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.)
What is A11 for the geometric sequence 3,072, −1,536, -768, −384...?
Answer:
3
Step-by-step explanation:
The general formula of the series is 3072/(-2)^(n-1). A11=3072/(-2)^10=3
A driveway is 2/5 kilometers long. Omar walks 1/3 of the driveway. How far does Omar walk?
Answer:
2/15 kilometers
Step-by-step explanation:
Multiply the distance by the fraction walked
2/5 kilometers * 1/3
2/5 * 1/3
2/15 kilometers
how many feet is in one centimeter and how many inches is in 1 feet?
Answer:
12 inches r in a foot
0 feet r in a centimeter
Step-by-step explanation:
Answer:
0.032 feet in a centimeter and 12 inches in 1 foot
Step-by-step explanation:
hope it helps pls mark as brainliest!
A tour bus is traveling along a triangular path. The three straight lines form a right triangle. One leg of the triangle represents a distance of 8 miles. The other leg of the triangle is 4 miles shorter than the hypotenuse. What is the length of the hypotenuse of this triangle? Of the other leg?
Answer:
Hypotenuse=10 miles.
Short leg=6 miles.
Step-by-step explanation:
Set up triangle, leg 8 miles, hypotenuse x miles, short leg x-4 miles.Input into Pythagoras theorem.Simplify.The average of two numbers is 5x. If one of the numbers is 2x + 3, find the other number.
Answer:
8x-3
Step-by-step explanation:
Average of 2 numbers means add the two numbers and divide by 2
(y+z)/2 = 5x
Let z = 2x+3
(y+2x+3)/2 = 5x
Multiply each side by 2
y+2x+3 = 10x
Subtract 2x from each side
y+3 = 10x-2x
y+3 = 8x
Subtract 3
y = 8x-3
The other number is 8x-3
How to divided 245 by 70
Show your work
Answer:
Step-by-step explanation:
Hello!
2 4 5 ∟ 70
-2 1 3, 5
------------------------
3 5 0
3 5 0
- --------------------------------
0 0 0
An automobile manufacturer has given its jeep a 51.3 miles/gallon (MPG) rating. An independent testing firm has been contracted to test the actual MPG for this jeep since it is believed that the jeep has an incorrect manufacturer's MPG rating. After testing 230 jeeps, they found a mean MPG of 51.1. Assume the population variance is known to be 6.25. A level of significance of 0.02 will be used. Make the decision to reject or fail to reject the null hypothesis.
Answer:
The p-value of the test is 0.2262 > 0.02, which means that the decision is to fail to reject the null hypothesis.
Step-by-step explanation:
An automobile manufacturer has given its jeep a 51.3 miles/gallon (MPG) rating.
At the null hypothesis, we test if the mean is of 51.3, that is:
[tex]H_0: \mu = 51.3[/tex]
An independent testing firm has been contracted to test the actual MPG for this jeep since it is believed that the jeep has an incorrect manufacturer's MPG rating.
This means that at the alternative hypothesis, we test if the mean is different of 51.3, that is:
[tex]H_0: \mu \neq 51.3[/tex]
The test statistic is:
[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
In which X is the sample mean, [tex]\mu[/tex] is the value tested at the null hypothesis, [tex]\sigma[/tex] is the standard deviation and n is the size of the sample.
51.3 is tested at the null hypothesis:
This means that [tex]\mu = 51.3[/tex]
After testing 230 jeeps, they found a mean MPG of 51.1. Assume the population variance is known to be 6.25.
This means that [tex]n = 230, X = 51.1, \sigma = \sqrt{6.25} = 2.5[/tex]
Value of the test statistic:
[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
[tex]z = \frac{51.1 - 51.3}{\frac{2.5}{\sqrt{230}}}[/tex]
[tex]z = -1.21[/tex]
P-value of the test and decision:
The p-value of the test is the probability of the sample mean differing from 51.1 by at least 0.2, which is P(|z| > 1.21), which is 2 multiplied by the p-value of z = -1.21.
Looking at the z-table, z = -1.21 has a p-value of 0.1131.
2*0.1131 = 0.2262
The p-value of the test is 0.2262 > 0.02, which means that the decision is to fail to reject the null hypothesis.
