The tangent to the curve at a point P (x, y) has slope dy/dx at that point. By the chain rule,
dy/dx = (dy/dθ) / (dx/dθ)
We're in polar coordinates, so
y (θ) = r (θ) sin(θ) ==> dy/dθ = dr/dθ sin(θ) + r (θ) cos(θ)
x (θ) = r (θ) cos(θ) ==> dx/dθ = dr/dθ cos(θ) - r (θ) sin(θ)
We're given r (θ) = 1 - sin(θ), so that
dr/dθ = -cos(θ)
Then the slope of the tangent to the curve at P is
dy/dx = (dr/dθ sin(θ) + r (θ) cos(θ)) / (dr/dθ cos(θ) - r (θ) sin(θ))
dy/dx = (-cos(θ) sin(θ) + (1 - sin(θ)) cos(θ)) / (-cos²(θ) - (1 - sin(θ)) sin(θ))
dy/dx = - (cos(θ) - sin(2θ)) / (sin(θ) + cos(2θ))
The tangent is horizontal if dy/dx = 0 (or when the numerator vanishes):
cos(θ) - sin(2θ) = 0
cos(θ) - 2 sin(θ) cos(θ) = 0
cos(θ) (1 - 2 sin(θ)) = 0
cos(θ) = 0 or 1 - 2 sin(θ) = 0
cos(θ) = 0 or sin(θ) = 1/2
[θ = π/2 + 2nπ or θ = 3π/2 + 2nπ] or [θ = π/6 + 2nπ or θ = 5π/6 + 2nπ]
where n is any integer.
In the interval 0 ≤ θ ≤ 2π, we get solutions of θ = π/6, θ = 5π/6, and θ = 3π/2. (We omit π/2 because the denominator is zero at that point and makes dy/dx undefined.) So the points where the tangent is horizontal are themselves (√3/4, 1/4), (-√3/4, 1/4), and (0, -2), respectively.
The tangent is vertical if 1/(dy/dx) = 0 (or when the denominator vanishes):
sin(θ) + cos(2θ) = 0
sin(θ) + (1 - 2 sin²(θ)) = 0
2 sin²(θ) - sin(θ) - 1 = 0
(2 sin(θ) + 1) (sin(θ) - 1) = 0
2 sin(θ) + 1 = 0 or sin(θ) - 1 = 0
sin(θ) = -1/2 or sin(θ) = 1
[θ = 7π/6 + 2nπ or θ = 11π/6 + 2nπ] or [θ = π/2 + 2nπ]
Then for 0 ≤ θ ≤ 2π, the tangent will be vertical for θ = 7π/6 and θ = 11π/6, which correspond respectively to the points (-3√3/4, -3/4) and (3√3/4, -3/4). (Again, we omit π/2 because this makes dy/dx non-existent.)
Please guys help to solve this problem
9514 1404 393
Answer:
300
Step-by-step explanation:
Since nobody failed, the number who passed one or the other was 100%.
P(O + W) = P(O) +P(W) -P(O&W)
100% = 80% +70% -P(O&W)
P(O&W) = 50% = 150 students
If 150 students are 50% of the examinees, then 100% will be 300 students.
Answer:
[tex]300[/tex]hope it helps
#CARRYONLEARNINGA boy leaves station X on a bearing of 035' to station Y. which is 21km away. He then travels to another station Z on a bearing of 125 degrees . If Z is directly East of X, what is the distance from X to his present position?
9514 1404 393
Answer:
36.6 km
Step-by-step explanation:
We assume the initial bearing of the boy is 35°. Then he will make a 90° turn to a heading of 125°. A diagram shows the distance of interest is the hypotenuse of a right triangle in which 35° is the angle opposite the side of length 21 km.
The relevant trig relation is ...
Sin = Opposite/Hypotenuse
sin(35°) = (21 km)/XZ
XZ = (21 km)/sin(35°) ≈ 36.61 km
The distance from X to Z is about 36.61 km.
_____
The attached diagram has the angles measured in the usual way for a Cartesian plane: CCW from the +x axis. This will correspond to bearing measures if we relabel the axes so that +x is North, and +y is East.
A multiple regression model is ^Y = 8.114 + 2.005X1 + 0.774X2. Which of the following values is the estimate for the intercept parameter?
a. 0.774.
b. 8.114.
c. 1.000.
d. 2.005.
