Answers
By transitive property, proved RQ is parallel to SU.
What are parallel lines?Parallel lines are those lines that are equidistant from each other and never meet, no matter how much they may be extended in either directions.
Given that,
RS║VT
∠6≅∠3
To prove: RQ║SU
RS║VT (Given)
∠6≅∠3 (Given)
∠5≅∠3 (Alternate interior angles between the parallels RS and VT)
Now, ∠6≅∠5 (Transitive property)
Since, ∠6 is congruent to ∠5, then they are alternate angles between RQ and SU.
Then, RQ║SU
Hence, proved RQ is parallel to SU.
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Related Questions
Can you guy solve it please
Answers
The area of the triangle in the diagram is one square foot.
How to find the area of the triangle?We know that the area of a triangle of base B and height H is given by the formula:
Area = B*H/2
On the diagram we can se that the measure of the base is 3 ft, and the height is 2/3 ft, then the area of that triangle will be:
Area = (3 ft)*(2/3 ft)/2 = 1 ft²
The area of the triangle is one square foot.
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1. Two points on k are (-4, 3) and (2, -1). Write a ratio expressing the slope of k
Answers
Answer: The slope of a line can be found using the formula:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line. Plugging in the given points, we get:
slope = (-1 - 3) / (2 - (-4)) = -4 / 6 = -2/3
So the ratio expressing the slope of line k is -2/3.
Step-by-step explanation:
Help me urgentt 16 pointsssss
Answers
Answer:
Step-by-step explanation:
Sorry did you already finish?
1. B. 2/3 = 8/12 b/c 2x4 = 8 and 3x4 = 12
2. D. 5/6 = 10/12 b/c 5x2 = 10 and 6x2 = 12
3. 2
4. A. simplified is 2/3 C. self-explanatory D. simplified is 3/4
If the volume of the cylinder is represented by 5x² +
15x + 2 and the height is 5x, which expression
represents the area of the base?
Answers
Answer:
its A. x² + 3x + 2/5x because it's the right answer
PLEASE HELP
I finished the first one but this website called edmentum /PLATO cant explain very well.
Answers
Step-by-step explanation:
It may be your last point
[tex]y = 2x + 6[/tex]
[tex]y = 2(2) + 6 = 10 \: not \: 8[/tex]
You have the point (2,8) when it should be (1,8)
Keep me updated
giving 50 points and brainliest to the best correct answer. thanks.
Answers
evidence of substituting into b =
correct substitution
2π (M1) period
e.g. period = 30 minutes, b = 2π A1
30
b=0.209(π ) A1 N2 15
substituting into h(t) (M1) e.g. h(0) = 1 , h(15) = 101 correct substitution A1 1=50sin(−π c)+51
METHOD 2
15 c=7.5 A1 N2
evidence of setting up a system of equations (M1)
two correct equations e.g.1=50sinb(0−c)+51,101=50sinb(15−c)+51 A1A1 attempt to solve simultaneously (M1)
e.g. evidence of combining two equations b=0.209(π ),c=7.5 A1A1 N2N2
evidence of recognizing the amplitude is the radius (M1)
e.g. amplitude is half the diameter
a=8 A1 2
a=4 AG N0
METHOD 2
evidence of recognizing the maximum height (M1) e.g. h = 6 , a sin bt + 2 = 6
correct reasoning
e.g. asinbt = 4 and sinbt has amplitude of 1 A1 a=4 AG N0
valid reasoning for their conclusion (seen anywhere) R1
e.g. h(t) < 0 so underwater; h(t) > 0 so not underwater
evidence of substituting into h (M1)
e.g. h(19.4) , 4 sin 19.4π + 2 15
correct calculation A1
e.g. h(19.4) = −1.19
correct statement A1 N0 e.g. the bucket is underwater, yes METHOD 2
valid reasoning for their conclusion (seen anywhere) R1 e.g. h(t) < 0 so underwater; h(t) > 0 so not underwater evidence of valid approach (M1)
e.g. solving h(t) = 0 , graph showing region below x-axis correct roots A1
e.g. 17.5, 27.5
correct statement A1 N0 e.g. the bucket is underwater, yes
I need help on this,
what's the vertex form of 8-2x=3(y+1)^2?
