Answers
The inequality -1/2 ≤ ab ≤ 1/2 implies [tex](ab)^2[/tex] ≤ 1/4, and squaring bοth sides οf the inequality gives us [tex](ab)^2[/tex] ≤ 1/4 as required.
Tο prοve that -1/2 ≤ ab ≤ 1/2 fοr [tex]a^2+b^2 = 1[/tex]and a, b ∈ ℝ, we can start by nοting that:
-1 ≤ a ≤ 1 (because [tex]a^2 \le a^2 + b^2 = 1[/tex], sο -1 ≤ a ≤ 1)
-1 ≤ b ≤ 1 (because [tex]b^2 \le a^2 + b^2 = 1[/tex], sο -1 ≤ b ≤ 1)
Multiplying these inequalities, we get:
-1 ≤ ab ≤ 1
Nοw, we need tο shοw that ab cannοt equal ±1. If ab = 1, then we have:
[tex]a^2 + b^2 = 1[/tex]
[tex]a^2 + 2ab + b^2 = 1 + 2ab[/tex]
[tex](a + b)^2 = 1 + 2ab[/tex]
Since a and b are bοth between -1 and 1, a + b is between -2 and 2, sο [tex](a + b)^2[/tex] is between 0 and 4. Therefοre, we have:
1 + 2ab ≤ 4
Simplifying, we get:
ab ≤ 3/2
This cοntradicts the fact that ab = 1, sο ab cannοt equal 1. Similarly, if ab = -1, we get:
[tex](a + b)^2 = 1 - 2ab[/tex]
Since [tex](a + b)^2[/tex] is nοnnegative, we have:
1 - 2ab ≥ 0
Simplifying, we get:
ab ≤ 1/2
This cοntradicts the fact that ab = -1, sο ab cannοt equal -1. Therefοre, we have -1 < ab < 1, which implies -1/2 ≤ ab ≤ 1/2.
Taking the square οf bοth sides οf -1/2 ≤ ab ≤ 1/2, we get:
[tex]1/4 \le a^2b^2 \le 1/4[/tex]
Adding [tex]a^2 + b^2 = 1[/tex]tο bοth sides, we get:
[tex]5/4 \le 1 + a^2b^2 \le 5/4[/tex]
Dividing by 2, we get:
[tex]5/8 \le (1 + a^2b^2)/2 \le 5/8[/tex]
Since [tex](1 + a^2b^2)/2[/tex] is the average οf [tex]a^2[/tex] and [tex]b^2[/tex], we have:
[tex]5/8 \le (a^2 + b^2)/2 \le 5/8[/tex]
Simplifying, we get:
5/8 ≤ 1/2 ≤ 5/8
Therefοre, the inequality -1/2 ≤ ab ≤ 1/2 implies [tex](ab)^2 \le 1/4[/tex], and squaring bοth sides οf the inequality gives us [tex](ab)^2 \le 1/4[/tex] as required.
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Related Questions
pls answer it will give you 10 pints
Answers
Answer:
the third option
Step-by-step explanation:
y=x^(2) Transformation: Reflects across x-axis Horizontal shift 5 units to the right Vertical shift up 4 units Vertical shrink by (1)/(2) Domain: 3,9
Answers
The domain of the function is y = -1/2(x-5)^2 + 4, for 3 ≤ x ≤ 9.
Domain of function explained.
To transform the function y=x^2 according to the given specifications, we can use the following steps:
Reflecting across the x-axis: This transformation reflects the function across the x-axis, which means that every y-coordinate is multiplied by -1. Therefore, the function becomes y = -x^2.Horizontal shift 5 units to the right: This transformation moves the entire function horizontally to the right by 5 units. To achieve this, we subtract 5 from the x-coordinate. Therefore, the function becomes y = -(x-5)^2.Vertical shift up 4 units: This transformation moves the entire function vertically up by 4 units. To achieve this, we add 4 to the y-coordinate. Therefore, the function becomes y = -(x-5)^2 + 4.Vertical shrink by (1/2): This transformation shrinks the function vertically by a factor of 1/2, which means that every y-coordinate is multiplied by 1/2. Therefore, the function becomes y = -1/2(x-5)^2 + 4.Domain: Finally, the domain of the transformed function is given as [3, 9], which means that the function is only defined for x-values between 3 and 9. Therefore, we need to restrict the domain of the function to this interval. Therefore, the final transformed function is:
Therefore the transformation is y = -1/2(x-5)^2 + 4, for 3 ≤ x ≤ 9.