Answer:
B. 8.114
Step-by-step explanation:
The intercept parameter is the zero-grade component of the multilinear equation, that is, the component independent from [tex]x_{1}[/tex] and [tex]x_{2}[/tex]. Hence, the intercept parameter of the multilinear regression is 8.114. (Correct answer: B)
someone help and tell me what the correct answer is? got it wrong and want to learn
In one U.S city, the taxi cost is $3 plus $0.80 per mile. If you are traveling from the airport, there is an additional charge of $5.50 for tolls. How far can you travel from the airport by taxi for $56.50?
Answer:
60 miles
Step-by-step explanation:
Create an equation where y is the total cost and x is the number of miles traveled.
0.8x will represent the cost from the miles traveled. 8.5 will be added to this to represent the taxi cost and additional charge from tolls:
y = 0.8x + 8.5
Plug in 56.50 as y and solve for x, the number of miles:
y = 0.8x + 8.5
56.5 = 0.8x + 8.5
48 = 0.8x
60 = x
So, you can travel 60 miles
The number 55 is attached to a two-digit number to its left and the formed 4-digit number is divisible by 24. What could be the 2-digit number? List all options
the answer will be 44 I think I hoped I helped if not sorry.
A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There were 1000 grams of the material 10 years ago. There are 980 grams right now. What will be the amount of the material right after 20 years
Answer:
x = 960.4
Step-by-step explanation:
980 = 1000[tex]e^{kt}[/tex]
.98 = [tex]e^{10 k}[/tex]
ln(.98) = 10k ln(e)
k = ln(.98)/10
k=-0.00202
~~~~~~~~~~~~~~
x = 1000[tex]e^{20 *-.00202}[/tex]
x = 960.4
The amount of the material right after 20 years will be x = 960.4.
What is an exponential expression?Powers can simply be expressed in concise form using exponential expressions. The exponent shows how many times the base has been multiplied. Since 2 is the "base" and 5 is the "exponent," it can be represented as 2x2x2x2=25 for the number 32. This phrase should be understood as "two to the fifth power."
Given that radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There were 1000 grams of the material 10 years ago. There are 980 grams right now.
The amount of the material will be calculated as,
980 = 1000
[tex]0.98 = e^{10k}[/tex]
ln(.98) = 10k ln(e)
k = ln(.98)/10
k=-0.00202
The value after 20 years will be,
[tex]x = 1000e^{20\times 0.00202}[/tex]
x = 960.4
Therefore, the amount of the material right after 20 years will be x = 960.4.
To know more about an exponential expression follow
https://brainly.com/question/2456547
#SPJ5
21.Factorize : x² +10x+21
22.Factorize: x² -2x-35
23.Solve : 2x+28 = 9x-56
Step-by-step explanation:
21.
x²+10x+21x²+3x+7x+21x(x+3)+7(x+3)(x+3)(x+7)22.
x²-2x-35x²-5x+7x-35x(x-5)+7(x-5)(x-5)(x+7)23.
2x+28=9x-5628+56=9x-2x84=7xx=84/7x=12hope it helps
stay safe healthy and happy..Answer:
21. (x+7)(x+3)
22. (x-7)(x+3)
23. 12
Step-by-step explanation:
X^2+10x+21
(X+7)(x+3)
(We know that 7*3 is equal to 21 and that 7+3= 10)
Now try the same thing for 22.
(X-7)(X+5)
(-7*5= -35 . and -7+5=-2)
23. Use SPOE to get like terms to one side:
-7x=-84
Then use DPOE to find the value of X
x=12
Is the random variable described discrete or continuous? The amount of rain during the next thunderstorm.
Answer:
continuous
rain does not fall in specific units like 1 inch , 2 inches etc... but 1.23456 etc..
Step-by-step explanation:
What is the value of k?
K=?
9514 1404 393
Answer:
k = 2
Step-by-step explanation:
The geometric mean theorem for the altitude tells you ...
ON = √(OL·OM)
ON² = OL·OM . . . . . square both sides
4² = 8·k . . . . . . . . substitute values
k = 16/8 = 2 . . . . divide by the coefficient of k
_____
Additional comment
The geometric mean theorem for the legs tells you ...
MN = √(MO·ML) ⇒ l = 2√5
LN = √(LO·LM) ⇒ m = 4√5
These relations come from the fact that corresponding sides of the right triangles are proportional. (All of the triangles are similar.)