If 2^x=3^y=12^z then prove it 2/x = 1/z -1/y.
[tex] \begin{array}{l} 2^x = 3^y = 12^z \\ 2^x = 3^y = 2^{2z} \cdot 3^z \\ \Rightarrow 3 = 2^{\frac{x}{y}} \\ \Rightarrow 2^x = 2^{2z} \cdot 2^{\frac{xz}{y}} \\ \Rightarrow x = 2z + \frac{xz}{y} \\ \Rightarrow xy = 2zy + xz \\ \Rightarrow 2zy = xy - xz \\ \text{Dividing both sides by }xyz,\text{ we get:} \\ \dfrac{2}{x} = \dfrac{1}{z} - \dfrac{1}{y} \end{array} [/tex]
if point B is the midpoint of points A and C, find the value of x and AC. AB= 5x - 2, BC= 9x -10
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Answer:
x = 2AC = 16Step-by-step explanation:
The midpoint divides the segment into two equal lengths:
AB = BC
5x -2 = 9x -10
8 = 4x
2 = x
AB = 5(2) -2 = 8
AC = 2AB = 2(8) = 16
The perimeter of a triangle is 83 centimeters. If two sides are equally long and the third side is 8 centimeters longer than the others, find the lengths of the three sides.
Answer:
25, 33
Step-by-step explanation:
let the length of the one with equal sides be x
third side = x+8
x+x+x+8 = 83
3x+8 = 83
3x = 75
x = 25
x+8 = 25+8 = 33
8x=3x²-1 plz help me show your work
Answer:
Step-by-step explanation:
3 times 8= 24 • 24 = 576 - 1 =575
or
3•8=24•2=48-1=47
not sure
Answer:
The answer is [tex]x=\frac{4(+-)\sqrt{19} }{3}[/tex] in exact form or [tex]x=2.7863[/tex], [tex]x=-0.1196[/tex] in decimal form.
Step-by-step explanation:
To solve this equation, start by moving all expression to the left side of the equation, which will include subtracting [tex]3x^2[/tex] and adding 1 to both sides of the equation. The equation will look like [tex]8x-3x^2+1=0[/tex].
Then, use the quadratic formula to find the solutions to the equation. The quadratic formula looks like [tex]\frac{-b(+-)\sqrt{b^2-4ac} }{2a}[/tex].
For this problem, the quadratic variables are as follows:
[tex]a=-3\\b=8\\c=1[/tex]
The next step is to substitute the values [tex]a=-3[/tex], [tex]b=8[/tex], and [tex]c=1[/tex] into the quadratic formula and solve for x. The quadratic formula will look like [tex]\frac{-8(+-)\sqrt{8^2-4(-3)(1)} }{2*-3}[/tex]. To simplify the equation, start by simplifying the numerator, which will look like [tex]x=\frac{-8(+-)2\sqrt{19} }{2*-3}[/tex]. Then, multiply 2 by -3 and simplify the equation, which will look like [tex]x=\frac{4(+-)\sqrt{19} }{3}[/tex]. The final answer is [tex]x=\frac{4(+-)\sqrt{19} }{3}[/tex] in exact form. In decimal form, the final answer is [tex]x=2.7863[/tex], [tex]x=-0.1196[/tex].
Evaluate the expression: y – y ÷ 1 + x Use x = 7 and y = 3
Hi ;-)
[tex]x=7 \ and \ y=3\\\\y-y:1+x=3-3:1+7=3-3+7=0+7=\boxed7[/tex]
Probability that a person is chosen at random
Answer:
152 / 370
Step-by-step explanation:
Total number of people
152+218 = 370
P( own a dog) = people said yes / total
= 152 / 370
If the terminal side of an angle (θ) goes through the point (4 , -3) what is (θ)?
Answer:
The family of directions of the given vector is represented by [tex]\theta = 323.130^{\circ} \pm 360\cdot i[/tex], [tex]\forall \,i\in \mathbb{N}_{O}[/tex].