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How many mg of drug are in 30 mL of a 60 mg/ 5 mL elixir?
Answers
There is 360 mg of drug in 30 mL of the 60 mg/5 mL elixir.
The amount of drug in a given volume of a medication can be calculated using the following formula: Amount of drug (mg) = Volume (mL) x Concentration (mg/mL). In this case, we have 30 mL of a 60 mg/5 mL elixir, so the amount of drug in 30 mL can be calculated as follows: Amount of drug (mg) = 30 mL x 60 mg/5 mL = 360 mg. Therefore, there is 360 mg of drug in 30 mL of the 60 mg/5 mL elixir.
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The sum of two numbers is 67 and the difference is 13. What are the numbers?
Answers
Answer:
40 and 27
Step-by-step explanation:
Answer:Let's call the two numbers x and y.
From the problem, we know that:
x + y = 67 (equation 1)
and
x - y = 13 (equation 2)
To solve for x and y, we can use the method of elimination. We can add equations 1 and 2 to eliminate y:
(x + y) + (x - y) = 67 + 13
Simplifying this equation, we get:
2x = 80
Dividing both sides by 2, we get:
x = 40
Now we can use equation 1 to solve for y:
x + y = 67
40 + y = 67
Subtracting 40 from both sides, we get:
y = 27
Therefore, the two numbers are 40 and 27.
Step-by-step explanation:
non-linear sequence is
100, 95, 90, .............
20, 12, 4, ...............
4, 5, 7, ...............
0.5, 1, 1.5, ...............
Answers
use your knowledge of the decision-making process to choose the correct location for each of the three missing labels in the following diagram.
Answers
The correct locations for the three missing labels in the diagram are "Analyze the Situation" in the top-left corner, "Identify Options" in the bottom-left corner, and "Make a Decision" in the bottom-right corner. We need to apply the decision-making process to the three missing labels in the diagram.
The first missing label is for the top-left corner of the diagram. The correct location for this label is "Analyze the Situation." This is because the decision-making process begins with an analysis of the situation to understand the problem and identify any potential constraints.
The second missing label is for the bottom-left corner of the diagram. The correct location for this label is "Identify Options." This is because the decision-making process involves identifying all of the available options for solving the problem.
The third missing label is for the bottom-right corner of the diagram. The correct location for this label is "Make a Decision."
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Decompose each of the numbers 72, 204, 1800, and 42336 as products of their prime factors. (4)
Answers
The numbers when decomposed using their prime factors are 72: 2³ × 3², 204: 2² × 3 × 17, 1800: 2³ × 3² × 5² and 42336: 2⁵ × 3 × 11² × 17
How the numbers can be decomposed using their prime factorsTo decompose a number into its prime factors, we need to find the prime numbers that multiply together to give the original number.
The numbers are given as 72, 204, 1800, and 42336
So, we have
Let's decompose each of the given numbers:
72 = 2 × 2 × 2 × 3 × 3
Prime factorization: 2³ × 3²
204 = 2 × 2 × 3 × 17
Prime factorization: 2² × 3 × 17
1800 = 2 × 2 × 2 × 3 × 3 × 5 × 5
Prime factorization: 2³ × 3² × 5²
42336 = 2 × 2 × 2 × 2 × 2 × 3 × 11 × 11 × 17
Prime factorization: 2⁵ × 3 × 11² × 17
Therefore, the prime factorization of each number is:
72: 2³ × 3²
204: 2² × 3 × 17
1800: 2³ × 3² × 5²
42336: 2⁵ × 3 × 11² × 17
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Use the graph to determine a. the function's domain; b. the function's range; c. the x-intercepts, if
any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks,
below.
f(-2)=? f(2)=?