2•^2=?
A) -4
B) 1/4
C) 4
Answer:
1/4.
Step-by-step explanation:
2^-2 = 1/2^2
= 1/4.
Need help
What is the domain shown in the graph
Answer:
A
Step-by-step explanation:
Which simplified fraction is equal to 0.53? Need answers now plz
Answer:
8/15
Step-by-step explanation:
Answer:
8/15
Step-by-step explanation:
when you divide 8/15 its 0.53
What is the number of degrees of freedom that should be used for finding the critical value
Answer:
tow degree of freedom.
this inform I know that from vibration method
In a group of 450 students, 250 like oranges 280 like apples and 40 dislike both
the fruits
Answer:
Could you please question correctly??
Step-by-step explanation:
Total Students:450
Oranges:250
Apples:280
Dislike:40
like:280-250+40
Answer
i dont see the question sorry
Suppose that you are offered the following "deal.
You roll a six-sided die.
If you roll a 6, you win $8.
If you roll a 3, 4 or 5, you win $1.
Otherwise, you pay $7.
Complete the Probability Distribution table shown below.
Let X represent your profit and list the X values from smallest to largest. Roond to 4 decimal places where
appropriate.
Probability Distribution
Table
Х
P(X)
Find the expected profit. $
(Round to the nearest cent)
Answer:
expected profit is - $0.50
Step-by-step explanation:
1 $(7.00) 0.166666667 $(1.17)
2 $(7.00) 0.166666667 $(1.17)
3 $1.00 0.166666667 $0.17
4 $1.00 0.166666667 $0.17
5 $1.00 0.166666667 $0.17
6 $8.00 0.166666667 $1.33
$(0.50)
Help me pls, BRAINEST AWARD
Answer:
x = 3.7
Step-by-step explanation:
By applying sine ratio for the given angle B,
sin(39°) = [tex]\frac{\text{Opposite side}}{\text{Hypotenuse}}[/tex]
sin(39°) = [tex]\frac{AD}{AB}[/tex]
0.6293 = [tex]\frac{AD}{7}[/tex]
AD = 4.41
By applying tangent ratio for the given angle C,
tan(50°) = [tex]\frac{\text{Opposite side}}{\text{Adjacent side}}[/tex]
1.19 = [tex]\frac{AD}{x}[/tex]
1.19 = [tex]\frac{4.41}{x}[/tex]
x = 3.7
Evaluate the expression
The exclamation mark at the end of any number indicates a factorial. You may be able to find this button in your calculator, though where it is varies.
A factorial is the product of the integer and all of the integers below it (greater than 0).
So, 9 factorial = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
6 factorial = 6 x 5 x 4 x 3 x 2 x 1
We can immediately eliminate the numbers that are the same on the top and bottom, since this is a fraction. That leaves us with the following multiplication problem to solve.
9 x 8 x 7 = 504
Hope this helps!
Mr. Alvarado bought a total of 20 pounds of grass seed at the nursery for $168. He paid $9 per pound for Kentucky blue grass and $6 per pound for Tall Fescue. Which system of equations can be used to find the amount x (in pounds) of Kentucky blue grass and the amount y (in pounds) of Tall Fescue Mr. Alvarado purchased?
Answer:
K+T=20
$9K + $6T = $168
K is the Kentucky blue grass in pounds
T is the Tall fescue in pounds
Step-by-step explanation:
You can start with the first equation. We don't know the exact amounts of each but we know that there was a total of 20 pounds, and there were 2 types of grass seeds, so we can get that the amount of pounds of Kentucky blue grass(K) and the pounds of Tal Fescue(T) has a sum of 20.