Step-by-step explanation:
According to the given information, vector stands in the 4th Quadrant ([tex]x > 0[/tex], [tex]y < 0[/tex]) and direction of the vector ([tex]\theta[/tex]) in sexagesimal degrees, is determined by following definition:
[tex]\theta = 360^{\circ} - \tan^{-1} \left(\frac{|y|}{|x|} \right)\pm 360\cdot i[/tex], [tex]\forall \,i\in \mathbb{N}_{O}[/tex]
Please notice that angle represents a function with a periodicity of 360°.
If we know that [tex]x = 4[/tex] and [tex]y = -3[/tex], then the direction of the vector is:
[tex]\theta = 360^{\circ}-\tan^{-1}\left(\frac{|-3|}{|4|} \right)\pm 360\cdot i[/tex]
[tex]\theta = 323.130^{\circ} \pm 360\cdot i[/tex]
The family of directions of the given vector is represented by [tex]\theta = 323.130^{\circ} \pm 360\cdot i[/tex], [tex]\forall \,i\in \mathbb{N}_{O}[/tex].
A bakery owner asked 150 customers to taste a new type of cookie and found that 60 people liked its taste.
Which of these is an example of descriptive statistics?
a.)
40% of the people in the city where the bakery is located like the taste of the cookie.
b.)
40% of the surveyed customers like the taste of the cookie.
c.)
40% of all the bakery's customers like the taste of the cookie.
d.)
40% of all people like the taste of the cookie.
Answer:[ 40% of the surveyed customers like the taste of the cookie is an example of descriptive statistics. ]
A bakery owner asked 150 customers to taste a new type of cookie and found that 60 people liked its taste. [ 40% of the surveyed customers like the taste of the cookie is an example of descriptive statistics. ]
An example of the disruptive statistics is 40% of the surveyed customers like the taste of the cookie is an example of descriptive statistics.
We have given that,
A bakery owner asked 150 customers to taste a new type of cookie and found that 60 people liked its taste.
We have to determine the
an example of descriptive statistics
What are the descriptive statistics?
Descriptive statistics is a set of brief descriptive coefficients that summarize a given data set representative of an entire or sample population.
A bakery owner asked 150 customers to taste a new type of cookie and found that 60 people liked its taste. 40% of the surveyed customers like the taste of the cookie is an example of descriptive statistics.
An example of the disruptive statistics is that 40% of the surveyed customers like the taste of the cookie is an example of descriptive statistics.
To learn more about the statistics visit:
https://brainly.com/question/3493733
#SPJ2
A 230 pound man, a 140 pound woman, a 750 pound crate of equipment, an 80 pound bag of concrete. What percent of the total weight was concrete?
What percent of the total weight was human?
plzzzzz helppp i will give brainlyist
Answer:
C. (2)
Step-by-step explanation:
an integer is a WHOLE NUMBER
have an amazing day :)
Answer:
2 is an integer
Step-by-step explanation:
An integer is a whole number, it does not have a fractional part
Please help I will mark brainliest to who ever is rigjt
Answer:
(1,0) and (0,4)
Step-by-step explanation:
Crosses the x axisWhen f(x) will cross the x axis, the y coordinate will turn 0, so 0=-5^(x)+5, 5=5^(x) Which is possible when x=1. So (1,0)
Crosses the y axisWhen f(x) will cross the y axis, the x coordinate will turn 0, so f(0)=-5^(0)+5, f(0)=-1+5=4. So (0,4)
Polynomials with odd degrees typically make a "u-shaped graph" and polynomials with even degrees typically make an "s-shaped" graph.
True
False
The statement that odd degree polynomials have a u-shaped graph and even degree polynomials have an s-shaped graph is FALSE.
What do odd degree polynomials look like on a graph?Odd degree polynomials have branches that go in opposing directions which means that they will form an s-shaped graph.
Even degree polynomials on the other hand, have graphs that go in the same direction which is why they form u-shaped graphs.
In conclusion, the above statement is false.
Find out more on polynomials at https://brainly.com/question/9696642.
Which best describes the relationship between the line that passes through the points (8, 2) and (3,
5) and the line that passes through the points (-3,-7) and (0, -12)?
