(a) Rectangular equation: [tex](x-3)^2/9 + y^2/4 = 1;[/tex] conic section: ellipse centered at (3, 0) with semi-major axis 3 and semi-minor axis 2.
(b) Integral for arclength: [tex]s = \int [0,\pi /2] \sqrt{(72 + 112 cos 2\theta )} d\theta[/tex].
(c) Equation for vertical tangents: θ = arctan(3/4) or θ = arctan(-4/3) + π, corresponding to points on the ellipse at (3+3cos(arctan(3/4)), 2sin(arctan(3/4))) and (3+3cos(arctan(-4/3)+π), 2sin(arctan(-4/3)+π)).
(a) To convert the polar equation to rectangular coordinates, we use the following relations:
x = r cos θ
y = r sin θ
Substituting r = 6 cos θ + 4 sin θ into these expressions, we get:
[tex]x = (6 cos \theta + 4 sin \theta) cos \theta = 6 cos^2 \theta + 4 sin \theta cos \theta[/tex]
[tex]y = (6 cos \theta + 4 sin \theta ) sin \theta = 6 sin \theta cos \theta + 4 sin^2 \theta[/tex]
Expanding these expressions using trigonometric identities, we get:
x = 3 + 3 cos 2θ
y = 2 sin 2θ
Thus, the rectangular equation of the curve is:
[tex](x - 3)^2/9 + y^2/4 = 1[/tex]
This is the equation of an ellipse centered at (3, 0) with semi-major axis 3 and semi-minor axis 2.
(b) To set up an integral for the arclength of the curve, we use the formula:
[tex]ds = \sqrt{(dx/d\theta ^2 + dy/d\theta ^2) d\theta }[/tex]
We have:
dx/dθ = -6 sin θ + 4 cos θ
dy/dθ = 6 cos θ + 8 sin θ
So,
[tex](dx/d\theta )^2 = 36 sin^2 \theta - 48 sin \theta cos \theta + 16 cos^2 \theta[/tex]
[tex](dy/d\theta )^2 = 36 cos^2 \theta + 96 sin \theta cos \theta + 64 sin^2 \theta[/tex]
Therefore,
[tex]dx/d\theta^2 = -6 cos \theta - 4 sin \theta[/tex]
[tex]dy/d\theta^2 = -6 sin \theta + 8 cos \theta[/tex]
And,
[tex](dx/d\theta^2)^2 = 36 cos^2 \theta + 48 sin \theta cos \theta + 16 sin^2 \theta[/tex]
[tex](dy/d\theta ^2)^2 = 36 sin^2 \theta - 48 sin \theta cos \theta + 64 cos^2 \theta[/tex]
Adding these expressions together and taking the square root, we get:
[tex]ds/d\theta = \sqrt{(72 + 112 cos 2\theta) }[/tex]
To find the arclength of the curve, we integrate this expression with respect to θ from 0 to π/2:
[tex]s = \int [0,\pi /2] \sqrt{(72 + 112 cos 2\theta )} d\theta[/tex]
(c) To find points on the curve with vertical tangents, we need to find values of θ where dy/dx is infinite.
Using the expressions for x and y in terms of θ, we have:
dy/dx = (dy/dθ)/(dx/dθ) = (6 cos θ + 8 sin θ)/(-6 sin θ + 4 cos θ)
Setting this expression equal to infinity, we get:
-6 sin θ + 4 cos θ = 0
Dividing both sides by 2 and taking the arctangent, we get:
θ = arctan(3/4) or θ = arctan(-4/3) + π
Plugging these values into the expressions for x and y, we get the corresponding points with vertical tangents.
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The local high school is hosting the last soccer game. They charged adults, x, 5 dollars to enter, and 3 dollars for students, y. It cost the school $300 to pay the referees for the game. The school wants to make a profit on the game, and 75 people attended. This is represented by the system:
5x + 3y > 300
x + y = 75
Which of the following points is a solution to the system?
(20, 15)
(25, 70)
(30, 45)
(40, 35)
IVE HEARD (30, 45) AND (40, 35). PLEASE HELP.
