Find the equations of the tangent line to the curve at the given point. y = 9cos x. (π/3,4.5)
Answers
The details (the curve and the tangent line) are provided on the attached picture.
The equation of the tangent line to the curve is
y = -9 ( √3 / 2 )x + [ ( 3π√3 + 9 ) / 2 ]
What is an Equation of a line?
The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept
And y - y₁ = m ( x - x₁ )
y = y-coordinate of second point
y₁ = y-coordinate of point one
m = slope
x = x-coordinate of second point
x₁ = x-coordinate of point one
The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Given data ,
Let the curve be y = 9 cos ( x )
To find the equation of the tangent line to the curve y , take the derivative of y , we get
dy / dx = 9 ( - sin ( x ) )
dy / dx = -9 sin ( x )
So , y' = -9 sin ( x )
when x = π/3
Now plug in your value for x into y'
-9 sin ( π/3 ) = -9 ( √3 / 2 )
This is the slope of the tangent line at x = π/3
And , To find the equation of the tangent line, we need a value for y. Simply plug your x value into the original equation for y
So ,
y = 9 cos x
y = 9 cos ( π / 3 )
y = 9 ( 1/2 )
y = 9/2
Now use point slope form to find the equation of the tangent line
y - y₁ = m ( x - x₁ )
Substituting the values in the equations , we get
y - 9/2 = -9 ( √3 / 2 ) ( x - π/3 )
On simplifying the equation , we get
y - 9/2 = -9 ( √3 / 2 )x + 9 ( √3 / 2 ) ( π/3 )
Adding 9/2 on both sides of the equation , we get
y = -9 ( √3 / 2 )x + [ ( 3π√3 + 9 ) / 2 ]
Hence , the equation is y = -9 ( √3 / 2 )x + [ ( 3π√3 + 9 ) / 2 ]
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Related Questions
cx+5c=7c solve for x assume c doesn't equal 0
Answers
[tex]\begin{aligned} x+5=7 \end{aligned}[/tex]
Subtracting [tex]5[/tex] from both sides gives [tex]x = \boxed{2}[/tex].
6200ft is <>= to 1mi 900ft
Answers
Convert 1 mile to feet. This would be 5280 feet. Add this to 900. 5280+900=6180. So 6200 is greater than 1 mile 900 feet, or 6200>1 mile 900 feet.
I hope this helps!
5280 + 900=6180
6200ft < 1 mi 900ft
The U.S.-based motorcycle manufacturer says that it expects to build 149000 motorcycles this year, up from 134000 last year. Find the percent of increase in production
Answers
15,000 / 134000 = 0.11
0.11 * 100 = 11
11 % increase
James gets headaches. the time between one headache and the next is an exponential random variable. he has noticed that, after having a headache, there is a 50% chance of having another headache within the next 4 days. james has not had a headache in 5 days. what is the probability that he will go for at least 5 more days before the next headache?
Answers
For an exponential distribution,
λ=rate of occurrence=mean number of headaches per day.
probability distribution function
=pdf(x, λ ) = λ e^(- λ x)
cumulative probability distribution function,
cdf(x, λ ) = 1 - e^(- λ x)
We are given probability of having no headaches in 4 days is 0.5,
this means x=4 in
cdf(4, λ ) = 0.5, =>
1 - e^(-4 λ )=0.5
Solve for λ
λ = -log(0.5)/4 = 0.1732868 [log(x)=natural log of x]
=>
interpretation: the mean number of headaches is 0.1732868 per day.