Answers
By using the graph, the key features of the function include the following:
The function's domain = {-∞, ∞}.The function's range = {-1, ∞}.The x-intercepts = (0, 0)The y-intercept = (0, 0)f(-2) = 3.f(2) = -1.How to determine the domain and range?In Mathematics, the horizontal extent of any graph of a function represents all domain values and they are read and written from smaller to larger numerical values, and from the left of a graph to the right.
By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:
Domain = {-∞, ∞} or {x|x ∈ R}.
Range = {-1, ∞}.
Based on the graph, the x-intercept and y-intercept of the function both occur at the origin (0, 0).
When x = -2, the y-value is given by:
f(-2) = 3.
When x = 2, the y-value is given by:
f(2) = -1.
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what is the volume of a right circular cone that has a height of 19.9 cm and a base with a radius of 9.6 cm. round your answer to the nearest tenth of a cubic centimeter.
Answers
The volume of the right circular cone is approximately 1819.1 cubic centimeters.
What is volume ?Volume is the measure of the amount of space occupied by a three-dimensional object or shape. It is expressed in cubic units, such as cubic centimeters, cubic meters, or cubic inches, depending on the system of measurement being used. The formula for volume varies depending on the shape of the object, but for many common three-dimensional shapes, such as cubes, spheres, cylinders, and cones, there are well-known formulas for calculating their volumes.
According to the given information :
The formula for the volume of a right circular cone is V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone.
Substituting the given values, we get:
V = (1/3)π(9.6 cm)²(19.9 cm)
V ≈ 1819.1 cubic centimeters (rounded to the nearest tenth)
Therefore, the volume of the right circular cone is approximately 1819.1 cubic centimeters.
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PLEASE SHOW WORK!!!!!!!!!
Answers
Based on the information provided, the probability of both events happening is 0.011 or 1.1% (option A).
How to calculate the probability in this case?The probability can be described as the proportion between the desired outcome happening and the total of possible outcomes. Using this principle, let's calculate the probability:
Probability of Gypsy finishing first: 1 / 10 = 0.1
Probability of Silver Rush finishing second: 1/ 9 = 0.11
Total probability 0.1 x 0.11 = 0.011 or 1.1% (0.011 x 100)
Therefore, the answer that matches the probability is option A.
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What is the vertex of the graph?
Answers
The calculated vertex of the graph represented in the figure is (-3, -3)
What is the vertex of the graph?Given that we have the graph
The curve on the graph is a quadratic function
As a general rule, the vertex of a graph is the minimum or the maximum point on a graph
From the graph, we have the minimum point to be
Minimum = (-3, -3)
This means that
Vertex = (-3, -3)
Hence, the vertex is (-3, -3)
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Help please and thank you
Answers
The reasons for the statements supporting the proof that ∡TV ≅ ∡ XW are given as follows;
A - Given
H - Vertical Angles
F- Angles that inscribe a half circle are congruent and 90°
C - Angle-Angle-Side Triangle Congruence
E - CPCTC
D - Reflexive Property
I - Hypotenuse Leg (H-L) Triangle Congruence
G - CPCTC
B - Arcs of a congruent angles are congruent
CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent". It is a theorem used in geometry to prove that two triangles are congruent.
An inscribed angle is half the measure of its intercepted arc. A half circle is a semicircle, which has an intercepted arc of 180°, so the inscribed angles are 1/2(180°) = 90° and congruent.
The reflexive property is a property of equality that states that any quantity is equal to itself. In other words, a = a. This property is used in many mathematical proofs.
The Hypotenuse-Leg (H-L) Triangle Congruence is a rule used to prove that two right triangles are congruent. If the hypotenuse and a leg of one triangle are congruent to the hypotenuse and the corresponding leg of another triangle, then the two triangles are congruent.
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1. convert 34/66 into percentage.
2. convert 32/66 into percentage
Answers
Answer:
1.) 51.515% 2.) 48.485%
Step-by-step explanation:
In this question we know 66 is 100% of the fraction, so we can start the process of converting a fraction into a percent, by figuring out how to adjust the fraction so that the denominator will be 100. First, divide 100 by the denominator.
100/66 = 1.515
Now we want to multiply denominator and numerator by 100.
34 × 1.515 = 51.515
66 × 1.515 = 100
Therefore, 34/66 as a percentage will be 51.515%
Now we want to convert 32/66. We will do the same for this equation.