The correct answer is (40, 35) because by plugging in, 40 +35 = 75 and
(40 x 5) + (35 x 3) > 300
How to solve Word Problem ?Word problem can be solved by interpreting the problem into its best fit equation. The equation may be linear, quadratic or simultaneous equations.
Given that
5x + 3y > 300
x + y = 75
Let us first assume that 5x + 3y = 300. That is,
5x + 3y = 300
x + y = 75
Eliminate y by multiplying equation 2 by 3
5x + 3y = 300
3x + 3y = 225
2x = 75
x = 75/2
x = 37.5
Substitute x in equation 2
37.5 + y = 75
y = 75 - 37.5
y = 37.5
The correct answer is (40, 35) because 40 +35 = 75 and
(40 x 5) + (35 x 3) > 300
Therefore, the solution to the system is (40, 35)
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Okey please help me out
Rewrite the second expression as
[tex]\dfrac1{1+x^m + x^{-n}} = \dfrac{x^{\ell+n}}{x^{\ell+n} + x^{\ell+m+n}+x^\ell} = \dfrac{x^{-m}}{x^{-m} + 1 + x^\ell}[/tex]
and the third expression as
[tex]\dfrac1{1 + x^n + x^{-\ell}} = \dfrac{x^\ell}{x^\ell + x^{\ell+n} + 1} = \dfrac{x^\ell}{x^\ell + x^{-m} + 1}[/tex]
so the fractions all have the same denominator.
Then combining the fractions gives the desired result,
[tex]\dfrac1{1+x^\ell+x^{-m}} + \dfrac1{1+x^m+x^{-n}} + \dfrac1{1+x^n+x^{-\ell}} = \dfrac{1+ x^\ell + x^{-m}}{1+x^\ell+x^{-m}} = \boxed{1}[/tex]
Explain why the following form linearly dependent sets of vectors. (solve this problem by inspection.)
u1 = (3, -l), u2 = (4, 5), u3 = (-4, 7) in r2
A set of vectors {v1,v2,...,vk} is linearly independent if the vector equation x1v1 + x2v2 + .......... + xkvk = 0 has only the trivial solution
x1 = x2 = .... = xk =0. Then the set {v1,v2,...,vk} is linearly dependent otherwise.
So putting in the formula we get
x1u1 + x2u2 + x3u3 = 0
x1u1 + x2u2 = -x3u3
au1 + bu2 = u3 ∵ (-x1/x3 = a & -x2/x3 = b)
On putting in the values
a(3,-1) + b(4,5) = (-4,7)
(3a+4b,-a+5b) = (-4,7)
On comparing we get
3a + 4b = -4 -(1)
-a + 5b = 7 -(2)
on solving these equations we get
b = 17/19 and a= -48/19
which is non trivial.
Thus the following form linearly dependent sets of vectors.
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ok more of a question not like a equation or anything but when do you know when there is a extraneous answer and how do you solve it (please provide an example equation)
One can know if an equation is extraneous if after plugging it in the original equation, it shows a false meaning or the value is undefined.
What is an extraneous equation?It should be noted that an extraneous equation means a root of a transformed equation that isn't the root of the original equation due to the fact that it's excluded from the domain of the original equation.
In this case, one can know if an equation is extraneous if after plugging it in the original equation, it shows a false meaning or the value is undefined.
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what should be added to 3x³-5x²+7x-4 to get 7x³+2x²-3x+9
Answer:
Let the number be a.
==: a + (5x³-2x²+3x+7) = (7x³+7x-5)
==: a = (7x³+7x-5) - (5x³-2x²+3x+7)
==: a = 7x³ + 7x - 5 - 5x³+ 2x² - 3x - 7
==: a = 2x³ + 2x² + 4x - 12
i guess this is the process if you don't understand i can explain more
Which of the measurements of triangleBAC are correct
Answer:
yes
no
yes
Step-by-step explanation:
sin C = opp/hyp = 9/15 yes
tan B = opp/adj = 12/9 no
tan B = 12/9; B = tan^-1 12/9 = 53.13°
help me please 10 points
Answer:160ft^3
Step-by-step explanation:
Volume=(bh/2)h
V=(5ftx8ft/2)8ft
V=20x8=160
Volume=160ft^3
HOPE YOU GET IT RIGHT:)!
