An ellipse has vertices along the major axis at (0, 1) and (0, −9). The foci of the ellipse are located at (0, −1) and (0, −7). The equation of the ellipse is in the form below.
Answers
now, the center is half-way between the vertices, therefore it'd be at 0, -4, like in the picture in red.
the distance from the center to either foci, is "c", and that's c = 3.
the "a" component of the major axis is 5 units, now let's find the "b" component,
[tex]\bf \textit{ellipse, vertical major axis} \\\\ \cfrac{(x- h)^2}{ b^2}+\cfrac{(y- k)^2}{ a^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h, k\pm a)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{ a ^2- b ^2} \end{cases}\\\\ -------------------------------[/tex]
[tex]\bf \begin{cases} a=5\\ c=3 \end{cases}\implies c=\sqrt{a^2-b^2}\implies c^2=a^2-b^2\implies b^2=a^2-c^2 \\\\\\ b=\sqrt{a^2-c^2}\implies b=\sqrt{5^2-3^2}\implies b=4\\\\ -------------------------------\\\\ \begin{cases} h=0\\ k=-4\\ a=5\\ b=4 \end{cases}\implies \cfrac{(x- 0)^2}{ 4^2}+\cfrac{[y-(-4)]^2}{ 5^2}=1 \\\\\\ \cfrac{(x- 0)^2}{ 16}+\cfrac{(y+4)^2}{25}=1[/tex]
The equation of the ellipse is required.
The required equation is [tex]\dfrac{x^2}{16}+\dfrac{(y+4)^2}{25}=1[/tex]
It can be see that the major axis is parallel to the y axis.
The major axis points are
[tex](h,k+a)=(0,1)[/tex]
[tex](h,k-a)=(0,-9)[/tex]
[tex]k+a=1[/tex]
[tex]k-a=-9[/tex]
Subtracting the equations
[tex]2a=10\\\Rightarrow a=5[/tex]
The foci are
[tex](h,k+c)=(0,-1)[/tex]
[tex](h,k-c)=(0,-7)[/tex]
[tex]k+c=-1[/tex]
[tex]k-c=-7[/tex]
Subtracting the equations
[tex]2c=6\\\Rightarrow c=3[/tex]
[tex]k+c=-1\\\Rightarrow k=-1-c\\\Rightarrow k=-1-3\\\Rightarrow k=-4[/tex]
Foci is given by
[tex]c^2=a^2-b^2\\\Rightarrow b=\sqrt{a^2-c^2}\\\Rightarrow b=\sqrt{5^2-3^2}=4[/tex]
The equation is
[tex]\dfrac{(x-h)^2}{b^2}+\dfrac{(x-k)^2}{a^2}=1\\\Rightarrow \dfrac{(x-0)^2}{4^2}+\dfrac{(y+4)^2}{5^2}=1\\\Rightarrow \dfrac{x^2}{16}+\dfrac{(y+4)^2}{25}=1[/tex]
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Related Questions
What is the exact value of the trigonometric expression in simplest form? 2 cos(3π/4)−4 sin(7π/6)
Answers
What is the exact value of the trigonometric expression in simplest form? 2 cos(3π/4)−4 sin(7π/6)
[tex] 2 cos(\frac{3\pi}{4})-4sin(\frac{7\pi}{6}) [/tex]
Let us find the value of cos([tex] \frac{3\pi}{4} [/tex]) and [tex] sin(\frac{7\pi}{6} ) [/tex]
[tex] cos(\frac{3\pi}{4} )=cos(\pi -\frac{\pi}{4} ) [/tex]
The angle [tex] \pi -\frac{\pi}{4} [/tex] lies in second quadrant.