100/66 = 1.515
32 × 1.515 = 48.485
66 × 1.515 = 100
Therefore 32/66 as a percentage will be 48.485
Hope this helps : )
What is 1/4+1/8 estimated answer
Answers
Answer:
3/8
Step-by-step explanation:
1/4+1/8
2/8+1/8
=
3/8
3/8
1+2 to equal 3, which is all the numerators above the least common denimonater 8
Which expression must be added to 3x - 7 to equal 0?
Answers
Answer:
To find out which expression must be added to 3x - 7 to equal 0, we need to solve the equation:
3x - 7 + a = 0
where "a" is the expression to be added.
To solve for "a", we can isolate "a" on one side of the equation by adding 7 to both sides:
3x - 7 + a + 7 = 0 + 7
Simplifying the left side, we get:
3x + a = 7
Therefore, the expression that must be added to 3x - 7 to equal 0 is:
a = -3x + 7
We can check that this is correct by substituting -3x + 7 for "a" in the original equation:
3x - 7 + (-3x + 7) = 0
Simplifying the left side, we get:
3x - 3x - 7 + 7 = 0
0 = 0
The equation is true, which confirms that the expression -3x + 7 must be added to 3x - 7 to equal 0.
Step-by-step explanation:
Joey has a storage cabinet in the shape of a rectangular prism with a base length of 3 feet, a base width of 2 feet, and a height of 8 feet.
A. Joey is going to paint the lateral area and the top of the storage cabinet, but he will not paint the bottom on which it sits. What is the surface area of the storage cabinet that Joey is going to paint?
B. What is the total surface area of the cabinet, which includes the bottom?
C. Find the volume of the cabinet to determine the amount of space Joey will have to use if he wants to fill the cabinet with supplies?
Answers
Answer: 592
Step-by-step explanation:
A=2(wl+hl+hw)
A=2(8*10+12*10+12*8)=592
A woman wants to collect exactly four litres of water from the well for her family. She only has two containers. One container can carry five litres and the other can carry seven litres. How can she measure out exactly four litres?
Answers
The can measure exactly 4 litres by having 1 2/3 in the 5litres container and 2 2/3 in the 7 litres container
What is word problem?A word problem is a math problem written out as a short story or scenario. Basically, it describes a realistic problem and been asked to imagine how you would solve it using math.
These word problems are interpreted into mathematical equation or expression.
If the woman wants to have the water in two containers with a good ratio, then we say;
The ratio of container 1 to container 2 is 5:7
therefore;
the amount of water in 5litres container = 5/12 × 4 5/3 = 1 2/3 litres
the amount of water in 7 litres = 7/12 × 4 = 7/3 = 2 2/3 litres
Therefore to measure it, she will have 1 2/3 in the 5litres container and 2 2/3 in the 7 litres container
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marbles a bag contains two red marbles, four green ones, one lavender one, four yellows, and four orange marbles. how many sets of four marbles include one of each color other than lavender?
Answers
The number of sets is 128 by using the multiplication principle.
The multiplication principle states that if there are k events, and if the first event can happen in n_1 ways, the second event can happen in n_2 ways, and so on, then the k events can happen in n_1 × n_2 × ... × n_k ways.
To find the number of sets of four marbles include one of each color other than lavender, A bag contains two red marbles, four green ones, one lavender one, four yellows, and four orange marbles.
There are two red marbles.
There are four green ones.
There are four yellows.
There are four orange marbles.
So, total of 2 + 4 + 1 + 4 + 4 = 15 marbles in the bag.
To find the total number of sets of four marbles including one of each color other than lavender, use the multiplication principle.
In this question, there are 5 events (choosing one of each color).
The first event can happen in 2 ways.
The second event can happen in 4 ways.
The third event can happen in 4 ways.
The fourth event can happen in 4 ways.
Therefore, by the multiplication principle, the total number of sets of four marbles includes one of each color other than lavender
= 2 × 4 × 4 × 4 ⇒ 128 sets.
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Make sure you explain why and show all your work.
In Triangle LMO, T is the midpoint of LM, J is the midpoint of MO, and P is the midpoint of LO. Also, LT = 9, JM = 12, and OL = 32.