K + T = 20
For the second equation we know that there is a sum of $168 so we'll start with that. Then, we know he paid $9 per pound of K so $9* the value of K is the amount paid for Kentucky blue grass total. This can be represented as 9K. We do the same for T, 6T. Since the sum of the cost of $9T and $6K must be $168 we can write this as:
$9K + $6T = $168
Romans Food Market, located in Saratoga, New York, carries a variety of specialty foods from around the world. Two of the store's leading products use the Romans Food Market name: Romans Regular Coffee and Romans DeCaf Coffee. These coffees are blends of Brazilian Natural and Colombian Mild coffee beans, which are purchased from a distributor located in New York City. Because Romans purchases large quantities, the coffee beans may be purchased on an as-needed basis for a price 10% higher than the market price the distributor pays for the beans. The current market price is $0.47 per pound for Brazilian Natural and $0.62 per pound for Colombian Mild. The compositions of each coffee blend are as follows: Blend
Bean Regular DeCaf
Brazilian Natural 75% 40%
Columbian Mild 25% 60%
Romans sells the regular blend for $3.60 per pound and the Decaf blend for $4.40 per pound. Romans would like to place an order for the Brazilian and Columbian coffee beans that will enable the production of 1000 pounds of Romans Regular coffee and 500 pound of Romans DeCaf coffee. The production cost is $0.80 per pound for the Regular blend. Because of the extra steps required to produce DeCaf, the production cost for the DeCaf blend is $1.05 per pound. Packaging costs for both products are $0.25 per pound. Formulate a linear programming model that can be used to determine the pounds of Brazilian Natural and Columbian Mild that will maximize the total contribution to profit. What is the optimal solution and what is the contribution profit?
Answer:
z(max) = 2996.13 $
x₁ = 968 x₂ = 430 ( quantities of regular and Decaf coffee respectevely)
Total quantity of BN = 898 pounds
Total quantity of CM = 500 pounds
Step-by-step explanation:
Cost of the beans
Brazilian natural = Price market + 10 % = 0.47 + 0.047
BN Cost = 0.517 $/lb
Clombian Mild = Price market + 10 % = 0.62 + 0.062
CM Cost = 0.682 $/lb
Composition of the coffee blend
Regular coffee 0.75 BN + 0.25 CM
De Caf coffee 0.40 BN + 0.60 CM
PRICES
Regular Roman = 3.60 $
Decaf = 4.40 $
Production costs:
Regular Roman = 0.80 $/lb
Decaf = 1.05 $/lb
Packaging costs: 0.25 $/Lb both
Profit = Price - cost
Profit of regular coffee = 3.60 - 0.80 - 0.25 -Cost of bean
for regular coffee cost of BN + CM
BN is : 0.75*BN cost = 0.75*0.517 = 0.38775 and
CM is : 0.25*0.682 = 0.1705
Profit of regular coffee = 1.99175 $
Profit for Decaf coffee = 4.4 - 1.05 - 0.25 - ( 0.517*0.4 + 0.6*0.682)
Profit for Decaf coffee = 4.4 - 1.30 - 0.616
Profit for Decaf coffee = 2.484 $
Let´s call x₁ pounds of regular coffee and x₂ pounds of Decaf coffee then:
Objective Function is:
z = 1.99175*x₁ + 2.484*x₂ to maximize
Subject to:
Availability of beans for 1000 pounds of Regular coffee means:
750 pounds of BN + 250 pounds of CM
Availability of beans for 500 pounds of Decaf coffee means
200 pounds of BN + 300 pounds of CM
Then 750 + 200 = 900 pounds of BN
And 250 + 300 = 550 pounds of CM
Availability of beans for 1000 pounds of Decaf coffee correspond to
0.75 *x₁ + 0.40*x₂ ≤ 900
Availability of beans for 500 pounds of Regular coffee correspond to
0.25*x₁ + 0.60*x₂ ≤ 500
Then the model is:
z = 1.99175*x₁ + 2.484*x₂ to maximize
Subject to:
0.75 *x₁ + 0.40*x₂ ≤ 900
0.25*x₁ + 0.60*x₂ ≤ 500
General constraints x₁ ≥ 0 x₂ ≥ 0 both integers
After 6 iterations optimal solution ( maximum z) is
z(max) = 2996.13 $
x₁ = 968 x₂ = 430
x₁ and x₂ are quantities of Regular and Decaf coffee respectively, to find out quantities of Brazilian Natural and Colombian Mild
we proceed as follows
Regular coffee is : 0.75*968 = 726 pounds of BN
Decaf coffee is : 0.40*430 = 172 pounds of BN
Total quantity of BN = 898 pounds
