if Marina does 0.8 of the job in two hours, each hour she will do half of it: 0.4 - and this is the correct answer 


Which point on the x-axis lies on the line that passes through point P and is perpendicular to line MN?

(0, 1)
(0, 4)
(1, 0)
(4, 0)

Answers

the answer is C. (1,0)

1. Points M and N have coordinates (-4,0) and (4,2), respectively.

Then vector [tex]\overrightarrow{MN}=(4-(-4),2-0)=(8,2)[/tex] is perpendicular to the neede line.

2. Write the equation of line that passes through the point P(2,-4) and is perpendicular to vector  [tex]\overrightarrow{MN}=(8,2):[/tex]

[tex]8(x-2)+2(y+4)=0,\\8x-16+2y+8=0,\\8x+2y-8=0,\\4x+y-4=0.[/tex]

3. Find the point on x-axis, that lies on the perpendicular line.

When y=0, then 4x-4=0, x=1 and point (1,0) lies on perpendicular line.

Answer: correct choice is C.

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