Using the cdf, for x=5 days,
cdf(5,0.1732868)=1-e^(-0.173868*5)=0.57955
=>
interpretation: probability of having a headache in 5 days = 0.57955
Therefore
probability of NOT having a headache in 5 days is (1-0.57955)=0.42045
need help plz I need this by today because it's due Tomorrow
Answers
.. income = 52*985 = 51,220
Dale's withholding was
.. tax paid = 0.16*51,220 = 8195.20
According to the 3rd row of the table, Dale owes 8,770 in taxes, so must pay an additional
.. $8770 -8195.20 = $574.80
Dale owes $574.80.
which statment is true
Answers
Answer: burger and fries
Step-by-step explanation:
How many lateral faces does a pentagonal prism have? Explain its actually 8 for anyone who needs this in the future
Answers
Explanation:
You can see the attached figure for better understanding.
A pentagonal prism has 2 bases (the bottom and top faces), each pentagonal, and 5 lateral faces.
The pentagon is a figure with 5 sides. Hence, from each side, the prism will have one lateral face that goes from one base to the other.
The figure attached shows a pentagonal prism where you can see what I explained.
A pentagonal prism has five lateral faces.
A pentagonal prism is a three-dimensional geometric figure with two parallel pentagonal bases and rectangular lateral faces. To determine the number of lateral faces:
(1) Visualize the pentagonal bases; a pentagon has five sides.
(2) Each side of the pentagon is perpendicular to a corresponding rectangle, forming a lateral face.
So, a pentagonal prism has five lateral faces, matching the number of sides of its base pentagon.
An octahedron, a completely different shape, has eight triangular faces, not to be confused with the prism.
What value of k makes the equation true? (5a2b3)(6akb)=30a6b4
2
3
4
8
Answers
Answer:
the answer is 4 I believe
Step-by-step explanation:
The equation below shows the total volume (V), in cubic units, of 4 identical boxes with each side equal to s units: V = 4s3 If s = 2.5 units, what is the value of V? (5 points) 25 cubic units 30 cubic units 62.50 cubic units 156.25 cubic units
Plz help
Answers
volume of one box is length * breadth * height
since its a cube all sides are equal
volume of one box = s³ = 2.5³
since there are 4 boxes,
total volume V = 4s³
V = 4 * 2.5³ = 4 * 15.625
= 62.50 cubic units
third response is correct
Answer:
Option 3 - V = 62.50 cubic units.
Step-by-step explanation:
Given : The equation below shows the total volume (V), in cubic units, of 4 identical boxes with each side equal to s units: [tex]V=4s^3[/tex] If s = 2.5 units.
To find : What is the value of V ?
Solution :
The equation of the total volume is
[tex]V=4s^3[/tex]
Substitute, s=2.5 units
[tex]V=4(2.5)^3[/tex]
[tex]V=4\times 15.625[/tex]
[tex]V=62.5[/tex]
Therefore, Option 3 is correct.
The value of V is 62.50 cubic units.
Movie A took in $142.44 million on it’s first weekend. This topped the previous opening weekend revenue set by movie B by $4.14 million. How much did movie B take in on its opening weekend?
Answers
According to the given statement, revenue of movie A was $4.14 million more than the revenue of movie B in their first weekends.
So, we can write:
Revenue of movie A = Revenue of movie B + $4.14 million
Revenue of movie B = 142.44 million - 4.14 million = $ 138.30 million.
Thus movie B take in $138.30 million on its opening weekend.
In a circle with a radius of 3635 cm, an arc is intercepted by a central angle of 2π7 radians. What is the arc length? Use 3.14 for π and round your final answer to the nearest hundredth.
Answers
C=θ/360πd
θ=2/7π=2/7*180=51 3/7°
d=2r=2×3635=7270 cm
hence the arc length will be:
C=(51 3/7)/360×π×7270
C=(1038+4/7)π cm
C=3,262.78 cm (to the nearest hundreth)
To the model estimated in table 8.1, add the interaction term, e401k · inc. estimate the equation by ols and obtain the usual and robust standard errors. what do you conclude about the statistical significance of the interaction term?