Rewrite each equation without absolute value for the given conditions.
[tex]y=/x-5/+/x+5/ if -5\ \textless \ x\ \textless \ 5[/tex]
Answer: 10
Step-by-step explanation:
If [tex]-5 < x < 5[/tex], then [tex]|x-5|=5-x[/tex] and [tex]|x+5|=x+5[/tex].
So, the expression is equal to 10.
The weight of football players is normally distributed with a mean of 180 pounds and a standard deviation of 20 pounds. what is the probability of a player weighing less than 215 pounds?
Answer:
Given:
mu = 200 lb, the mean sigma = 25 the standard deviation
For the random variable x = 250 lb, the
z-score is z = (x - mu) / sigma = (250 - 200) / 25 = 2
From standard tables for the normal distribution, obtain
P(x < 250) = 0.977
Answer: 0.977
In ΔPQR, find the measure of ∡P. Hypotenuse is 57.6 and the opposite side is 33.8.
Step-by-step explanation:
pythogoras theorem = Q²=P²+R²
P=P
Q= 57.6
R=33.8 =
(57.8)²=(P)²+(33.8)²
=3,340.84=(P)²+1142.44
=3340.84-1142.44=P²
=2,198.4=P²
[tex] = \sqrt{2198.4} = p \\ = 46.8 = p[/tex]
When figuring out the sign of a fraction as a
whole, you use the rules for dividing negatives.
A polypay sheep produces an average $9.9$ pounds of wool annually. This wool is cleaned, spun into yarn and then packaged into skeins of $175$ yards of yarn each. One pound of wool makes $10.95$ miles of yarn. If farmer Bill has a flock of $200$ polypay sheep, to the nearest thousand skeins, how many skeins of yarn is his flock expected to produce in one year
Answer:
218,048. skeins
Step-by-step explanation:
Break down the process into proper order.This question is built to throw you off by not telling you about the information in order.
First, the polypay sheep produces 9.9 pounds of wool.
Then, each pound produces 10.95 miles of yarn.
And finally, the yarn is packaged into skeins that are 175 yards each.
Step 1: Outline the workEach polypay sheep is 9.9 pounds. And we will use that total weight to find the amount of yarn in miles.
Then we will split up that yarn into equal pieces to make skeins.
BUT, the pieces are measured in yards, so we will have to convert the miles of yarn into an equivalent amount in yards as well. (Unmatched units of measurements don't mix well).
So, our equation will look like this:
(#of sheep) [tex]\times[/tex] (9.9 pounds per sheep) [tex]\times[/tex] (10.95 miles per yarn) [tex]\times[/tex] (1760 yards per mile) [tex]\div[/tex] (175 skeins per yard)
Step 2: Plug and ChThe number of sheep Bill has is 200.
So, we plug that into the number model we made.
[tex]200\times9.9\times10.95\times1760\div175[/tex]
We end up with:
218,048.914
The decimal indicates that there is a remaining amount of yarn that couldn't be properly packaged into a skein.
Since, that remainder is not a complete skein, we must ignore its existence.
So the number of skeins expected is:
218,048
The number of skeins of yarn is Bill's flock expected to produce in one year is 1819.
What is the unitary method?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Given that, a polypay sheep produces an average 9.9 pounds of wool annually.
Farmer Bill has a flock of 200 polypay sheeps
So, total weight of wool is 9.9×200
= 1980 pounds
One pound of wool makes 10.95 miles of yarn.
So, the number of yarns = 1980/10.95
= 180.82 miles of yarn
We know that, 1 mile =1760 yards
Then, 180.82 miles
= 318243.2 yards
This wool is cleaned, spun into yarn and then packaged into skeins of 175 yards of yarn each.
Number of skeins = 318243.2/175
= 1819
Therefore, the number of skeins of yarn Bill's flock expected to produce in one year is 1819.
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The points (-3,2) (5,2) and (5,1) are three vertices of a rectangle what is the fourth vortex?
The inverse variation equation shows the relationship between wavelength in meters, x, and frequency, y. y = startfraction 3 x 10 superscript 8 baseline over x endfraction what are the wavelengths for x-rays with frequency 3 × 1018?
