Answer:[tex]\Large\boxed{(2+\sqrt{2},~2-\sqrt{x} )~~and~~ (2-\sqrt{2},~2+\sqrt{x} )}[/tex]
Step-by-step explanation:
Given the system of equations
[tex]1)~xy=2[/tex]
[tex]2)~x+y=4[/tex]
Divide x on both sides of the 1) equation
[tex]xy=2[/tex]
[tex]xy\div x=2\div x[/tex]
[tex]y=\dfrac{2}{x}[/tex]
Current system
[tex]1)~y=\dfrac{2}{x}[/tex]
[tex]2)~x+y=4[/tex]
Substitute the 1) equation into the 2) equation
[tex]x+(\dfrac{2}{x} )=4[/tex]
Multiply x on both sides
[tex]x\times x+\dfrac{2}{x}\times x=4\times x[/tex]
[tex]x^2+2=4x[/tex]
Subtract 4x on both sides
[tex]x^2+2-4x=4x-4x[/tex]
[tex]x^2-4x+2=0[/tex]
Use the quadratic formula to solve for the x value
[tex]x=\dfrac{-(-4)\pm\sqrt{(-4)^2-4(1)(2)} }{2(1)}[/tex]
[tex]x=2\pm\sqrt{2}[/tex]
Substitute the x value into one of the equations to find the y value
[tex]xy=2[/tex]
[tex](2+\sqrt{2} )y=2[/tex]
[tex]y=2-\sqrt{2}[/tex]
[tex]OR[/tex]
[tex]xy=2[/tex]
[tex](2-\sqrt{2} )y=2[/tex]
[tex]y=2+\sqrt{2}[/tex]
Therefore, the points of intersection are
[tex]\Large\boxed{(2+\sqrt{2},~2-\sqrt{x} )~~and~~ (2-\sqrt{2},~2+\sqrt{x} )}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Answer:
(2 - [tex]\sqrt{2}[/tex] , 2 + [tex]\sqrt{2}[/tex] ) and (2 + [tex]\sqrt{2}[/tex], 2 - [tex]\sqrt{2}[/tex] )
Step-by-step explanation:
xy = 2 → (1)
x + y = 4 ( subtract x from both sides )
y = 4 - x → (2)
substitute y = 4 - x into (1)
x(4 - x) = 2
4x - x² = 2 ( multiply through by - 1 )
x² - 4x = - 2
using the method of completing the square
add ( half the coefficient of the x- term)² to both sides
x² + 2(- 2)x + 4 = - 2 + 4
(x - 2)² = 2 ( take square root of both sides )
x - 2 = ± [tex]\sqrt{2}[/tex] ( add 2 to both sides )
x = 2 ± [tex]\sqrt{2}[/tex] , that is
x = 2 - [tex]\sqrt{2}[/tex] , x = 2 + [tex]\sqrt{2}[/tex]
substitute these values of x into (2) for corresponding values of y
x = 2 - [tex]\sqrt{2}[/tex] , then
y = 4 - (2 - [tex]\sqrt{2}[/tex])
= 4 - 2 + [tex]\sqrt{2}[/tex]
= 2 + [tex]\sqrt{2}[/tex] ⇒ (2 - [tex]\sqrt{2}[/tex] , 2 + [tex]\sqrt{2}[/tex] ) ← 1 point of intersection
x = 2 + [tex]\sqrt{2}[/tex] , then
y = 4 - (2 + [tex]\sqrt{2}[/tex] )
= 4 - 2 - [tex]\sqrt{2}[/tex]
= 2 - [tex]\sqrt{2}[/tex] ⇒ (2 + [tex]\sqrt{2}[/tex] , 2 - [tex]\sqrt{2}[/tex] ) ← 2nd point of intersection
……hellppppppppp nowwwe
Answer:
18 miles
Step-by-step explanation:
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What is the answer to this equation?
Answer:W ^10
Step-by-step explanation:
The solution is in the
PLEASE I NEED THIS FAST a three dight number has one more ten than it has hundreds, and it also has one more than twice as many units as tens the sum of the number and that number reversed is 31 less than 10 cubed find the reverse number
The reverse number of the three-digit number is 732
How to determine the reverse of the number?Let the three-digit number be xyz.