Answers
Signed General Penguin
P - potassium
S - sodium
L - lithium
M - magnesium
C - calcium
A - aluminium
Z - zinc
I - iron
L - lead
H - hydrogen
C - copper
S - silver
G - gold
P - platinum
Carson wants to put her 232 stickers into a book that has 36 Pages if she puts 8 stickers on each page how many pages will be blank
Answers
A treasure chest lies 100 meters below sea level. Write a positive or negative integer that represents the situation.
Answers
Hope this helped!
An integer that represents the situation is -100.
What are integers?
Any positive or negative number without fractions or decimal places is known as an integer, often known as a "round number" or "whole number."
Integers are positive and negative whole numbers.
In this case, we are given a situation where a treasure chest is located 100 meters below sea level.
The sea level can be considered as zero, so anything below sea level would be negative.
Therefore, we use a negative integer to represent this situation, and the negative integer that represents 100 meters below sea level is -100.
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a rectangular field has dimensions of (2x-3) yards and (x+4) yards. the area of the field is 285 square yards. what are the length and width of the field?
Answers
285 = (2x-3)(x+4)
285 = 2x^2 + 8x - 3x - 12
285 = 2x^2 + 5x - 12
2x^2 + 5x - 12 = 285
2x^2 + 5x - 297 = 0
(2x^2 + 27)(x - 11) = 0
x = -27/2 (rejected, length cannot be negative) or 11
x = 11
2x - 3 = 2(11) - 3 = 19
x + 4 = 11 + 4 = 15
So the length and width are 19 yards and 15 yards
Jonas jogged up the hill at an average rate of of a 1/12 mile per minute and then walked down the hill at an average rate of of a 1/16 mile per minute. The round trip took him 42 minutes. What is the missing value in the table that represents the distance of the trip down the hill?
Table
Rate Time Distance
Up the Hill 1/12 x 1/12(x)
Down the Hill 1/16 42-x ?
Answers
Answer - [tex] \frac{1}{16} (42-x)[/tex] (In miles)
EXPLANATION
We know that distance traveled is calculated by multiplying average speed by total time.
According to the table,
Uphill average speed = [tex] \frac{1}{12} [/tex] (in miles per minute)
Uphill travel time = x (in minutes)
Total distance uphill = [tex] \frac{1}{12} x[/tex] (In miles)
The same way,
Downhill average speed = [tex] \frac{1}{16} [/tex] (in miles per minute)
Downhill travel time = 42 - x
Total distance downhill = Speed * Time = [tex] \frac{1}{16} (42-x)[/tex] (In miles)
The formula for distance is d = rt; Where r = rate; t = time
d =? ; r = 1/16; t = 42-x
d = rt
= 1/16 * 42-x
= (42-x) / 16
a helicopter hovers at an altitude that is 1000 feet above a mountain peak of altitude 5210 feet. A second,taller peak is viewed frim both the mountaintop and the helicopter. From the helicopter, the angle of depression is 43°, and from the mountaintop, the angle of elevation is 18°. How far apart are the mountain peaks? What is the altitude of the taller peak?
Answers
B.) 5468.39 ft.
mr samuel buys a sheet of stamps. there are 4 rows with 7 stamps in each row. how many stamps does mr samuel's buy?
Answers
Mr Samuel buys 28 stamps.
Hope this helps !
Photon
karla has read 85% of her book, which amounts to 238 pages. how long is the book?
Answers
1% = 238 ÷ 85 = 2.8
100% = 2.8 x 100 = 280
The book has 280 pages
Karla's book is 280 pages long because 85% of the book represents 238 pages, which means the entire book is 100% of that amount calculated by dividing 238 by 0.85. The additional questions can't be addressed accurately without more context or information.
Explanation:If Karla has read 85% of her book and that represents 238 pages, we can calculate the total length of the book. To find the full length of the book, we need to determine 100% of the book when 85% equals 238 pages. We can set up a proportion or use the formula Part = Percent \times Whole to solve for the whole, which is the total length of the book.