Answers
Answer:
The vertex form of 8-2x=3(y+1)^2 is: y= -1/6x^2 + 4x - 8.
Step-by-step explanation:
Answer:
Step-by-step explanation:
15 more than twice a number is 37. What is the number??
Answers
Answer:
11
Step-by-step explanation:
lets suppose the number is x
twice the number would be 2x and 15 more than that will be
2x + 15
and this is equal to 37
2x + 15 = 37
2x = 37 - 15
2x = 22
x = 22/2
x = 11
2. Use the linear combination method to solve the system of equations. Explain each step of your solution.
2x – 3y = 13
3x + y = -8
Answers
To use the linear combination method to solve the system of equations:
2x – 3y = 13
3x + y = -8
We can multiply the second equation by 3 to get:
9x + 3y = -24
Now we can add this equation to the first equation to eliminate the y term:
2x – 3y + 9x + 3y = 13 - 24
Simplifying the left-hand side and the right-hand side of the equation:
11x = -11
Finally, we can solve for x:
x = -1
Substitute x = -1 in either of the original equations:
2x – 3y = 13
2(-1) - 3y = 13
-2 - 3y = 13
-3y = 15
y = -5
Therefore, the solution to the system of equations is x = -1 and y = -5.
Explanation: We used the linear combination method to eliminate one of the variables by adding the two equations together. Then, we solved for the remaining variable and substituted the value back into one of the original equations to solve for the other variable.
A paperweight, the shape of a hemisphere, fits snugly inside a cylinder-shaped box with a radius of 9 cm. A cone fits snugly inside the paperweight hemisphere.
What is the volume of the cylinder in terms of
? Show all work.
What is the volume of the cone in terms of
? Show all work.
Estimate the volume of the hemisphere in terms of
by calculating the average of the volumes of the cylinder and cone. Show all work.
Answers
The volume of the cylinder is 162πx, the volume of the cone (1/3)πx^3 and the volume of the hemisphere is (162πx + (1/3)πx^3)/2
Volume of the cylinder:
The volume of a cylinder can be calculated using the formula V = πr^2h, where r is the radius and h is the height of the cylinder.
In this case, the height of the cylinder is equal to the diameter of the paperweight hemisphere, which is twice the radius of the cone.
Let's call the radius of the cone "x". The diameter of the paperweight hemisphere is equal to 2x, so the height of the cylinder is equal to 2x. The radius of the cylinder is 9 cm, so we can calculate the volume of the cylinder as follows:
V = πr^2h = π * 9^2 * 2x = 162πx
Volume of the cone:
The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius and h is the height of the cone. In this case, the height of the cone is equal to the radius of the cone, so we can calculate the volume of the cone as follows:
V = (1/3)πr^2h = (1/3)π * x^2 * x = (1/3)πx^3
Volume of the hemisphere:
The volume of a hemisphere can be calculated using the formula V = (2/3)πr^3, where r is the radius of the hemisphere. To estimate the volume of the hemisphere, we can calculate the average of the volumes of the cylinder and cone.
V_hemisphere = (V_cylinder + V_cone)/2 = (162πx + (1/3)πx^3)/2
This expression represents the estimated volume of the hemisphere in terms of x, the radius of the cone.
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A salesman earns 6% commission on all the merchandise that he sells. Last month he sold $6000 worth of merchandise. How much
Answers
The commission he earned last month was amount of $360.
What is the commission?The commission is determined by the product of the commission rate and the total sales.
A salesman receives a 6% commission on all merchandise sold.
He sold $6000 in merchandise last month.
Salesman earns 6% = 6/100 = 0.06
Now, we have to evaluate 6% of $6000
= 0.06 x 6000
Apply the multiplication operation, and we get
= 360$
Thus, the commission he earned last month was the amount of $360.
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The question seems incomplete, the complete question would be as:
A salesman earns 6% commission on all the merchandise that he sells. Last month he sold $6000 worth of merchandise. How much commission (in dollars) did he earn last month?