So, the reverse is zyx
This means that
Number = 100x + 10y + z
Reverse = 100z + 10y + x
From the question, we have the following parameters:
y = x + 1
z = 1 + 2y
The sum is represented as:
100x + 10y + z + 100z + 10y + x = 10^3 - 31
100x + 10y + z + 100z + 10y + x = 969
Evaluate the like terms
101x + 101z + 20y = 969
Substitute y = x + 1
101x + 101z + 20(x + 1) = 969
101x + 101z + 20x + 20 = 969
Evaluate the like terms
101x + 101z + 20x = 949
121x + 101z = 949
Substitute y = x + 1 in z = 1 + 2y
z = 1 + 2(x + 1)
This gives
z = 2x + 3
So, we have:
121x + 101z = 949
121x + 101* (2x + 3) = 949
This gives
121x + 202x + 303 = 949
Evaluate the sum
323x = 646
Divide by 323
x = 2
Substitute x = 2 in z = 2x + 3 and y = x + 1
z = 2*2 + 3 = 7
y = 2 + 1 = 3
So, we have
x = 2
y = 3
z = 7
Recall that
Reverse = 100z + 10y + x
This gives
Reverse = 100*7 + 10*3 + 2
Evaluate
Reverse = 732
Hence, the reverse number of the three-digit number is 732
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The following diagrams show six lines with equations of the form y =mx +c
I have the answers but I do not know how to FIND the right one. Please help!
Answer:
understand the meaning of m > 0 and c > 0.
Step-by-step explanation:
The slope-intercept form of the equation of a line tells you its slope and its y-intercept.
y = mx +c . . . . line with slope m and y-intercept c
Slope (m)The slope is the rate of increase moving from left to right. It is positive (m>0) when the line slants upward to the right. It is negative (m<0) when the line slants down to the right.
m > 0: lines 3, 5, 6m < 0: lines 1, 2, 4Y-intercept (c)The y-intercept is where the line crosses the (vertical) y-axis. It is positive (c>0) when the line crosses above the x-axis, where y-values are positive. It is negative (c<0) when the line crosses below the x-axis. If the line passes through the origin, then c=0.
c > 0: lines 4, 5c < 0: lines 1, 3c = 0: lines 2, 6What is the length of the hypotenuse of a 45°-45°-90° triangle with legs that are 8 cm long?
Answer:
Step-by-step explanation:
Use your 45-45-90 special triangle with side lengths 1, 1, sqrt(2).
The given triangle has legs length 8 (given) which is 8 times the size of our special 1-1-sqrt(2) triangle.
Therefore the length of the hypotenuse is 8 times the length of the hypotenuse in our special triangle
= 8 sqrt(2)
6. A cube of side 10 cm is melted down
and made into ten identical spheres.
Calculate the surface area of one of the
spheres.
The surface area of the Spheres is 104.19 cm².
According to the statement
we have given that the
there is a cube with side 10 cm and convert into ten spheres and we have to find the surface area of a sphere.
So,
we know that the cube has a 6 sides, each one 10 x 10 cm
So,
A cube with sides of 10 cm will have a surface area of 600 cm².
And The volume of the cube will be 1,000 cm³ ( = 10 x 10 x 10).
If you melt it down to form ten identical spheres, each of them will have a volume of:
1,000 cm³ / 10 = 100 cm³
And the volume of a sphere is:
[tex]V = 4/3 (pi)r ^{2}[/tex]
So you can calculate the radius of the spheres:
r³ = 3V / 4pi = (300) / (12.5664) = 23.8732 cm³
r = 2.8794 cm
Now, you can use the radius to find the surface area of the spheres:
As = 4pi*r² = (12.5664)(2.8794)^2 = 104.19 cm²
So, The surface area of the Spheres is 104.19 cm².
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A Big Year
By Bob Kowalski
Would you go to the ends of the earth to see a bird? What if it were a really special bird: one with beautiful feathers, an entrancing call, or a silly dance? What if seeing that one special bird would allow you to win a contest?
If that contest doesn't get you on television or win you any cash prizes, would you still do it? For those who participate in the "Big Year," the honor of beating the previous record is the only reward they get or even want.
A "Big Year" is a year in which a person attempts to see as many different species of birds as possible within a particular region. For most in North America who participate in a "Big Year," this region is the lower 48 American states, plus Alaska, Canada, and a couple of French islands off the Canadian coast.