What is the perimeter of the Triangle TJP?
Answers
First, we can find the length of TJ. Since T is the midpoint of LM, we know that LT = TM. Therefore, we can find TM by subtracting the length of LT from the length of OL: TM = OL - LT = 32 - 9 = 23. Since J is the midpoint of MO, we know that JM = JO. Therefore, we can find JO by subtracting the length of JM from the length of OL: JO = OL - JM = 32 - 12 = 20. Finally, we can find the length of TJ by using the Pythagorean theorem on right triangle LJT: TJ^2 = LT^2 + JT^2. Since we know that LT = 9 and JT = JM/2 = 6, we can solve for TJ: TJ^2 = 9^2 + 6^2 = 81 + 36 = 117. Taking the square root of both sides, we get TJ = sqrt(117) = 3sqrt(13).
Next, we can find the length of JP. Since P is the midpoint of LO, we know that JP is half the length of LO: JP = LO/2 = 32/2 = 16.
Finally, we can find the length of TP. Since T, J, and P are all midpoints of their respective sides, we know that triangle TJP is similar to triangle LMO, with a scale factor of 1/2. Therefore, the length of TP is half the length of LM: TP = LM/2 = TM/2 = 23/2.
Now we can add up the lengths of the three sides to find the perimeter of triangle TJP: TJ + JP + TP = 3sqrt(13) + 16 + 23/2 = 3sqrt(13) + 32 + 11.5 = 3sqrt(13) + 43.5. Therefore, the perimeter of triangle TJP is 3sqrt(13) + 43.5.
A side of the triangle below has been extended to form an exterior angle of 66°. Find the value of x.
Answers
Answer:
x = 114 degrees
Step-by-step explanation:
Angle x and the exterior angle form a straight line, which is 180 degrees. Because of this, we can subtract 66 from 180, equaling 114.
PLEASE HELPPPP!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
Answer and Explanation:
1. corresponding: bottom right graph
→ the angles are in the same place in relation to the parallel lines
2. alternate interior: bottom left graph
→ the interior angles are on opposite sides of the transversal
3. alternate exterior: top left graph
→ the exterior angles are on opposite sides of the transversal
4. none of these: top middle graph
→ there are no parallel lines
Frank has $50 dollars to buy party supplies. He wants to buy chips and dip. He has already bought seven bags of chips that costs him $32. The dip he wants to buy has a unit cost of $2.25 per container. Write and solve an equation, or inequality, to determine the number of containers of dip Frank can buy.
Help-
Answers
Answer: 32+2.25(8)=50
Step-by-step explanation:
If you have $50 and waste $32 that leaves you with $18.
If each dip cost 2.25 you can have a total of 8 dips.
32+2.25(x)=50
32+2.25(8)=50
Select all ordered pairs that correspond to input-output pairs of the function y = -7x + 3,
A
(4,0)
B.
(
639)
DO
C (
853)
D. (5,32)
Select all order pairs to that correspond to input-output pairs of the function y=-7x+3
Answers
The ordered pairs that correspond to input-output pairs of the function y=-7x+3 are (5,32).
The input-output pairs of the function y=-7x+3 can be determined by plugging in the input value x into the function and solving for y. The formula y=-7x+3 can be used to calculate the output value corresponding to a given input value x. For example, if x=4, then y=-7(4)+3=-28+3=5, so the input-output pair is (4,5). Similarly, if x=5, then y=-7(5)+3=-35+3=32, so the input-output pair is (5,-32). The ordered pairs that correspond to input-output pairs of the function y=-7x+3 and (5,32).
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A company can make a hollow rubber ball for $0.02 per square foot. each ball costs the company $1. what is the diameter of a ball to the nearest tenth of a foot?
Answers
Answer:
4.0 feet
Step-by-step explanation:
You want the diameter of a rubber ball that costs $1, if that cost is $0.02 per square foot.
AreaThe surface area of a sphere is given by ...
A = 4πr²
In terms of diameter, this is ...
A = 4π(d/2)² = πd²
CostThe cost will be ...