The wavelength of X-rays for the given frequency is [tex]$1 \times 10^{-10} \mathrm{~m}$[/tex].
What is the wavelength of light ?The distance between the two crests or troughs of the light wave is known as the wavelength of light.
The color of light is determined by its wavelength, and the pitch of sound is determined by its wavelength. The visible spectrum of light has wavelengths between around 700 nm (red) to 400 nm (violet). The range of audible sound wavelengths is roughly 17 mm to 17 m.
To calculate the wavelength of light, we use the equation:
[tex]$\lambda=\frac{c}{\nu}$[/tex]
where,
λ = wavelength of the light
c = speed of light = [tex]$3 \times 10^{8} \mathrm{~m} / \mathrm{s}$[/tex]
ν= frequency of light = [tex]$3 \times 10^{18} s^{-1}$[/tex]
Putting the above value in equation,
[tex]$\lambda=\frac{3 \times 10^{8} \mathrm{~m} / \mathrm{s}}{3 \times 10^{19} \mathrm{~s}-1}=1 \times 10^{-10} \mathrm{~m}$[/tex]
Hence, the wavelength of X-rays for the given frequency is [tex]$1 \times 10^{-10} \mathrm{~m}$[/tex].
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Use trig to find the height of the hot air balloon. Write an equation and show work. Round answer to the nearest tenth
Answer:
x = 753.6 ft
Step-by-step explanation:
One leg is x.
The other leg is given.
The trig function that relates the two legs of a right triangle is the tangent.
tan A = opp/adj
tan 37° = x/1000ft
x = 1000 ft × tan 37°
x = 753.6 ft
Answer: 753.6 feet
Step-by-step explanation:
The angle, theta, is 37 degrees. However, we're not given the length of the hypotenuse in this case. So sine would be x by the hypotenuse and cosine would be 1,000 by the hypotenuse. In this case, we can use tangent, which is sine/cosine. In this case, tan 37 = x/1,000.
Hence,
tan 37 = x/1,000
x = tan 37 * 1000
x is approximately 753.6, rounded to the nearest tenth.
Nani needs to buy 8 cups of fresh pineapple for a fruit cocktail. the table shows the unit prices of pineapple at different stores. if nani only has $4, from which stores could she purchase her pineapple?
Nani could purchase her pineapple from stores A, D, and E.
Given Information
It is given that Nani needs to purchase 8 cups of pineapple.
The unit prices of pineapple at each store is as follows,
Store A : 0.49
Store B : 0.51
Store C : 0.55
Store D : 0.48
Store E : 0.45
Amount that Nani has = $4
Reason Behind Purchasing From Either A, D, or E
Since Nani has amount of $4, she can purchase pineapple only from the stores where the total cost of 8 pineapple cups adds up to less than $4. The purchase of 8 pineapple cups at each store would be given as,
Store A : 0.49(8) = $3.92
Store B : 0.51(8) = $4.08
Store C : 0.55(8) = $4.40
Store D : 0.48(8) = $3.84
Store E : 0.45(8) = $3.60
As it can be seen, Nani can purchase from the store A, D, and E as it falls within her budget of $4.
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Express tan D as a fraction in simplest terms. 40 B 24 32 C
Answer:
[tex]\sf Tan \ D = \dfrac{4}{3}[/tex]
Step-by-step explanation:
Trigonometry ratios:
[tex]\sf \boxed{\bf \tan \ D =\dfrac{opposite \ side \ of \ \angle \ D}{adjacent \ side \ of \ \angle D}}[/tex]
[tex]\sf =\dfrac{BC}{DC}\\\\ =\dfrac{32}{24}\\\\=\dfrac{4}{3}[/tex]
B=(2x+3)(4x^2-6x+9)-2(4^3-1)
Please help me
better be worth my 57 points
Answer: a
Step-by-step explanation:
The slope is the rate at which a line goes up or down (i.e., how steep it is). It is measured by the slope formula, which is [tex]\frac{y_2-y_1}{x_2-x_1}[/tex], where (x1, y1) and (x2, y2) are two points on the line.
Below, (x1, y1) are (c, d) and (x2, y2) are (a, b).