You may be thinking that looking at birds is silly, but just think about the numbers of the recent record holders and the commitment it takes to get these numbers. One recent "Big Year" winner managed to see 744 birds in one year, missing the record by just one bird. Big Year birders travel by train, plane, boat, car, bicycle, and of course, by foot. They can cover over 150 thousand miles to get numbers of sightings this high. They can also spend a small fortune.
Just to clarify, the birds these contestants are counting are the number that they see in a particular year. You see, the contest is based on an honor system. No pictures or other evidence is required as proof of a sighting. Most birders take great pride in their reputation and their abilities to see or hear and then identify a bird. Usually, important sightings of the rare birds needed to get counts in the 700s are visited by hundreds of birders. It is pretty hard to cheat your way to a record-breaking year, but in general, few are interested in cheating.
This honesty comes from the fact that most people who want to break such a record know the greatest rewards are not necessarily in winning. Such rewards are in being able to commit a year of your life to doing something you love. Rewards are found in seeing amazing, inspiring creatures like the California Condor or the Magnificent Frigate bird. Rewards also come in spending time with people who, like you, want to spend their time looking to the skies and trees for glimpses of emerald, crimson, or cerulean blue feathered jewels.
You don't have to be able to travel a continent to have a big birding experience though. Have a big month. Or a big weekend. Set a personal record, learn to identify the species that live in your part of the world, or try to learn the calls of just two species of birds. You will soon find looking at birds isn't such a strange way to spend your time.
Extra! Extra! Backyard Birding
Many schools, families, and young birders across the country participate in the "Great Backyard Bird Count." While not as long as a "Big Year," the "Great Backyard Bird Count" happens every year. It depends on birders and families across the country to watch feeders and other areas in their yards and count the number of birds they see. Unlike the "Big Year," the goal is not to see who can count the most birds. Instead, participants in this event work together to help bird experts get a good idea of how birds are doing. Participants are given checklists and enter their sightings on a website. Called a "citizen-science" project, this event is open to anyone, requires no travel, and happens every year over one weekend in February.
Read this sentence from the article:
You may be thinking that looking at birds is silly, but just think about the numbers of the recent record holders and the commitment it takes to get these numbers.
Why would the author suggest that looking at birds is silly?
To create a sense of doubt in the reader
To distance himself from what seem to be rather strange people
To show readers that some people do not think birding is interesting
To suggest he has never birded himself but might someday
Ill give brainliest
The inference is that the author suggest that looking at birds is silly A. to create a sense of doubt in the reader.
What is an inference?It should be noted that that an inference simply means the conclusion that can be deduced based on the information given in a story.
In this case, A Big Year" is a year in which a person attempts to see as many different species of birds as possible within a particular region. For most in North America who participate in a "Big Year," this region is the lower 48 American states, plus Alaska, Canada, and a couple of French islands off the Canadian coast.
The inference is that the author suggest that looking at birds is silly is to create a sense of doubt in the reader. Therefore, the correct option is A.
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Use synthetic division to determine which of the following is a factor of [tex]2x^3-3x^2-5x+6\\[/tex]
Option A. x + 6
Option B. x - 2
Option C. x - 3
Option D. x + 2
The factor of the polynomial function 2x^3 - 3x^2 - 5x + 6 is (b) x -2
How to determine the factor using the synthetic division?The polynomial is given as:
2x^3 - 3x^2 - 5x + 6
Next, we test the factors
Option A. x + 6
Set the factor to 0
x + 6 = 0
Solve for x
x = -6
So, we set up the division as follows:
-6 l 2 - 3 -5 6
Bring down the leading coefficient
-6 l 2 - 3 -5 6
2
Multiply 2 and -6
-6 l 2 - 3 -5 6
-12
2
Add -3 and -12
-6 l 2 - 3 -5 6
-12
2 -15
Repeat this process
-6 l 2 - 3 -5 6
-12 72 -402
2 -15 67 -396
-396 is the remainder of the above division.
This means that x + 6 is not a factor of the polynomial
Option B. x - 2
Set the factor to 0
x - 2 = 0
Solve for x
x = 2
So, we set up the division as follows:
2 l 2 - 3 -5 6
Bring down the leading coefficient
2 l 2 - 3 -5 6
2
Multiply 2 and 2
2 l 2 - 3 -5 6
4
2
Add -3 and -12
2 l 2 - 3 -5 6
4
2 1
Repeat this process
2 l 2 - 3 -5 6
4 2 -6
2 1 -3 0
0 is the remainder of the above division.