A balloon archway has been ordered for decorating high school gym for a 1950 style prom. The archway will contain 275 balloons each holding 1. 2 liters of helium. Each balloon has a mass of 27 milligrams when empty and all the string and fixings that hold the balloons together total 45 grams. If helium has a density of 0. 1785 grams per liter , what is the mass of the entire archway
Answers
We could calculate the weight of individual balloons from given density and empty mass. From that calculate the total weight of balloons and added to the weight of fixings. So the weight of archway becomes 111.33g.
Volume of helium each balloon is holding = 1.2L
Density of helium = 0.1785
Density = mass/volume
Mass = density/volume
So mass of helium in each balloon = 1.2×0.1785 = 0.2142 g = 214.2 mg
Mass of empty balloon = 27 mg
So the total mass of a balloon after filled with helium = 214.2 + 27 = 241.2 mg
Mass of 275 such balloons = Mass of 1 balloon × 275
= 241.2 ×275 = 66330 mg = 66.33 g
Mass of all strings and fixings = 45 g
Total mass of archway = mass of 275 balloons + mass of fixings and strings = 66.33 + 45 = 111.33 g
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what is a1 in a geometric series for which Sn=21825, an=125, r=15.
Answers
The a1 of the sequence is 28715.
How do you find the sum of geometric series?The sum of a geometric series can be found using the formula:
S = a / (1 - r)
where:
S is the sum of the series
a is the first term of the series
r is the common ratio, which is the ratio of any term to its preceding term
For a1;
Sn= a1 - anr/ r - 1
Where;
Sn = 21825
n = 125
r = 15
Thus;
2185 = a1 - (125 * 15)/15 -1
2185 * 14 = a1 - 1875
-30590 = a1 - 1875
a1 = 30590 + 1875
a1 = 28715 as we cans see
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You have been hired to design the next Desla Electric Advertisement. Use your work from today to create a persuasive and accurate advertisement. Include any additional information that you think is important when buying a vehicle.
Answers
Kweku would save $180 in a year buying electricity for the Desla electric compared to buying gas for the Deswagon.
To make up for the higher sale price,
it will take him more than 66 years.
And hybrid cars save money in the long term.
What is an equation?Two algebraic expressions having the same value and symbol '=' in between are called an equation.
Given:
Deselectric costs $11900 more than the Deswagon.
Unit cost:
Deselectric: $0.11 per mile.
Deswagon: $0.12 per mile.
Kweku drives about 18,000 miles a year.
That means
the cost for each car;
Deselectric: 18000 x 0.11 = $1980
Deswagon: 18000 x 0.12 = $2160
The more cost is,
= $2160 - $1980
= $180 of savings in a year.
Now, the make-up for the higher sale price,
= 11900/180
= 66.11 in years.
And hybrid cars save money in the long term.
Therefore, it will take him more than 66 years.
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solve the system of equations below
Answers
Answer:
y = 2
Step-by-step explanation:
2x + y = 8 → (1)
x - 3y = - 3 → (2)
multiplying (1) by 3 and adding will eliminate y
6x + 3y = 24 → (3)
add (2) and (3) term by term to eliminate y
7x + 0 = 21
7x = 21 ( divide both sides by 7 )
x = 3
substitute x = 3 into either of the 2 equations and solve for y
substituting into (1)
2(3) + y = 8
6 + y = 8 ( subtract 6 from both sides )
y = 2
Answer:
y = 2
Step-by-step explanation:
The question has given us two equations with two unknown variables and told us to solve the system of equations.
To do this, there are multiple methods. The method used in this solution will involve taking any one of the variables and making it the subject of both equations and then equating the resulting equations.