Regular coffee is : 0.25*968 = 242 pounds of CM
Decaf coffee is : 0.6*430 = 258 pounds of CM
Total quantity of CM = 500 pounds
If someone can pls give the answer with steps that would be greatly appreciated :)
Answer:
1.Sentence examples for that would be greatly appreciated from inspiring English sources. If Norton or Symantec or anyone else can provide any info that would be greatly appreciated!! In all sincerity, if Hillary supporters at The Daily Beast and Daily Banter can enlighten us, that would be greatly appreciated.Hope It helps
a+in=√1+i÷1-i,prove that a^2+b^2=1
Answer with Step-by-step explanation:
We are given that
[tex]a+ib=\sqrt{\frac{1+i}{1-i}}[/tex]
We have to prove that
[tex]a^2+b^2=1[/tex]
[tex]a+ib=\sqrt{\frac{(1+i)(1+i)}{(1-i)(1+i)}}[/tex]
Using rationalization property
[tex]a+ib=\sqrt{\frac{(1+i)^2}{(1^2-i^2)}}[/tex]
Using the property
[tex](a+b)(a-b)=a^2-b^2[/tex]
[tex]a+ib=\sqrt{\frac{(1+i)^2}{(1-(-1))}}[/tex]
Using
[tex]i^2=-1[/tex]
[tex]a+ib=\frac{1+i}{\sqrt{2}}[/tex]
[tex]a+ib=\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}[/tex]
By comparing we get
[tex]a=\frac{1}{\sqrt{2}}, b=\frac{1}{\sqrt{2}}[/tex]
[tex]a^2+b^2=(\frac{1}{\sqrt{2}})^2+(\frac{1}{\sqrt{2}})^2[/tex]
[tex]a^2+b^2=\frac{1}{2}+\frac{1}{2}[/tex]
[tex]a^2+b^2=\frac{1+1}{2}[/tex]
[tex]a^2+b^2=\frac{2}{2}[/tex]
[tex]a^2+b^2=1[/tex]
Hence, proved.
what is the equation of the directx for the following parabola -8(x-5)=(y+1)^2
Answer:
x=7
Step-by-step explanation:
The directrix of a parabola is the vertical line found by subtracting
p from the x-coordinate h
of the vertex if the parabola opens left or right.
x=h-p
Substitute the known values of
p and h
into the formula and simplify.
x=7
Can someone help me out here? Not sure how to solve this problem or where to start either?
Answer:
19.3 miles per gallon
Step-by-step explanation:
First, subtract 54,042.8-53,737.7. The answer is 305.1
Then, find the unit rate. 305.1/15.8
You get 19.31012658. The prompt says to round to the nearest tenth, so round, and you get 19.3.
That's your answer!
What are the coordinates of vertex F" of ΔF"G"H"?
Can someone please help?
Find the value of x for the right triangle.
Answer:
5
Step-by-step explanation:
cos60° = x / 10
10cos60° = x
10cos60° = 5
A driver must decide whether to buy a new car for $24,000 or lease the same car over a four-year period. Under the terms of the lease, she can make a down payment
of $3000 and have monthly payments of $150. At the end of the four years, the leased car has a residual value (the amount she pays if she chooses to buy the car at
the end of the lease period) of $11,000. Assume she can sell the new car at the end of the four years at the same residual value. Is it less expensive to buy or
to lease?
Answer:
3000 is the answer this question.
help
Does this graph represent a function? Why or why not?
A- no because it fails the vertical line test
B- yes because it passes the vertical line test
C-yes because it passes the horizontal line test
D- no because it fails the horizontal line test
Answer: No, it fails the vertical line test.
Step-by-step explanation:
Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. (Enter your answers as a comma-separated list.)
f(x) = 7/(1+x), a = 2
Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.]
f(x) = e−5x
f(x)=
[infinity]
n = 0
=
Find the associated radius of convergence R.
R =
Answer:
A) [ 7/3, (-7/9)(x/2), 7/27(x-2)^2, (-7/81)(x-2)^3 ]
B) attached below
Step-by-step explanation:
A) Using the definition of a Taylor series
The first four nonzero terms of the series for f(x) = 7/ (1 +x), a = 2
= [ 7/3, (-7/9)(x/2), 7/27(x-2)^2, (-7/81)(x-2)^3 ]
attached below is the detailed solution
B) Finding Maclaurin series for f(x)
f(x) = e^-5x
attached below
Associated radius of convergence = ∞ ( infinity )
PLEASE HELP!!! WILL GIVE BRAINLIEST!!!!!
Step-by-step explanation:
[tex]g(x) = 3^{\frac{x}{2}}[/tex]
For [tex]x = -2[/tex], we get
[tex]g(-2) = 3^{\frac{-2}{2}} = 3^{-1} = \frac{1}{3}[/tex]