This means that x - 2 is a factor of the polynomial
There is no need to check for the remaining options
Hence, the factor of the polynomial function 2x^3 - 3x^2 - 5x + 6 is (b) x -2
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what is the value of x when the richter scale rating scale is 8.4?
The value of x when the richter scale rating scale is 8.4 is 251188643.151
What is a richter scale?A richter scale is a numerical scale that is used for expressing the magnitude of an earthquake in a geographical area
How the richter scale works?The richter scale takes the amplitude of the earthquake's largest seismic as an input and calculate (and display) the magnitude of an earthquake in the geographical area using a logarithmic function
How to determine the value of x?The formula of a richter scale is expressed as:
log(x) = n
Where n represents the scale rating of the scale
In this case, n = 8.4.
So, we have:
log(x) = 8.4
Express as an exponent
x = 10^8.4
Evaluate the exponent
x = 251188643.151
Hence, the value of x when the richter scale rating scale is 8.4 is 251188643.151
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Justin recently drove to visit his parents who live 270
miles away. On his way there his average speed was 11 miles per hour faster than on his way home (he ran into some bad weather). If Justin spent a total of 9 hours driving, find the two rates.
Answer:
66 mph to visit55 mph to homeStep-by-step explanation:
An equation can be set up and solved based on the relation between the two speeds and the relation between time, speed, and distance.
Setuptime = distance/speed
Let x represent the (slower) speed on the way home. Then the total time for the round trip was ...
time going + time coming home = total time
270/(x +11) +270/x = 9
Solution30x +30(x +11) = x(x +11) . . . . . . multiply by x(x+11)/9
x^2 -49x -330 = 0 . . . . . . . . rewrite in standard form
(x -55)(x +6) = 0 . . . . . . . . factor
x = 55 or x = -6 . . . . . . . solutions to this equation; x < 0 is extraneous
Justin's rate on the way there was 66 mph; on the way home, it was 55 mph.
points a,b, and c make the triangle ABC and are at the coordinates A(-2,9), B(-33,13) and c(-21,25) point D is the midpoint of BC and AD is a median of ABC, the equation of the median can be given by ax+by=c where A B and C
what is most simple way to answer this
Answer:
[tex]2x+5y=41[/tex]
Step-by-step explanation:
Median of a triangle: A line segment that connects a vertex of a triangle to the midpoint of the opposite side.
Vertex: The point where any two sides of a triangle meet.
Given vertices of a triangle:
A = (-2, 9)B = (-33, 13)C = (-21, 25)Step 1Find the midpoint of BC (Point D) by using the Midpoint formula.
Midpoint between two points
[tex]\textsf{Midpoint}=\left(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\right)\quad \textsf{where}\:(x_1,y_1)\:\textsf{and}\:(x_2,y_2)\:\textsf{are the endpoints}}\right)[/tex]
Define the endpoints:
[tex]\text{Let }(x_1,y_1)=\sf B=(-33,13)[/tex][tex]\text{Let }(x_2,y_2)=\sf C=(-21,25)[/tex]Substitute the defined endpoints into the formula:
[tex]\textsf{Midpoint of BC}=\left(\dfrac{-21-33}{2},\dfrac{25+13}{2}\right)=(-27,19)[/tex]
Therefore, D = (-27, 19).
Step 2Find the slope of the median (line AD) using the Slope formula.
Define the points:
[tex]\textsf{let}\:(x_1,y_1)=\sf A=(-2,9)[/tex][tex]\textsf{let}\:(x_2,y_2)=\sf D=(-27,19)[/tex]Substitute the defined points into the Slope formula:
[tex]\implies \textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{19-9}{-27-(-2)}=-\dfrac{2}{5}[/tex]
Therefore, the slope of the median is -²/₅.
Step 3Substitute the found slope and one of the points into the Point-slope formula to create an equation for the median.
[tex]\implies y-y_1=m(x-x_1)[/tex]
[tex]\implies y-9=-\dfrac{2}{5}(x-(-2))[/tex]
Simplify and rearrange the equation so it is in standard form Ax+By=C:
[tex]\implies 5(y-9)=-2(x+2)[/tex]
[tex]\implies 5y-45=-2x-4[/tex]
[tex]\implies 2x+5y-45=-4[/tex]
[tex]\implies 2x+5y=41[/tex]
ConclusionTherefore, the equation of the median is:
2x + 5y = 41
Consider a circle whose equation is x2 + y2 – 2x – 8 = 0. Which statements are true? Select three options. The radius of the circle is 3 units. The center of the circle lies on the x-axis. The center of the circle lies on the y-axis. The standard form of the equation is (x – 1)² + y² = 3. The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
The statements which are correct about the equation of circle [tex]x^{2} +y^{2}-2x-8=0[/tex] are 1) The radius of the circle is 3 units,2) The center of the circle lies on the x axis,5) The radius of this circle is the same the radius of the circle whose equation is [tex]x^{2} +y^{2} =9[/tex].