So, [tex] cos(\frac{3\pi}{4} )=-cos(\frac{\pi}{4} ) =-\frac{1}{\sqrt{2}} [/tex]
[tex] So, cos(\frac{3\pi}{4} )=-\frac{1}{\sqrt{2}} [/tex]
Now, Let us find the value of [tex] sin(\frac{7\pi}{6} ) [/tex]
[tex] sin(\frac{7\pi}{6} ) =sin(\pi +\frac{\pi}{6} )=-sin(\frac{\pi}{6} ) =\frac{-1}{2} [/tex]
[tex] sin(\frac{7\pi}{6} ) =\frac{-1}{2} [/tex]
[tex] 2 cos(\frac{3\pi}{4})-4sin(\frac{7\pi}{6}) [/tex]=2*[tex] \frac{-1}{\sqrt{2}} [/tex]-4*[tex] \frac{-1}{2} [/tex]
=[tex] \frac{-2}{\sqrt{2}} +\frac{4}{2} [/tex]
=[tex] \frac{-2\sqrt{2}}{2} +\frac{4}{2} [/tex]
=[tex] \frac{-1\sqrt{2}}{1} +\frac{2}{1} [/tex]
=[tex] -\sqrt{2} +2 [/tex]
=[tex] 2-\sqrt{2} [/tex]
[tex] 2 cos(\frac{3\pi}{4})-4sin(\frac{7\pi}{6})=2-\sqrt{2} [/tex]
Tell whether x and y show direct variation 5y=x
Answers
Direct variation is the relationship between variables such that increase in one variable causes an increase in the other variable and vicevarsa.
In this case, y is directly proportional to x where 5 is a constant
Jax solved an equation and justified his steps as shown below. What reason would justify step 2?
3x + 5 > 2 Step 1: Given
3x > -3 Step 2: ?
x> -1 Step 3: Division Property of Equality
Combine Like Terms
Subtraction Property of Equality
Division Property of Equality
Multiplication Property of Equality
Answers
i believe the answer is
Subtract Property of Equality
I'm not 100% sure but this is what i would pick
I will make up the BRAINLIEST!!!ls tell me the ......distinguish between expression and equation algebraically by providing an example of an algebraic expression and an example of algebraic equation.pls answer this I beg you !!!!
Answers
For example, 2+3 is a math expression
To make it an algebraic expression, we replace one of the numbers with a variable. Often this is the letter x. So let's say we replace the 2 with x. That would have the expression turn into x+3. What we're saying is "some number plus 3"
Normally 2+3 = 5 which is a true equation. Replace the 2 with x and we have x+3 = 5. So we're saying "some number plus 3 is 5". You can guess and check to find that the "some number" has to be 2
--------------------------------------
In summary, the expression example was x+3 and the equation was x+3 = 5
Note: an equation is simply two expressions set equal to each other.
which statement best describes the function below?
f(x)=2x^2-3x+1
A.It is a one-to-one function.
B.It fails the vertical line function.
C.It is a many-to-one function.
D.It is not a function.
Answers
f(x)=2x^2-3x+1
the graph of the function is attached with the answer
As we can see that equation is quadratic and its graph is a parabola facing upward.
As we can see it passes the Vertical line test
As it passes the Vertical line test , So it is a function
And two values of x has one value of y
So it is a many to one function.
A triangle has an area of 30 cm². The base and height are scaled by a factor of 4.
What is the area of the resulting triangle?
Enter your answer in the box.
Answers
b--------------> the base of triangle
h--------------> the height of triangle
Area A1=b*h/2--------------- > 30 cm²
if the base and height are scaled by a factor of 4
we have that
b=4b
h=4h
A2=(4b)*(4h)/2--------> 16(b*h/2)=16A1
A2=16A1---------> 16*30=480 cm²
the area of resulting triangle is 480 cm² (16 times the original area)
Answer:
480cm². Verified by correct test results.
Solve the inequality.
3 + |t + 4| < 10
A. t > 3 or t < –11
B. –3 < t < 11
C. –11 < t < 3
D. t < 3 and t > 11
Answers
Answer:
Just took the test Its C
A cylindrical container holding sugar has a height of 15 inches and a diameter of 4 inches. Which of the following is the closest to the volume of the sugar container?
A.
707 cubic inches
B.
94 cubic inches
C.
754 cubic inches
D.
188 cubic inches
Answers
say pi=3.14
h=15
d/2=r=4/2=2
so
v=(15)(3.14)(2)^2
v=(15)(3.14)(4)
v=(60)(3.14)
v=188.4
so answer is D
Using the formula V = πr²h, the volume of the cylinder is calculated as 188.4954 cubic inches, which can be rounded to 188 cubic inches, corresponding to answer choice D.