Answer:
C
Step-by-step explanation:
They are neither perpendicular nor parallel since line 1 has slope=-3/5 and line 2 has slope=-5/3. They are neither equal nor have a product equal to - 1.
evaluate the function f(x)=4x^2-7x+7 find f(7)
please I need the answer soon!
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Answer:
f(7) = 154
Step-by-step explanation:
The basic idea is you put 7 where x is and do the arithmetic.
Polynomial evaluation is sometimes easier if you rewrite it to Horner form.
f(x) = (4x -7)x +7
f(7) = (4·7 -7)(7) +7 = 21(7) +7 = 147 +7
f(7) = 154
x = either 100 , 140 , or 120
A dodgeball team at Lincoln Elementary School needs a team of 4 in order to compete against other schools. If there are 9 kids that want to be part of the team, how many different ways can you pick a team of 4
Answer:
3 ways
Step-by-step explanation:
The percent of data between z=0.23 and z = 1.27 is
(Round to two decimal places as needed.)
Answer:
0.40905 - 0.10204 = .30701 = 30.7 %
Step-by-step explanation:
0.23 0.40905
1.27 0.10204
You flip a coin that is not fair, the prbability of heads on each flip is 0.7. if the coin shows heads, you draw a marble from urn h with 1 blue and 4 red marbles. if the coin shows tails, you draw a marble from urn t with 3 blue and 1 red marble. Find the following probabilities:
a. The probability of choosing a red marble.
b. The probability of choosing a blue marble, given that the coin showed heads.
c. The probability that the coin showed tails, given that the marble was red.
Solution :
P(H) = 0.7 ; P(T) = 0.3
If heads, then Urn H, 1 blue and 4 red marbles.
If tails, then Urn T , 3 blue and 1 red marbles.
a).
P ( choosing a Red marble )
= P (H) x P( Red from Urn H) + P (T) x P( Red from Urn T)
[tex]$=0.7 \times \frac{4}{5} + 0.3 \times \frac{1}{4}$[/tex]
= 0.56 + 0.075
= 0.635
b). If P (B, if coin showed heads)
If heads, then marble is picked from Urn H.
Therefore,
P (Blue) [tex]$=\frac{1}{5}$[/tex]
= 0.2
c). P (Tails, if marble was red)
[tex]$=P (T/R) = \frac{P(R/T)}{P(R)} \ P(T)$[/tex]
Where P (R/T) = P ( red, if coin showed tails)
[tex]$=\frac{1}{4}$[/tex]
= 0.25 (As Urn T is chosen)
P (R) = P (Red) = 0.635 (from part (a) )
P (T) = P (Tails) = 0.3
∴ [tex]$P(T/R) = \frac{0.25 \times 0.3}{0.635}$[/tex]
= 0.118
what is the area of triangle JHK?
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Answer:
4.18 square units
Step-by-step explanation:
The area is given by the formula ...
A = 1/2bh
where b is the length of the base, and h is the perpendicular distance from the base to the opposite vertex.
A = 1/2(2.2)(3.8) = 4.18 . . . square units
what is the slope and point
Answer:
Step-by-step explanation:
Find the volume of the figure. If necessary, round the answer to the nearest whole number.
Answer:
V = 108 ft³
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightGeometry
Volume of a Rectangular Prism Formula: V = lwh
l is lengthw is widthh is heightStep-by-step explanation:
Step 1: Define
Identify variables
l = 4 ft
w = 3 ft
h = 9 ft
Step 2: Find Volume
Substitute in variables [Volume of a Rectangular Prism Formula]: V = (4 ft)(3 ft)(9 ft)Evaluate [Order of Operations]: V = 108 ft³ulwazi's Father offered to pay for Ani's wedding ring, which cost R1349 excluding 14%VAT calculate the selling price
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Answer:
₹1537.86
Step-by-step explanation:
With the 14% tax added, the final cost is ...
₹1349 × (1 +14%) = ₹1349×1.14 = ₹1537.86
Find the value of this expression
Answer:
[tex] \frac{(3) ^{2} + 3}{3 - 1} [/tex]
[tex] \frac{9 + 3}{3 - 1} [/tex]
[tex] \frac{12}{2} [/tex]
= 6