Given the equation of circle be [tex]x^{2} +y^{2}-2x-8=0[/tex].
We are required to find the appropriate statements related to the equation [tex]x^{2} +y^{2}-2x-8=0[/tex].
[tex]x^{2} +y^{2}-2x-8=0[/tex] can be written as under:
[tex]x^{2} +y^{2} -2x+1-9=0[/tex]
[tex]x^{2} +1^{2}-2x+y^{2} -9=0[/tex]
[tex](x-1)^{2}+y^{2}[/tex]-9=0
[tex](x-1)^{2} +y^{2} =9[/tex]
[tex](x-1)^{2}+y^{2} =3^{2}[/tex]
Equation of a circle usually in the form [tex]x^{2} +y^{2} =a^{2}[/tex] in which a is radius.
From the comparison of both the equations we get that radius is 3 units.
From the equation point will be (1,0). It is on the x axis.
Hence the statements which are correct about the equation of circle [tex]x^{2} +y^{2}-2x-8=0[/tex] are 1) The radius of the circle is 3 units,2) The center of the circle lies on the x axis,5) The radius of this circle is the same the radius of the circle whose equation is [tex]x^{2} +y^{2} =9[/tex].
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Answer: 1,2,5 on edge
Step-by-step explanation:
see above ans for work
find the square (4x-6y^3)^2
domain of f(x)=(1/4)^x
What is the domain of f(x)
O A. x>0
OB. All real numbers
O C. y>0
O D. x<0
? Need help asap
Answer: B. All real numbers
Step-by-step explanation:
See attached image.
Patty pays $235 in advance on her account at the athletic club. Each time she uses the club, $6 is deducted from the account. Write an equation that represents the value remaining in her account after x visits to the club. Find the value remaining in the account after 5 visits.
a. V=235 - 6x; $1415
b. V= 6 - 235x; $205
c. V=235 - 6x; $1401
d. V=235 - 6x; $205
*show your work*
Answer:
205
Step-by-step explanation:
We start with 235, this will be our y-intercept.
6 Is deducted from our value every visit, making our slope -6.
Put the two into slope intercept form:
[tex]y=235-6x[/tex]
To find the value after 5 visits substitute x for 5:
[tex]y=235-6(5)[/tex]
Solve
[tex]y=235-6(5)\\y=235-30\\y=205[/tex]
Therefore you have 205 left in your account after 5 visits.
What is the probability that the top-three finishers in the contest will
all be seniors?
Type in the correct answer in each box. Use numerals instead of
words. If necessary, round your answers to the nearest tenth.
Answer: Just guessing, try.. or dont.
Step-by-step explanation:
There are 3 seniors. To make up 100% we would have to do 33.33%. I would say that the probabilty is around 33.33, and don't know the different orders possible.
Factorise: a8 - b8
please solve this
Answer:
(a - b)(a + b)(a² + b²)([tex]a^{4}[/tex] + [tex]b^{4}[/tex] )
Step-by-step explanation:
by repeated use of the difference of squares factorising method, that is
a² - b² = (a - b)(a + b)
given
[tex]a^{8}[/tex] - [tex]b^{8}[/tex]
= ([tex]a^{4}[/tex] )² - ([tex]b^{4}[/tex] )²
= ([tex]a^{4}[/tex] - [tex]b^{4}[/tex] )([tex]a^{4}[/tex] + [tex]b^{4}[/tex] ) ← factor left parenthesis using difference of squares
= ( a² - b²)(a² + b²)([tex]a^{4}[/tex] + [tex]b^{4}[/tex] ) ← factor left parenthesis by difference of squares
= (a - b)(a + b)(a² + b² )([tex]a^{4}[/tex] + [tex]b^{4}[/tex] )
A binomial experiment is given. Decide whether you can use the normal distribution to approximate the binomial distribution. If you can, find the mean and standard
deviation. If you cannot, explain why.