Let's take the first equation:
[tex]2x + y = 8[/tex] ,
and rearrange it to make x the subject:
[tex]2x + y = 8[/tex]
⇒ [tex]2x = 8 - y[/tex] [Subtracting y from both sides]
⇒ [tex]x = \frac{8 - y}{2}[/tex] ------(i) [Dividing both sides by 2]
Now let's take the second equation and rearrange it to also make x its subject:
[tex]x - 3y = -3[/tex]
⇒ [tex]x = -3+3y[/tex] ------(ii) [Adding 3y to both sides]
As both equations (i) and (ii) are equal to [tex]x[/tex], we can say:
(i) = (ii)
⇒ [tex]\frac{8-y}{2} = -3+3y[/tex]
⇒ [tex]8 - y = 2(-3+3y)[/tex] [Multipying both sides by 2]
⇒ [tex]8-y = -6 +6y[/tex]
⇒ [tex]8 - y - 6y=-6[/tex] [Subtracting 6y from both sides]
⇒ [tex]8-7y=-6[/tex]
⇒ [tex]-7y = -6-8[/tex] [Subtracting 8 from both sides]
⇒ [tex]-7y = -14[/tex]
⇒ [tex]y = \frac{-14}{-7}[/tex] [Dividing both dies by -7]
⇒ [tex]y = \bf 2[/tex]
Therefore, the answer to the question is 2.
We can now easily calculate the value of x by substituting y = 2 into either equation (i) or (ii).
Substituting into (i):
[tex]2x + 2= 8[/tex]
⇒ [tex]2x = 6[/tex]
⇒ [tex]x = \bf 3[/tex]
How do you find the least common denominator with variables?
Answers
Answer: To find the least common denominator (LCD) of fractions with variables, you need to determine the smallest number that all the denominators will divide into evenly.
One method to find the LCD is to multiply all the denominators together and simplify the expression. For example, if the denominators are x and y, the LCD would be xy. Another method is to find the greatest common factor (GCF) of all the denominators, and then divide each denominator by the GCF to obtain the LCD.
In either case, it may be necessary to factor the denominators to determine the GCF and simplify the expression for the LCD.
Step-by-step explanation:
Please hurry
Choose the best method for solving
and solve. Show your work.
y=x-7
y=x+7
Answers
The system of equations has no solution
How to determine the solutionFrom the question, we have the following parameters that can be used in our computation:
y = x - 7
y = x + 7
By substituting the first equation in the second equation
So, we have the following representation
x + 7 = x - 7
Evaluate the like terms
7 = -7
The above equation is false
Hence, the system has no solution
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Solve the equation round the result to the nearest hundredth 13.7t-4.7=9.9+8.1t
Answers
Answer:
t ≈ 2.61
Step-by-step explanation:
13.7t - 4.7 = 9.9 + 8.1t ( subtract 8.1t from both sides )
5.6t - 4.7 = 9.9 ( add 4.7 to both sides )
5.6t = 14.6 ( divide both sides by 5.6 )
t ≈ 2.61 ( to the nearest hundredth )
A parallelogram has sides of 25 cm and 32 cm. If the distance between the longer sides is15 cm, find the distance between the shorter sides.
Answers
Step 1: Calculate the length of the diagonal. To calculate the length of the diagonal, use the Pythagorean Theorem: a2 + b2 = c2, where a is the length of one side and b is the length of the other side. In this case, a is 25 cm and b is 32 cm, so we have 252 + 322 = c2, which gives us c2 = 900 + 1024 = 1924. Taking the square root of this gives us c = 43.8 cm.
Step 2: Calculate the distance between the shorter sides. To calculate the distance between the shorter sides, subtract the length of the diagonal (43.8 cm) from the length of the longer side (32 cm), which gives us a total of 8.2 cm.
Therefore, the distance between the shorter sides is 8.2 cm.
Algebraic word problems involve converting sentences into equations and then resolving those equations. and only one variable. The variable typically represents an unknowable quantity in a real-world situation.
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D(-5, -6), E(5,2), F(4, -4), G(-6, -12) (Distance & Slope Formulas)
Answers
The distance between D(-5, -6) and E(5,2) is 12.8, and the slope is 0.8. The distance between E(5,2) and F(4, -4) is 6.08, and the slope is 6. The distance between F(4, -4) and G(-6, -12) is 12.8, and the slope is 0.8.