Let's use the formula to find the total length of the book:
First, we express 85% as a decimal, which is 0.85.Next, we rearrange the formula to solve for Whole: Whole = Part / Percent.Substitute the given part (238 pages) and the percent (0.85) into the formula: Whole = 238 / 0.85.When we calculate that, we get the total length of the book: Whole = 238 / 0.85 = 280 pages.Therefore, the total length of Karla's book is 280 pages.
Regarding the additional questions:
Without any calculations, it's not likely that the professor will need to read a total of more than 1,050 pages if we are only referencing the pages that Karla has read. However, if we are considering a separate piece of text or assignment for the professor, a calculation would be necessary to determine the likelihood.
The idea of dividing the total pages by the number of available days is a good strategy for managing reading assignments. This technique helps to break down the task into manageable daily portions and allows better time management.
As for the probability that the professor will need to read more than 1,050 pages, we cannot calculate this without additional information about the total number of assignments and the length of each assignment. Moreover, it is unlikely that the average length of the papers will be less than 12 pages if we consider typical lengths for academic papers, which often exceed this length.
The price of apples went from $1.99 per lb to $3.19 per lb in four years. Find the rate of change of the price of apples.
Answers
1.20/4 = 0.30
rate of change is 0.30 per year (30 cents per year)
Plzzzz help for 12 points
Answers
Answer 3rd option from up to down
convert the following equation into standard form
y= 3/14 - x/7
Answers
Answer:
[tex]\displaystyle 2x + 14y = 3[/tex]
Step-by-step explanation:
y = 3⁄14 - ⅐x
+ ⅐x + ⅐x
___________
⅐x + y = 3⁄14 ► [Multiply the equation by 14 to eradicate the fraction.]
14[⅐x + y = 3⁄14] → [tex]\displaystyle 2x + 14y = 3[/tex]
I am joyous to assist you at any time.
What is another name for an indirect proof? Picture included
Answers
Which is the correct way to write three and one-tenth of a milliliter as an Arabic number?
Answers
three and one-tenth------> 3 1/10--------> (3*10+1)/10--------> 31/10-----> 3.1
the answer is 3.1 ml
Juanita is cutting a piece of construction paper in the shape of a parallelogram. Two opposite sides of the parallelogram have lengths (5n − 6) cm and (3n − 2) cm. The third side measures (2n + 3) cm. What are the lengths of two adjacent sides of the parallelogram? 2 cm and 2 cm 4 cm and 7 cm 7 cm and 9 cm 13 cm and 19 cmJuanita is cutting a piece of construction paper in the shape of a parallelogram. Two opposite sides of the parallelogram have lengths (5n − 6) cm and (3n − 2) cm. The third side measures (2n + 3) cm. What are the lengths of two adjacent sides of the parallelogram? 2 cm and 2 cm 4 cm and 7 cm 7 cm and 9 cm 13 cm and 19 cm
Answers
Answer:
Correct option is 2. The lengths of two adjacent sides of the parallelogram are 4 cm and 7 cm.
Step-by-step explanation:
The opposite sides of a parallelogram are equal.
[tex]5n-6=3n-2[/tex]
[tex]2n=4[/tex]
Divide both sides by 2.
[tex]n=2[/tex]
The value of n is 2.
The lengths of two adjacent sides of the parallelogram are
[tex]5n-6=5(2)-6=10-6=4[/tex]
[tex]2n+3=2(2)+3=7[/tex]
Therefore the correct option is 2. The lengths of two adjacent sides of the parallelogram are 4 cm and 7 cm.
Is 90 g less than, greater than, or equal 9kg
Answers
9 kg = 9000 grams since 1 kg is equal to 1000 grams (kilo = thousand)
So comparing "90 g to 9kg" is the same as "90 g to 9000g"
90g is smaller than 9000 g
therefore, 90 g is smaller than 9 kg
Find constants a and b such that the function y = a sin(x) + b cos(x) satisfies the differential equation y'' + y' − 7y = sin(x).