To calculate the volume of the cylindrical container holding sugar, we can use the formula for the volume of a cylinder, which is V = πr²h. Here, π is a constant (approximately 3.14159), r is the radius of the cylinder, and h is the height of the cylinder.
The diameter of the container is given as 4 inches, so the radius (r) is half of the diameter, which is 2 inches. The height (h) is given as 15 inches. Plugging these values into the volume formula, we get:
V = π(2 inches)²(15 inches)
= π(4 square inches)(15 inches)
= 60π cubic inches
Using 3.14159 as the value for π, the volume calculates to:
V = 60 * 3.14159 cubic inches = 188.4954 cubic inches
So, the closest estimate to the volume of the sugar container would be 188 cubic inches, which corresponds to answer choice D.
Four years ago, Tom’s age was 3/4 of the age he will be in 6 years. How old is he now?
Answers
Sue wakes up at 7:45
a.m. and jogs for 40 minutes. she then spends 45 minutes dressing for school before she catches the bus. at what time does sue get on the bus?
Answers
8:25 plus 45 mins = 9:10
She catches it at 9:10
The base of an isosceles triangle is 70 centimeters long. The altitude from the vertex angle to the base is 75 centimeters long. Find, to the nearest whole degree, the measure of the base angles of this triangle.
Answers
We get two similar right angled triangles ADC and BDC. For triangle ADC, AD = 35cm, DC = 75cm. Tangent of angle DAC can be written as:
[tex]tan(DAC)= \frac{CD}{DA} \\ \\ tan(DAC)= \frac{75}{35}= \frac{15}{7} [/tex]
Angle DAC is equal to tangent inverse of 15/7 which equals 65 degrees, rounded of to nearest whole degree.
Angle DBC = DAC = 65 degrees
Thus the base angles for the given isosceles triangle will be 65 degrees each, rounded of to nearest whole degree.
Write an numerical expression to show the product of the difference of 12 and 4 and the sum of 3 and 2.
Answers
Product ⇒ Multiplication
Difference ⇒ Subtraction
Sum ⇒ Addition
40 PTS if u do this correctly!
Algebra 1
Do #'s 4,5,6,7,8,9,10,12,13,14
Answers
n^2+4n=32
--------------------------
Subtract 32 from each side
n^2+4n-32=32-32
n^2+4n-32=0
--------------------------
Factor left side of equation
(n-4)(n+8)=0
--------------------------
Now set the factors to equal 0
n-4=0
or/and
n+8=0
--------------------------
n=4 and/or n=-8
====================================================================
5.)
t^2-26t=56
Subtract 56 from each side
t^2-26t-56=56-56
t^2-26t-56=0
----------------------------
Factor left side
(t+2)(t-28)=0
--------------------------
Set factors equal to 0
t+2=0
or/and
t-28=0
t = -2 and/or t = 28
===================================================================
As you may have noticed you need to subtract to get 0 on the right side, then you factor the left side and then switch the symbols from the numbers. Negative to Positive and Positive to Negative. I'll keep provided the work for the next 3 or 4 questions
===================================================================
6.)
z^2-14z=72
----------------
Subtract 72 from each side
z^2-14z-72=72-72
z^2-14z-72=0
----------------
Factor left side
(z+4)(z-18)=0
----------------
Set factors equal to 0
z+4=0 or z-18=0
Invert
z = -4 and z = 18
===================================================================
7.)
80+a^2=18a
-------------------------
Simplify
a^2+80=18a
------------------------
Subtract 18a from each side
a^2+80-18a=18a-18a
a^2-18a+80=0
------------------------
Factor left side
(a-8)(a-10)=0
-----------------------
Set factors to equal 0
a-8=0 or a-10=0
Invert
a = 8 and a = 10
===================================================================
8.)
u^2=16u+36
--------------------------
Subtract 16u+36 from both sides.
u^2-(16u+36)=16u+36-(16u+36)
u^2-16u-36=0
--------------------------
Factor left side
(u+2)(u-18)=0
--------------------------
Set factors to 0
u+2=0 or u-18=0
Invert
u = -2 and u = 18
====================================================================
9.)