When all variables are plugged in, the slope is
[tex]\frac{b-d}{a-c}[/tex]
Find the surface area of the composite figure. Round to the nearest tenth if necessary.
The surface area of the composite figure is equal to 233.6 square centimeters.
How to determine the surface area of a solid
The surface area is the area of all faces of a solid. Since the surface area of the figure is a combination of triangles and quadrilaterals, we must sum all the areas to determine the surface area of the figure. Now we proceed to determine this:
A = 4 · (1/2) · (3 cm) · (2 cm) + 2 · (6 cm) · (3 cm) + 2 · (8 cm) · (6 cm) + 2 · (8 cm) · (2 cm) + 2 · (8 cm) · (3.6 cm)
A = 233.6 cm²
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The slope-intercept form of the equation of a line that passes through point (-3, 8) is y = -2x + 6. What is the
point-slope form of the equation for this line?
O y-3=-%(x + 8)
y+3=-2(x-8)
y+8=-%(x-3)
D y-8 = -2(x + 3)
Answer: [tex]y-8=-2(x+3)[/tex]
Step-by-step explanation:
See attached image, it's direct formula substitution.
suppose cos(A) = 4/5. use the trig identity sin^2(A)+cos^2(A)=1 to find sin(A) in quadrant IV. round to the ten-thousandth.
a. -0.3954
b. -0.6000
c. 0.6485
d. 0.4500
In quadrant IV, [tex]\sin(A)[/tex] is negative. So
[tex]\sin^2(A) + \cos^2(A) = 1 \implies \sin(A) = -\sqrt{1-\cos^2(A)} = -\dfrac35 = \boxed{-0.6000}[/tex]
Find the measure of each numbered angle.
Step-by-step explanation:
18z+10z+40z+5+1=360
68z+6=360
68z=360-6
68z=364
z≈5.353
A company makes rubber rafts. 12% of them develop cracks within the first month of operation. 27 new rafts are randomly sampled and tested, by being used for one month, under standardized conditions that mimic typical operating conditions. Calculate the probability that the number of tested rafts that develop cracks is no more than 3. Round your answer to four decimal places.
The probability that the number of tested rafts that develop cracks is no more than 3 is .00006.
The true proportion, p for the population is given to 0.12.
Thus, the mean, μ, for the sample = np = 27*0.12 = 3.24.
The sample size, n, given to us is 27.
Thus, the standard deviation, s, for the sample can be calculated using the formula, s = √{p(1 - p)}/n.
s = √{0.12(1 - 0.12)}/27 = √0.003911 = 0.0625389.
We are asked to calculate the probability that the number of tested rafts that develop cracks is no more than 3, that is, we need to calculate P(X ≤3).
P(X ≤ 3)
= P(Z ≤ {(3 - 3.24)/0.0625389) {Using the formula z = (x - μ)/s}
= P(Z ≤ -3.8376114706)
= .00006 {From table}.
Thus, the probability that the number of tested rafts that develop cracks is no more than 3 is .00006.
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Select the correct answer from each drop-down menu.
Consider the function represented by this graph.
As the value of x increases, the value of f (x)........ increases or decreases
The x-intercept is the point......
As the value of x increases and the function of f(x) also increases since it is moving in the upward direction. The x-intercept is (-3,0)
What is the function?A function is a set of input values into a program that produces the desired output. In mathematical terms, we can say a function is a set of x values that maps to a set of y values.
Functions can also be represented on the graph called graphical representation.
From the given graphical information, as the value of x increases, the function of f(x) also increases since it is moving in the upward direction.
The x-intercept is the value where the line cuts the horizontal x-axis when y is zero, therefore the x-intercept is (-3,0)
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In a right triangle the length of the hypotenuse is a and the measurement of one of the
acute angle is a. Find the measurement of the other acute angle and the lengths of the
legs.
Answer:
The measure of the acute angle = 90 - a
Leg 1 = a cos(a)
Leg 2 = a sin(a)
Step-by-step explanation:
In a right triangle ,the two acute angles are complementary
We are given that the measure of one of the acute angles is ‘a’
Then
The measure of the other one is ‘90 - a’
Now, we use the law of sine to determine the length of the legs :
Let x and y be respectively the length of the two legs.