A survey of adults found that 66% have used a multivitamin in the past 12 months. You randomly select 40 adults and ask them if they have used a multivitamin in the
past 12 months.
Select the correct answer below and, if necessary, fill in the answer boxes within your choice.
OA. No, because np < 5.
B. No, because nq <5.
OC. Yes, the mean is_______
(Round to two decimal places as needed.)
and the standard deviation is _____
Yes, the mean is 26.4 and the standard deviation is 3
How to find the mean in binomial experiment?We are given;
n = 40, p = 66% = 0.66, and q = 1 – 0.66 = 0.34
If np ≥ 5 and np ≥ 5, then the binomial random variable, x is approximately normally distributed, then;
Mean; µ = np
Standard deviation; σ = √npq
where;
n is the sample size.
p is the population proportion.
q = 1 – p.
Calculating np and nq gives;
np = (40)(0.66) = 26.4
nq = (40)(0.34) = 13.6
Both np and nq are greater than 5, the normal distribution can be used to approximate the binomial distribution.
Thus, µ = np
µ = 40(0.66) = 26.4
Standard deviation;
σ = √(40 * 0.66 * 0.34)
σ = 3
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NO LINKS!! What is the equation shown in the smaller circle shown in the figure?
[tex] {\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
Here we go ~
The general equation of ellipse is :
[tex]\qquad \sf \dashrightarrow \: \cfrac{ {x}^{2} }{ {a}^{2} } + \cfrac{ {y}^{2} }{ {b}^{2} } = 1[/tex]
[ because it's a horizontal ellipse with centre at origin ]
so, let's equate given equation with the standard equation ~
[tex]\qquad \sf \dashrightarrow \: \cfrac{ {x}^{2} }{64} + \cfrac{ {y}^{2} }{36} = 1[/tex]
se get :
a² = 64 ; a = 8b² = 36 ; b = 6As we know, length of minor axis is : 2b = 2 × 6 = 12 units
and, the smaller circle has diameter = 2b = 12 units
so, it's radius = 6 units ~
Now, let's write the equation of circle with origin as centre and radius = 6 units
[tex]\qquad \sf \dashrightarrow \: {(x - h)}^{2} + (y - k) {}^{2} = {r}^{2} [/tex]
[ h = 0, k = 0, since circle has centre at origin ]
[tex]\qquad \sf \dashrightarrow \: {x}^{2} + {y}^{2} = 36[/tex]
Answer:
[tex]x^2+y^2=36[/tex]
Step-by-step explanation:
Equation of the red eclipse (taken from your previously posted question):
[tex]\dfrac{x^2}{64}+\dfrac{y^2}{36}=1[/tex]
General equation of an ellipse
[tex]\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1[/tex]
where:
(h, k) is the center[tex](h \pm a,k)\: \textsf{and}\:(h, k \pm b)\: \textsf{are the vertices}[/tex]As the center of the red ellipse is (0, 0) ⇒ h = 0, k = 0.
[tex]\implies a^2=64 \implies a=\sqrt{64}=\pm8[/tex]
[tex]\implies b^2=36 \implies a=\sqrt{36}=\pm6[/tex]
The Major Axis is the longest diameter of an ellipse.
Therefore, using the formula for the vertices, the vertices of the major axis of the red ellipse are (8, 0) and (-8, 0).
The Minor Axis is the shortest diameter of an ellipse.
Therefore, using the formula for the vertices, the vertices of the minor axis of the red ellipse (0, 6) and (0, -6).
Therefore, the radius of the larger circle is 8 units and the radius of the smaller circle is 6 units (half the respective axis, since the radius of a circle is half its diameter).
Equation of a circle
[tex](x-a)^2+(y-b)^2=r^2[/tex]
(where (a, b) is the center and r is the radius)
Given:
center = (0, 0)radius = 6 unitsThe equation of the smaller circle is:
[tex]\implies (x-0)^2+(y-0)^2=6^2[/tex]
[tex]\implies x^2+y^2=36[/tex]
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A cone is 10cm high and has a base radius of 8cm.
Find the radius and height of a cylinder that is inscribed in the cone such that the volume of the cylinder is a maximum. Find the maximum volume of the cylinder and leave the answer in exact form.