Distance Formula:
The distance between two points A(x1, y1) and B(x2, y2) is calculated using the following formula:
d = √((x2-x1)² + (y2-y1)²)
Slope Formula:
The slope of the line between two points A(x1, y1) and B(x2, y2) is calculated using the following formula:
m = (y2-y1)/(x2-x1)
Distance between D(-5, -6) and E(5,2):
d = √((5-(-5))² + (2-(-6))²)
d = √((10)² + (8)²)
d = √(100 + 64)
d = √164
d = 12.8
Slope between D(-5, -6) and E(5,2):
m = (2-(-6))/(5-(-5))
m = (8)/(10)
m = 0.8
Distance between E(5,2) and F(4, -4):
d = √((4-5)² + (-4-2)²)
d = √((-1)² + (-6)²)
d = √(1 + 36)
d = √37
d = 6.08
Slope between E(5,2) and F(4, -4):
m = (-4-2)/(4-5)
m = (-6)/(-1)
m = 6
Distance between F(4, -4) and G(-6, -12):
d = √((-6-4)² + (-12-(-4))²)
d = √((-10)² + (-8)²)
d = √(100 + 64) …
d = √164
d = 12.8
Slope between F(4, -4) and G(-6, -12):
m = (-12-(-4))/(-6-4)
m = (-8)/(-10)
m = 0.8
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The slope is 0.8 and the distance between D(-5, -6) and E(5,2) is 12.8.
The slope is 6 and the distance between E(5,2) and F(4, -4) is 6.08.
The slope is 0.8 and the distance between F(4, -4) and G(6, -12) is 12.8.
The formula for Distance:
The following formula is used to determine the distance that separates points A (x1, y1) and B (x2, y2) from one another:
d = √((x2-x1)² + (y2-y1)²)
Slope Method:
The following formula can be used to determine the slope of the line that connects the two points A (x1, y1) and B (x2, y2):
m = (y2-y1)/(x2-x1)
The distance that separates D(-5, -6) from E(5,2):
d = √((5-(-5))² + (2-(-6))²)
d = √((10)² + (8)²)
d = √(100 + 64)
d = √164
d = 12.8
Slope from D(-5, -6) to E(5,2):
m = (2-(-6))/(5-(-5))
m = (8)/(10)
m = 0.8
Between E(5, 2) and F(4, -4):
d = √((4-5)² + (-4-2)²)
d = √((-1)² + (-6)²)
d = √(1 + 36)
d = √37
d = 6.08
Slope between E(5,2) and F(4, -4):
m = (-4-2)/(4-5)
m = (-6)/(-1)
m = 6
Distance between F(4, -4) and G(-6, -12):
d = √((-6-4)² + (-12-(-4))²)
d = √((-10)² + (-8)²)
d = √(100 + 64) …
d = √164
d = 12.8
Slope between F(4, -4) and G(-6, -12):
m = (-12-(-4))/(-6-4)
m = (-8)/(-10)
m = 0.8
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Fill in the answers on the right
Long's Coffee Shop sells a refill mug for $8. 95. Each refill costs $1. 50. Last month, Jalissa spent $26. 95 on a mug and refills. How many refills did she buy?
constant:
Rate:
Equation:
Solve:
Answers
Constant: Mug price = $8.95, Refill price = $1.50, Rate: Refill rate = $1.50, Equation: 8.95 + 1.50x = 26.95, Solve: x = 15 refills. Jalissa purchased 15 refills last month, paying $26.95 for the mug and refills combined.
To calculate the number of refills Jalissa purchased last month, we must first define the constants of the equation. The cost of the mug is $8.95 and the cost of the refill is $1.50. After establishing these constants, we can create an equation using the rate of refills ($1.50) and the total amount that Jalissa spent ($26.95). The equation is 8.95 + 1.50x = 26.95. This equation can then be solved to find the value of x, which is the number of refills Jalissa purchased. To solve the equation, subtract 8.95 from both sides to get 1.50x = 18. This equation can then be divided by 1.50 to find the value of x, which is 15. This means that Jalissa purchased 15 refills last month, paying $26.95 for the mug and refills combined.