Cost = (cost/square foot)(area)
1.00 = 0.02(πd²)
DiameterSolving for d, we have ...
d² = 1.00/(0.02π) ≈ 15.915
d ≈ √15.915 ≈ 3.989 ≈ 4.0 . . . . feet
The diameter of the ball is about 4.0 feet.
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A $9. 00 album is marked up 6 1/4% and then is offer for 14% off. If the sales tax is 5% what is the total amount if tax that must be paid for this album
Answers
A $9. 00 album is marked up 6 1/4% and then is offered for 14% off. If the sales tax is 5%. The total amount of tax that must be paid for this album is $0.41.
Let's break this problem down into steps:
Find the markup price of the album after it is marked up by 6 1/4%.Markup price = $9.00 + 6.25% of $9.00
Markup price = $9.00 + $0.56
Markup price = $9.56
Find the sale price of the album after it is discounted by 14%.Sale price = Markup price - 14% of Markup price
Sale price = $9.56 - 14% of $9.56
Sale price = $9.56 - $1.34
Sale price = $8.22
Add the sales tax of 5% to the sale price.Tax amount = 5% of the Sale price
Tax amount = 5% of $8.22
Tax amount = $0.41
Therefore, the total amount of tax that must be paid for this album is $0.41.
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A gym charges $59 per month plus an initiation fee. The total cost for initiation and one year of membership is $742. 50. Write an equation to find the cost y of initiation and x months of membership. Then find the total cost for 18 months of membership
Answers
Therefore the total cost for 18 months of membership is[tex]$742.50[/tex].The equation to find the cost of initiation and x months of membership is y + 59x = 742.50
The equation to find the cost of initiation and x months of membership is y + 59x = 742.50. Since the cost for one year of membership is known, the cost for initiation can be found by subtracting 59x from both sides of the equation, resulting in y = 742.50 - 59x. To find the total cost for 18 months of membership, we plug x = 18 into the equation, resulting in y + 59(18) = 742.50. This can be simplified to y + 1062 = 742.50, which means that y = -319.50. Therefore, the total cost for 18 months of membership is[tex]$742.50 - $319.50[/tex], which equals [tex]$423.00[/tex]. This calculation can be represented by the equation y + 59x = 742.50, where y is the cost of initiation and x is the number of months of membership. Plugging in the number of months of membership (x = 18) gives us the total cost for 18 months of membership (y + 1062 = 742.50). Simplifying this equation yields y = -319.50, and therefore the total cost for 18 months of membership is[tex]$742.50.[/tex]
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Find the minimum and maximum value of the functiony=(x−9)²+9. Enter infinity or -infinity if thefunction never stops increasing or decreasing.Maximum value =Minimum value =
Answers
The minimum value of the function is 9, and the function never stops increasing.
The given function is y=(x−9)²+9.
We need to find the minimum and maximum value of the function. The given function is a quadratic function whose graph is a parabola. Since the coefficient of x² is positive, the graph of the quadratic function will be in the form of an upward parabola whose vertex is at the point (h,k).
The vertex form of the quadratic function is given byy = a(x - h)² + k, where(h,k) is the vertex of the parabola.
a is the coefficient of (x - h)².
In the given function,y = (x-9)² + 9, the vertex of this function is (9,9) and a=1.
Therefore, the minimum value of the function is 9 at x=9, which is the vertex of the parabola.
The function y=(x−9)²+9 is an upward parabola, and hence it never stops increasing, which means that the function has no maximum value.
Thus, the minimum value of the function is 9, and the function never stops increasing.
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Can someone find the median
Answers
Answer: 17.5
Step-by-step explanation:
First find the two numbers (data values) that are in the middle. Thats 15 and 20. When you have two numbers in the middle of a data set you average them. (Buy adding 15 and 20 then dividing by 2) So the answer is 17.5
[tex]\sqrt { (340)^2^/^3 *\frac{1}{{(0.09)^1^/^2}}* \sqrt{81{(0.09)^1^/^2}[/tex]
Answers
in 30 seconds, 180 cm^3 of oxygen diffuse through a porous plate. How long will it take 300cm3 of chlorine to diffuse through the same pot
Answers
Answer:
50s
Step-by-step explanation:
180cm³= 30s
300cm³=x s
Thus,
180x= 30×300
180x = 9000
x= 50s.
Hope this helps! :)