The maximum volume of the cylinder in exact form is; V_max = 2560π/27
How to maximize the volume of a cylinder?Let us define the variables as:
Radius of cone = r
Radius of cylinder = x
Height of Cone = h
Height of Cylinder = y
The general equation for volume of the cylinder is;
V = πx²y
Taking a strip of both figures and relating x, y, r & h as well as using ratios of similar triangles, we have:
Height of triangle above cylinder/Base of Triangle above cylinder = Height of full triangle/Base of full triangle
This gives;
(h - y)/2x = h/2r
Making y the subject gives us;
h – y = hx/r
y = h – hx/r
y = h(1 – x/r)
Plug in y into our Volume equation:
V(x) = πx²h(1 – x/r)
V(x) = πh(x² - x³/r)
To get maximum volume, find the derivative of the volume and solve for when the derivative equals zero:
V'(x) = πh(2x - 3x²/r)
V'(x) = 0
Thus;
(2x - 3x²/r) = 0
x( 2r – 3x) = 0
Thus;
x = 0 or x = 2r/3
Put x = 2r/3 in our volume equation to find V_max
V_max = πh((2r/3)² – (2r/3)³/r)
V_max = πh(4r²/9 - 8r²/27)
V_max = πh(12r²/27 - 8r²/27)
V_max = 4πhr²/27
Thus, at r = 8 and h = 10, we have;
V_max = 4π(10)8²/27
V_max = 2560π/27
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The table below shows the growth of algae cells within the Chesapeake Bay. Which of the following functions models the concentration, C(d), of algae cells per milliliter in d days?
The exponential function that represent the concentration of algae cells per milliliter in d days is [tex]C(d) = 10(2)^d[/tex]
What is an equation?An equation is an expression that shows the relationship between two numbers and variables.
The standard form of an exponential funtion is in the form:
y = abˣ
Where a is the initial value and b is the multiplication factor.
Let C(d) represent the concentration of algae cells per milliliter in d days
At point (1, 20)
20 = ab (1)
At point (2, 40)
40 = ab² (2)
From equations 1 and 2:
b = 2, a = 10
The exponential function that represent the concentration of algae cells per milliliter in d days is [tex]C(d) = 10(2)^d[/tex]
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The human eye contains approximately 9.1 x 10^7 rod cells and 4.5 x 10^6 cone cells . Samuel subtracts 9.1-4.5 and says there are about 46,000,000 more rod cells than cone cells in the human eye . Use the drop - down menus to explain why Samuel is incorrect
Samuel is incorrect because he didn't subtract the power of the numbers.
How many more rod cells are there?The number of rod and cone cells in the human eye is written in scientific notation. Scientific notation is used in expressing large numbers in smaller numbers. To write a number in scientific notation, the number is written as a decimal number, between 1 and 10 and multiplied by a power of 10. For example, 1 x 10² is equivalent to 100
(9.1 x 10^7) - (4.5 x 10^6)
= (9.1 - 4.5) x 10^(7 - 6)
4.6 x 10 = 46
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HOW MANY DIFFERENT ARRANGEMENTS CAN BE MADE WITH THE NUMBERS
28535852
Answer:
1,680
Step-by-step explanation:
8 positions with basically 8 choices.
that is 8! arrangements.
but 2 is there 2 times.
8 is there 2 times.
5 is there 3 times.
only 3 is a single digit.
so, we need to eliminate every arrangement, where the 2s trade places, where the 5s trade places, and where the 8s trade places, because they are the same numbers.
that means we have to divide the 8! by 2!, then again by 2!, and again by 3!.
8! / (2! × 2! × 3!) = 8! / 24 = 1,680
HELP!!!! Which of the following statements must be true to prove lines m and n are parallel? Question 19 options: A) ∠1 ≅ ∠6 B) m∠3 + m∠5 = 180° C) m∠2 + m∠6 = 180° D) ∠4 ≅ ∠8
Answer: B) m∠3 + m∠5 = 180°
Step-by-step explanation:
Concept:
For this question, we will be looking at each answer choice and eliminating the incorrect ones
A) ∠1 ≅ ∠6
∠1 is an exterior angle
∠6 is an interior angle
They are in an alternative position
Since they are neither both exterior nor interior, they are not congruent
[tex]\large\boxed{FALSE}[/tex]
B) m∠3 + m∠5 = 180°
∠3 is an interior angle
∠5 is an interior angle
They are in a same-side position
Since they are both interiors, they fulfill the same-side interior angle theorem, which states: that if a transversal cuts two parallel lines, then the interior angles on the same side of the transversal are supplementary, which means their sum adds up to 180°.