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The show johnny want to binge is 61 episodes long, it takes him 5 hours to complete 12 episodes, and 10 hours to complete 24 episodes. If johnny were to watch 12 episodes a day, how many days would it take johnny to finish the show?
Answers
Answer:
6 days
Step-by-step explanation:
This is an easy problem if you think about it. Let’s start by identifying unnecessary information. I’ll go ahead and point this out: we don’t need to know how long it takes Johnny to watch the episodes. The question already says he’s only watching 12 episodes per day, so the time information is useless. If you rewrite the problem with only message information, you get something like this:
“The show Johnny wants to binge watch is 61 episodes long. If Johnny were to watch 12 episodes a day, how many days would it take him to finish the show?”
That makes it seem so much easier! Let’s start by dividing the show into sets of 12 episodes. 61 divided by 12 is 5 R1. Since he needs another day to finish the show, even if he only watches one episode, the answer is 6 days.
I need to know what is the distance between the points
Answers
Answer:9
Step-by-step explanation:
add
9. Triangle LMN with vertices L(-7, 4), M(-3, 0),
and N(-8, 1):
(a) Translation: (x,y) → (x + 5, y-6)
(b) Reflection: in the line y=-x
L’
М'
N'
10
Answers
The transformation of the vertices of the triangle is a. L' = (-7 + 5, 4 - 6) = (-2, -2), M' = (-3 + 5, 0 - 6) = (2, -6), N' = (-8 + 5, 1 - 6) = (-3, -5). b. L' = (-7, 4) = (-4, 7), M' = (-3, 0) = (0, 3), and N' = (-8, 1) = (-1, 8).
What are transformations?The transformation, or f: X X, is the name given to a function, f, that maps to itself. After the transformation, the pre-image X becomes the picture X. Any operation, or a combination of operations, such as translation, rotation, reflection, and dilation, can be used in this transformation.
The vertices of the triangle are L (-7, 4), M(-3, 0) and N(-8, 1).
a. Translation: (x,y) → (x + 5, y-6) is as follows:
L' = (-7 + 5, 4 - 6) = (-2, -2)
M' = (-3 + 5, 0 - 6) = (2, -6)
N' = (-8 + 5, 1 - 6) = (-3, -5)
b. Reflection in line y = -x is as follows:
Here, the value of y coordinate is negative x coordinate and x coordinate is negative y coordinate.
L' = (-7, 4) = (-4, 7)
M' = (-3, 0) = (0, 3)
N' = (-8, 1) = (-1, 8)
Hence, the transformation of the vertices of the triangle is a. L' = (-7 + 5, 4 - 6) = (-2, -2), M' = (-3 + 5, 0 - 6) = (2, -6), N' = (-8 + 5, 1 - 6) = (-3, -5). b. L' = (-7, 4) = (-4, 7), M' = (-3, 0) = (0, 3), and N' = (-8, 1) = (-1, 8).
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In a cookie jar, 1/5 of the cookies are chocolate chip and 1/2 of the rest are peanut butter. what fraction of all the cookies are peanut butter?
Answers
Answer: 3/10
Step-by-step explanation: Convert given fractions into like terms. In this case 1/5 becomes 2/10 and 1/2 becomes 5/10. Add them up and you get 7/10. Subtract from the total of 10/10 and the result is 3/10. Since the final fraction can't be simplified, this is the final answer.
How does g(x)=8x change over the interval from x=3 to x=5?
Answers
The function g(x) = 8x is increasing over the interval from x=3 to x=5.
The graph of the function g(x) = 8x is shown below.
We can observe that the graph of g(x) is a staright line passing through origin (0, 0)
This means that the function g(x) is a linear function.
For x = 3, the value of function would be,
g(3) = 8 × (3)
g(3) = 24
And for input value x = 5, the value of function would be,
g(5) = 8 × (5)
g(5) = 40
This means that the value of function g(x) increases from x = 3 to x = 5
Also, from the graph the function g(x) is increasing over the interval [3, 5]
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Find four ordered pair solutions to the given equation by completing the table.