3h^2+6h=105
--------------------------
Subtract 105 from each side
3h^2+6h-105=105-105
3h^2+6h-105=0
--------------------------
Factor left side
3(h-5)(h+7)=0
--------------------------
Set factors to equal 0
h-5=0 or h+7=0
h = 5 and h = -7
===================================================================
10.)
a^2+14a=-45
---------------------------
Subtract -45 from each side
a^2+14a-(-45)=-45-(-45)
a^2+14a+45=0
--------------------------
Factor left side
(a+5)(a+9)=0
-------------------------
Set factors to equal 0
a+5=0 or a+9=0
Invert
a = -5 and a = -9
===================================================================
12.)
w=5 and w=6
===================================================================
13.)
k=6 and k=-9
===================================================================
14.)
h=5 and h=12
Person 1 can do a certain job in fifteen minutes, and person 2 can do the same job in thirty minutes. When completing the job together, which expression would be used to represent the amount of work done by person 1?
A.1/15
B.x/15
C.15
pls asap
Answers
Let time taken by first person =x
So x = 15
Let time taken by next two persons be y
So 2y=30 or y=15
So x= y = 15
Time taken = 1/x=1/y=1/15
Answer is a
Which stem-and-leaf plot represents the data 80, 81, 91, 92, 66, 55, 54, 30, 55, 79, 78?
Answers
30,54,55,55,66,78,79,80,81,91,92
Here the minimum value is 30 and maximum is 92.
Total counts=11
Now plot the data.
Stem | Leaf
3 | 0
5 | 4 5 5
6 | 6
7 | 8 9
8 | 0 1
9 | 1 2
This is the required stem leaf plot.
Answer:The answer is below Hope this help and tthanjs for answer please like and vote excellent thanks and have a good day.
Step-by-step explanation
3 0
4
5 455
6 6
7 89
8 01 9 12What are the angle measures of the triangle 6 12 6sqrt3?
Answers
Answer:
The measure of angle A is 30 degree, measure of angle B is 90 degree and measure of angle C is 60 degree.
Step-by-step explanation:
The measures of given sides are 6, 12, [tex]6\sqrt{3}[/tex].
According to the Law of Cosine:
[tex]\cos A=\frac{b^2+c^2-a^2}{2bc}[/tex]
Using Law of Cosine, we get
[tex]\cos A=\frac{(12)^2+(6\sqrt{3})^2-(6)^2}{2(12)(6\sqrt{3})}[/tex]
[tex]\cos A=\frac{216}{144\sqrt{3}}[/tex]
[tex]\cos A=\frac{3}{2\sqrt{3}}[/tex]
[tex]\cos A=\frac{\sqrt{3}{2}[/tex]
[tex]A=30^{\circ}[/tex]
Similarly,
[tex]\cos B=\frac{a^2+c^2-b^2}{2ac}[/tex]
[tex]\cos B=\frac{(6)^2+(6\sqrt{3})^2-(12)^2}{2(6)(6\sqrt{3})}[/tex]
[tex]\cos B=\frac{0}{36\sqrt{3}}[/tex]
[tex]\cos B=0[/tex]
[tex]B=90^{\circ}[/tex]
Therefore, measure of angle A is 30 degree and measure of angle B is 90 degree.
According to angle sum property, the sum of interior angles of a triangle is 180 degree.
[tex]A+B+C=180^{\circ}[/tex]
[tex]30^{\circ}+90^{\circ}+C=180^{\circ}[/tex]
[tex]C=60^{\circ}[/tex]
Therefore the measure of angle C is 60 degree.
The triangle with sides 6, 12, and 6√3 has angles of 30 degrees, 60 degrees, and 90 degrees.
Given a triangle with sides 6, 12, and 6√3, we can find the angles using the Law of Cosines.
First, we use the formula for the Law of Cosines,
c² = a² + b² - 2ab * cos(C).
Let a = 6, b = 6√3, and c = 12.
We substitute these values into the formula to find angle C,
cos(C) = (6² + (6√3)² - 12²) / (2 * 6 * 6√3)
Compute the values inside the formula:
6² = 36
(6√3)² = 108
12² = 144
cos(C) = (36 + 108 - 144) / 72
cos(C) = 0
Therefore, angle C = 90 degrees.