[tex]\Huge\boxed{TRUE}[/tex]
C) m∠2 + m∠6 = 180°
∠2 is an exterior angle
∠6 is an interior angle
They are in a same-side position
Since they are on the same side, they fulfill the corresponding angle theorem which states: the angles that occupy the same relative position at each intersection are congruent to each other.
However, they are only congruent, they don't add up to 180°
[tex]\large\boxed{FALSE}[/tex]
D) ∠4 ≅ ∠8
∠4 is an interior angle
∠8 is an exterior angle
They are in an alternative position
Since they are neither both exterior nor interior, they are not congruent
[tex]\large\boxed{FALSE}[/tex]
Hope this helps!! :)
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Graph the solution set of the system of inequalities. 2x - 6y≤ 12 4x-2y > 8 Use the graphing tool to graph the system. Click to enlarge graph.
The graph is shown in the attached image.
THE NICHOLS ARE BUYING A HOUSE SELLING FOR $245,000. THEY PAY A DOWN
PAYMENT OF $45,000 FROM THE SALE OF THEIR CURRENT HOUSE. TO OBTAIN A 15-
YEAR MORTGAGE AT 4.5% INTEREST, THEY MUST PAY 1.5 POINTS AT THE TIME OF
CLOSING. WHAT IS THE AMOUNT OF THE MORTGAGE, AND WHAT IS THE COST OF
THE 1.5 POINTS
The amount of the mortgage on this house is given as $300000, while the 1.5 points on the house is given as 3000 dollars
How to solve for the mortgage that is on this houseThe data from the questions says that the cost of the house = $245000
The down payment amount amount is 45000
Given that they already paid 45000 from the cost of the house, the mortgage would be 245000 - 45000
= $200000
The cost of 1.5 points is the same as the cost of 1.5% of the mortgage of this house.
This is calculated as 0.015 x 200000
= $3000
The conclusion is that the amount of the mortgage on this house is given as $300000, while the 1.5 points on the house is given as 3000 dollars
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The sum of the two rational equations is equal to (3 · n² + 5 · n - 10) / (3 · n³ - 6 · n²).
How to simplify the addition between two rational equations
In this question we must use algebra definitions and theorems to simplify the addition of two rational equations into a single rational equation. Now we proceed to show the procedure of solution in detail:
(n + 5) / (n² + 3 · n - 10) + 5 / (3 · n²) Given(n + 5) / [(n + 5) · (n - 2)] + 5 / (3 · n²) x² - (r₁ + r₂) · x + r₁ · r₂ = (x - r₁) · (x - r₂)1 / (n - 2) + 5 / (3 · n²) Associative and modulative property / Existence of the multiplicative inverse[3 · n² + 5 · (n - 2)] / [3 · n² · (n - 2)] Addition of fractions with different denominator(3 · n² + 5 · n - 10) / (3 · n³ - 6 · n²) Distributive property / Power properties / ResultTo learn more on rational equations: https://brainly.com/question/20850120
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Knowing the first day Anna and Tamara stop for sightseeing and lost 2 hours of travel time and the second day, they gain 1 hour because they did not stop for lunch, write an algebraic expression that represents how many hours Anna and Tamara traveled the first two days. Let x be the number of hours driven.
The algebraic expression that represents the number of hours Anna and Tamara traveled the first two days is (2x - 1) hours.
How to write Algebraic Word Problems?
We are told that;
On the first day, Anna and Tamara stop for sightseeing and lost 2 hours of travel time.
On the second day, they did not stop for lunch and gained 1 hour.
Now, we are told that x is the number of hours driven per day. Thus, for two days, number of hours driven is expressed as 2x.
Now, since they lost 2 hours on the first day, then total hours travelled is;
2x - 2 hours
On the second day, they gained 1 hour. Thus;
Final total number of hours traveled = 2x - 2 + 1
Final total number of hours traveled = (2x - 1) hours
Thus, we can conclude that the algebraic expression that represents how many hours Anna and Tamara traveled the first two days is (2x - 1) hours.
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Four seniors and six juniors are competing for four places on a quiz bowl team.
What is the approximate probability that all four seniors will be chosen at random?
what’s the % ?
Answer:
.00476
Step-by-step explanation:
Answers for number 32 and 31
Answer:
32: X=54
31: Xx54+74