Answers
Using the equation given, the four ordered pair solutions are:
x | y
0 5
-2 -5
6 35
-8 -35
How to Find the Ordered Pair Solutions to an Equation?To find the ordered pair solutions to the given equation as shown in the table, plug in the known value into y = 5x + 5 to find the other coordinate pair.
Therefore:
When x = 0:
y = 5(0) + 5
y = 5
When y = -5:
-5 = 5x + 5
-5 - 5 = 5x
-10 = 5x
-10/5 = x
-2 = x
x = -2
When x = 6:
y = 5(6) + 5
y = 35
When y = -35:
-35 = 5x + 5
-35 - 5 = 5x
-40 = 5x
-40/5 = x
-8 = x
x = -8
The four ordered pair solutions are:
x | y
0 5
-2 -5
6 35
-8 -35
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Consider the line 8x-6y=6
What is the slope of a line perpendicular to this line?
What is the slope of a line parallel to this line?
Answers
Answer:
Perpendicular: = -3/4
Parallel= 4/3
Hope this helps
Determine the center of the ellipse as well as the lengths of the major and minor axes:
5x2 +y2 −3x+40=0.
Answers
The center of the ellipse will be (3, 0). And the lengths of the major and minor axes will be 4.4721 and 2, respectively.
What is an ellipse?Ellipse is a locus of a point that moves in a plane such that the sum of its distances from the two points called foci adds up to a constant. It is taken from the cone by cutting it at an angle.
The equation of the ellipse is given below.
5x² + y² − 30x + 40 = 0
Convert the ellipse into standard form. Then we have
5x² + y² − 30x + 40 = 0
5[x² − 6x] + y² + 40 = 0
5[x² − 6x + 9] − 45 + y² + 40 = 0
5(x − 3)² + y² = 5
(x − 3)² + y² / 5 = 1
The equation center of the ellipse is (3, 0). Then the major axis is given as,
Major axis = 2 √5
Major axis = 4.4721
Then the minor axis is given as,
Minor axis = 2 x 1
Minor axis = 2
The center of the ellipse will be (3, 0). And the lengths of the major and minor axes will be 4.4721 and 2, respectively.
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The correct equation is given below.
5x² + y² − 30x + 40 = 0.
587 Consider the equation y =2/3|x-5|-3. Complete the following without using a graphing
tool.
(a) What are the coordinates of the vertex of this graph?
(b) Find the coordinates of all axis intercepts of the graph.
(c) Sketch the graph by hand.
(d) Using each of these points and the vertex, compute the slope of each side of the graph.
How are these slopes related?
Answers
a) So x = 5 is the critical point, and the vertex is at (5, -3).
b) the y-intercept is (0, -3).
c) y = 2/3 * (x - 5) - 3 for x >= 5
y = -2/3 * (x - 5) - 3 for x < 5
d) The slopes of the two pieces are 2/3 and -2/3 respectively.
Step by step solution(a) To find the vertex, set the derivative of the equation to 0 and solve for x:
dy/dx = 2/3 * d/dx |x-5| = 2/3 * sign(x-5) = 0, where sign(x-5) = 1 if x > 5, sign(x-5) = -1 if x < 5, sign(x-5) = 0 if x = 5.
So x = 5 is the critical point, and the vertex is at (5, -3).
(b) The axis intercepts are the points where the graph crosses the x-axis and y-axis.
The x-intercepts occur when y = 0:
2/3 * |x-5| - 3 = 0
2/3 * |x-5| = 3
|x-5| = 9/2
x = 5 +/- 9/2 = 5 + 9/2 = 7 or x = 5 - 9/2 = 2.5
So the x-intercepts are (7, 0) and (2.5, 0).
The y-intercept occurs when x = 0:
y = 2/3 * |0-5| - 3 = -3.
So the y-intercept is (0, -3).
(c) The graph is a piecewise linear function with two pieces:
y = 2/3 * (x - 5) - 3 for x >= 5
y = -2/3 * (x - 5) - 3 for x < 5
(d) The slopes of the two pieces are 2/3 and -2/3 respectively. These slopes are related because the graph is symmetrical about the vertical line x = 5, with the two pieces being reflections of each other across the line.
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