To find the other angles, we use the Law of Sines,
(sin A / a) = (sin B / b) = (sin C / c).
We already know C = 90°, so (sin C / 12) = 1/12.
For angle A,
sin(A) / 6 = 1/12
sin(A) = 6/12 = 1/2
A = 30 degrees
For angle B:
B = 180° - 90° - 30° = 60 degrees
Therefore, the angles of the triangle are 30 degrees, 60 degrees, and 90 degrees.
8j -k+148j−k+148, j, minus, k, plus, 14 when j=0.25j=0.25j, equals, 0, point, 25 and k=1k=1
Answers
.. 8j -k +14
for (j, k) = (0.25, 1).
Put the numbers where the letters are. If you cannot do the math, let your calculator do it.
.. 8*0.25 -1 +14
.. = 2 -1 +14
.. = 1 +14
.. = 15
The value of the expression is 15 at (j, k) = (0.25, 1).
Answer:
15. It was right for me, not sure for anyone else
Step-by-step explanation:
Given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease. y=8700(1.04)^4
Answers
Using exponential function concepts, it is found that it represents growth of 4%.
Exponential function:An exponential function is modeled by:
[tex]y(t) = y(0)(1 + r)^t[/tex]
In which:
y(0) is the initial value.r is the rate of change. If it is positive, it represents growth, and if it is negative, it represents decay.In this problem, the equation is:
[tex]y = 8700(1.04)^t[/tex]
Hence:
[tex]1 + r = 1.04[/tex]
[tex]r = 0.04[/tex]
Since r = 0.04 is positive, it represents a growth of 4%.
To learn more about exponential function concepts, you can take a look at https://brainly.com/question/24757163
Final answer:
The function y = 8700(1.04)^4 represents an exponential growth with a rate of increase of 4% per time period over four periods.
Explanation:
The given function is y = 8700(1.04)^4. This represents an exponential growth because the base of the exponent, 1.04, is greater than 1. To determine the percentage rate of increase, we observe that the base, 1.04, when written as a percentage, is 104%. This means that the amount increases by 4% over each time periodperiod. Since the exponent is 4, this growth occurs over four time periods. However, the overall increase is not simply 4% multiplied by 4, as exponential growth compounds wicompoundsth each time period.
Classify the model as exponential growth or exponential decay. Identify the growth or decay factor AND the percent of increase or decrease per time period.
y=16(7/6)^t
Plz help!
Answers
The growth factor is 7/6.
The percentage increase in each time period is (7/6 -1)*100% ≈ 16.67%.
Let d represent the number of degrees in one angle of a regular polygon, and n represent the number of sides of the polygon to tessellate. which equation can be used to determine if the polygon will tessellate? 360° · n = d 360° · d = n 360° ÷ d = n n + d = 360° brainly
Answers
Answer:
360°/d = kn.
Step-by-step explanation:
To know if a regular polygon can tessellate, we divide 360 by the interior angle of the polygon and the result needs to be an integer, but not in all cases you get the number of sides of the polygon, you get a multiple of the number of sides. So, none of the options are correct, the correct would be
360/d = kn, where k is an integer.
the length of a rectangular prism is 4 feet, and the width is 3 feet. The volume of the prism is 9 cubic feet. In inches, what is the height of the prism?
A) 6 inches
B) 8 inches
C) 9 inches
D) 10 inches
Answers
9 = 4(3)(h)
9 = 12h
Now we cancel the 12 by dividing:
9/12 = 12h/12
3/4 = h
However, this is in feet. We must convert to inches:
3/4 (12) = 3/4(12/1) = 36/4 = 9 inches.
Answer:
C 9 inches
Step-by-step explanation:
Margo has a coin collection. She was given 1 coin as a gift, and she adds 3 coins per year. Which of the graphs below represents Margo's coin collection? (1 point) Select one: a. graph of line going through 1, 2 and 3, 4 b. graph of line going through 1, 1 and 3, 3 c. graph of line going through 1, 4 and 3, 10 d. graph of line going through 1, 2 and 2, 5
Answers
y=total amount of coins
x=year
y=3y+1
Hope this helps you
~True boss
option: c. graph of line going through (1, 4) and (3, 10)
Step-by-step explanation:It is given that:
Margo has a coin collection.