Answers
The constants a = 1 and b = 1 satisfy the differential equation y'' + y' - 7y = sin(x).
Explanation:We can find the constants a and b by comparing the coefficients of the trigonometric functions in the given differential equation with the functions in the function y = a sin(x) + b cos(x).
In the differential equation, the coefficient of y'' is 1, the coefficient of y' is 1, and the coefficient of y is -7. In the function y = a sin(x) + b cos(x), the coefficient of sin(x) is a and the coefficient of cos(x) is b.
So, we equate the coefficients:
1 = a1 = b-7 = 0 (from the term -7y in the differential equation)Therefore, the constants a = 1 and b = 1 satisfy the differential equation.
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The value of a is [tex]\frac{-8}{\ 65}[/tex] and value of b is [tex]\frac{-1}{\ 65}[/tex].
Given the function (y = a sin(x) + b cos(x)), we need to find constants a and b such that the function satisfies the differential equation:[tex]\frac{-1}{\ 65}[/tex]
y'' + y' − 7y = sin(x)
First, find the first derivative y':
y' = a cos(x) - b sin(x)
Next, find the second derivative y'':
y'' = -a sin(x) - b cos(x)
Substitute y, y', and y'' into the differential equation:
(-a sin(x) - b cos(x)) + (a cos(x) - b sin(x)) − 7(a sin(x) + b cos(x)) = sin(x)
Combine like terms:
(-a − 7a)sin(x) + (a cos(x)) + (-b - 7b)cos(x) - b sin(x) = sin(x)
Simplify the equation:
-8a sin(x) + a cos(x) + (-8b cos(x)) - b sin(x) = sin(x)
Separate the equation into two parts, one for sin(x) and one for cos(x):
(-8a - b)sin(x) + (a - 8b)cos(x) = sin(x)
Match coefficients to make equal:
-8a - b = 1a - 8b = 0Solve the system of equations:
From a - 8b = 0, we get a = 8bSubstitute a = 8b into -8a - b = 1:-8(8b) - b = 1 -64b - b = 1-65b = 1[tex]b = \frac{-1}{\ 65}[/tex]Then a = 8b[tex]a = 8(\frac{-1}{\ 65})[/tex][tex]a = \frac{-8}{\ 65}[/tex]
The constants are [tex]a = \frac{-8}{\ 65}[/tex] and [tex]b = \frac{-1}{\ 65}[/tex].
The spiral in the picture is made from 45° right triangles. The smallest triangle (red) has a side length of 1. You can use the Pythagorean theorem to find the length of its hypotenuse. The next (orange) triangle has a side length exactly matching the hypotenuse of the smallest triangle, and so on with the larger triangles of other colors. The lengths of the hypotenuses of all the triangles (starting with the smallest) make a geometric sequence. What is the ratio of the sequence?
Answers
(√(1²+1²))/1 = √2
2 × (3 – 1) + 3 simplify
Answers
So 2×2+3
Next you multiply.
So 2×2=4 + 3
Then you add.
So your answer will be 7.
Hope this helps!
A sphere has a radius of 9mm. What is the volume of the sphere, rounded to the hundredths place ?
Answers
V=4/3* π(9)^3
V=4/3* π(243)
V= 324 * π
(rounded): V= 1017.88 mm
The volume of the sphere will be 1017.88 cubic mm.
What is the volume of the sphere?The volume is defined as the space occupied by the solid in a three-dimensional plane. `The volume of the sphere is defined as the space occupied by the sphere in a three-dimensional space.
The volume of the sphere is calculated by the formula below:-
Volume = V=4/3 x πr³
Given that a sphere has a radius of 9mm. Use the formula in calculating the volume of the sphere.
Volume of a sphere:
V=4/3 x πr³
V=4/3 x π(9)³
V=4/3 x π(243)
V= 324 x π
Therefore, the volume of the sphere will be 1017.88 cubic mm.
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