[tex]\frac{x}{\sin \left( 90-a\right) } =\frac{a}{\sin \left( 90\right) }[/tex]
[tex]\Longleftrightarrow \frac{x}{\sin \left( 90-a\right) } =a[/tex]
[tex]\Longleftrightarrow x = a \times\sin \left( 90-a\right)[/tex]
[tex]\Longleftrightarrow x = a \times\cos \left( a\right)[/tex]
On the other hand,
[tex]\frac{y}{\sin \left( a\right) } =\frac{a}{\sin \left( 90\right) }=a[/tex]
Then
[tex]y = a \times \sin(a)[/tex]
In the 30-60-90 triangle below, side s has a length of
length of
30
90°
S
60
OA. 16√3, 5
B. 4√2, 4√2
OC. 4, 4√3
OD. 8.5, 16
OE. 16√3, 16√3
OF. 4, 8√3
and side q has a
Answer:
4, 4√3
Step-by-step explanation:
In a 30-60-90 triangle:
The leg opposite of the 30-degree angle is 1/2 of the hypotenuse: 8/2 = 4
The longer leg is √3 times the shorter leg: 4*√3 = 4√3
Check work: 4^2+(4√3)^2 = 8^2 --> 16+48 = 64 --> 64 = 64
The vertex of a parabola is (0,0)and the focus is (1/8, 0) . What is the equation of the parabola?
The equation of a parabola whose vertex is (0, 0) and focus is (1 / 8, 0) is equal to x = 2 · y².
How to derive the equation of the parabola from the locations of the vertex and focus
Herein we have the case of a parabola whose axis of symmetry is parallel to the x-axis. The standard form of the equation of this parabola is shown below:
(x - h) = [1 / (4 · p)] · (y - k)² (1)
Where:
(h, k) - Coordinates of the vertex.p - Distance from the vertex to the focus.The distance from the vertex to the focus is 1 / 8. If we know that the location of the vertex is (0, 0), then the standard form of the equation of the parabola is:
x = 2 · y² (1)
The equation of a parabola whose vertex is (0, 0) and focus is (1 / 8, 0) is equal to x = 2 · y².
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Susie enjoys spending her afternoons kayaking. One afternoon she was kayaking on a river 3 miles upstream, and 3 miles downstream in a total of 4 hours. In still water, Susie can Kayak at an average speed of 2 miles per hour. Based on this information, what a reasonable estimation of the current measured in miles per hour.
Using the relation between velocity, distance and time, it is found that a reasonable estimation of the current is of 1 mph.
What is the relation between velocity, distance and time?Velocity is distance divided by time, hence:
[tex]v = \frac{d}{t}[/tex]
Upstream, against the current, he traveled 3 miles in t hours, hence the equation is:
[tex]2 - c = \frac{3}{t}[/tex]
[tex]c = 2 - \frac{3}{t}[/tex]
Downstream, with the current, he traveled 3 miles in 4 - t hours, hence the equation is:
[tex]2 + c = \frac{3}{4 - t}[/tex]
Hence:
[tex]c = \frac{3}{4 - t} - 2[/tex]
Then, taking the two equal equations:
[tex]2 - \frac{3}{t} = \frac{3}{4 - t} - 2[/tex]
[tex]\frac{3}{4 - t} + \frac{3}{t} = 4[/tex]
[tex]\frac{3t + 12 - 3t}{t(4 - t)} = 4[/tex]
12 = -4t² + 16t
4t² - 16t + 12 = 0
t² - 4t + 3 = 0
(t - 3)(t - 1) = 0.
The current is positive, hence:
[tex]c = 2 - \frac{3}{3}[/tex]
c = 2 - 1
c = 1 mph.
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Given the graph of the function, f(x) , what is the value of f (-2)?
The value of f(-2) is 0
How to evaluate the function?The graph of the function is added as an attachment
The graph is given as:
y = f(x)
When x =-2. we have
y = f(-2)
From the graph, we have:
x = -2 and y = 0
So, we have
f(-2) = 0
Hence, the value of f(-2) is 0
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