She was given 1 coin as a gift, and she adds 3 coins per year.
i.e. she started with a initial of one coin and add 3 coins each year.
That is this situation can be represented with the help of a function:
C(t) which represents the number of coins she has in 't' years.
Now,
C(0)=1
Also,
C(t)=3t+1-------------(A)
Hence, we will check each of the given functions whether at the value of t they passes through that point or not.
a)
it is given that:
Graph passes through (1,2) and (3,4)
i.e. when t=1 C(t)=2
and when t=3 C(t)=4
but on putting the value of t in equation (A) it does not hold true.
same could be done for option (b) and option (d).
But in option (c)
It hold true.
c. graph of line going through 1, 4 and 3, 10Since,
when t=1
C(t)=3×1+1=4
when t=3
C(t)=3×3+1=10
Hence, the correct answer is:
c.) graph of line going through 1, 4 and 3, 10
which pair of ratios form a true proportion
a) 1/2=6/10
b) 6/8=5/20
c) 3/5=6/10
d) 3/9=7/80
Answers
I'd say your answer would most likely be Option C
Because you can multiply the [tex] \frac{3}{5} [/tex] by [tex]2[/tex] (numerator and denominator by [tex]2[/tex] ) to get the fraction [tex] \frac{6}{10} [/tex]
[tex] \frac{3(2)}{5(2)} = 2(3) = 6 = 5(2)= 10 = \frac{6}{10} [/tex]
Good luck on your assignment and enjoy your day!
~[tex]LoveYourselfFirst:)[/tex]
Use the triangle pictured to calculate the following measurements.
Perimeter: p = ___ units Semiperimeter: s = ___units Area: A ≈ ___square
triangle is 8 by 12 by 10
Answers
The perimeter of the triangle is the sum of its sides.
We have then:
P = 8 + 12 + 10
P = 30 units
Semi-perimeter:
In geometry, the semiperimeter of a polygon is half its perimeter.
s = P / 2
s = 30/2
s = 15 units.
Area:
Knowing the semiperimeter and the sides, the area is:
A = root (s * (s-a) * (s-b) * (s-c))
where,
s: semi-meter
a, b, c are sides.
A = root (15 * (15-8) * (15-12) * (15-10))
A = 39.68626967
A = 40 units ^ 2
Answer:
Step-by-step explanation:
A baker used the equation y = kx to find how many cupcakes he needs to fill 22 identical boxes. He found that he would need 176 cupcakes. How many cupcakes fit into each box?
Answers
If BE is parallel to CD, the. The value of x is:
A. 10
B. 6
C. 9
Answers
x/15 = 12/(12+8) = 3/5
x = 15*3/5
x = 9
Given that f(x) = x2 − 3x + 3 and g(x) = the quantity of x minus one, over four , solve for f(g(x)) when x = 5. (1 point)
Answers
f (x) = x ^ 2 - 3x + 3
g (x) = ((x-1) / (4))
We now make the composition of functions:
f (g (x)) = ((x-1) / (4)) ^ 2 - 3 (((x-1) / (4))) + 3
We now evaluate for x = 5:
f (g (5)) = ((5-1) / (4)) ^ 2 - 3 (((5-1) / (4))) + 3
f (g (5)) = 1
Answer:
f (g (5)) = 1
Answer:
The answer is 1
Step-by-step explanation:
PLEASE HELP!!
is it right??
Answers
△BQH∼△SFN
What is FN ?
Answers
Or 16.81 or 16.8 simplified
Answer:
23.4 in
Step-by-step explanation:
The only side of SFN we know the measurement of is FS, which is 18.2. We see from the similarity statement that FS corresponds to QB, which is 14. This gives us the ratio
14/18.2.
From the similarity statement, we see that FN corresponds to QH; this gives us the ratio
18/x
Together this gives us the proportion
14/18.2 = 18/x
Cross multiplying gives us
14(x) = 18.2(18)
14x = 327.6
Divide each side by 14:
14x/14 = 327.6/14
x = 23.4
What is the relative minimum of the function?
Answers
relative minimum is -2 when x is -3