The equation of the quadratic function is f(x) = x²+ 2/3x - 1/9
How to determine the quadratic equation?From the question, the given parameters are:
Roots = (-1 - √2)/3 and (-1 + √2)/3
The quadratic equation is then calculated as
f(x) = The products of (x - roots)
Substitute the known values in the above equation
So, we have the following equation
[tex]f(x) = (x - \frac{-1-\sqrt{2}}{3})(x - \frac{-1+\sqrt{2}}{3})[/tex]
This gives
[tex]f(x) = (x + \frac{1+\sqrt{2}}{3})(x + \frac{1-\sqrt{2}}{3})[/tex]
Evaluate the products
[tex]f(x) = (x^2 + \frac{1+\sqrt{2}}{3}x + \frac{1-\sqrt{2}}{3}x + (\frac{1-\sqrt{2}}{3})(\frac{1+\sqrt{2}}{3})[/tex]
Evaluate the like terms
[tex]f(x) = x^2 + \frac{2}{3}x - \frac{1}{9}[/tex]
So, we have
f(x) = x²+ 2/3x - 1/9
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AB, CD, and EF intersect at point O. Find m AOC, m BOF, m COF, and m COE.
Answer:
Step-by-step explanation:
Based on the given proof that the triangles are congruent
Answer:
They are congruent by SAS:
Side: AB = CB
Angle: ∠ABD = ∠CBD
Side: BD = BD
Explanation:
Two triangles are congruent if they have the same interior angles and the same length of their corresponding sides.
If two of the angles are equal, we can say that all the interior angles are equal.
So, based on the given information we get:
1. AC ⊥ BD means that AC is perpendicular to BD, therefore,
∠ADB = ∠CDB = 90°
2. It is given that ∠A = ∠C
3. Since both of their interior angles are equal, we can say that all the interior angles are equal and:
∠ABD = ∠CBD
4. The triangle ABC is isosceles because two of their interior angles are equal, therefore AB = CB
5. BD = BD because it is the same segment.
6. By SAS (Side-angle-Side) we can say that ΔABD = ΔCBD
Because the side AB is equal to side CB, the angle ABD is equal to angle CBD, and the side BD is equal to itself.
Tara's office recycled a total of 60 kilograms of paper over 3 weeks. How many weeks will it take Tara's office to recycle a total of 80 kilograms of paper? Solve using unit rates.
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can someone help me understand the working of how to get the answer??
A) The ingredient she needed = Fruit juice = 21 litres
Lemonade = 14 litres
B) The extra punch she can make = 20litres
C) The ratio of fruit juice and lemonade for second batch = 7:6
What is Ratio?
A ratio in mathematics demonstrates how many times one number is present in another. For instance, if a dish of fruit contains eight oranges and six lemons, the ratio of oranges to lemons is eight to six. Similarly, the ratio of oranges to the overall amount of fruit is 8:14, while the ratio of lemons to oranges is 6:8.
A)
Let 3x be fruit juice and 2x be the lemonade
3x + 2x = 35litres
5x = 35
x = 35/5
x = 7
Then, required ingredient she needed
Fruit juice = 3x7 = 21 litres
Lemonade = 2x7 = 14 litres
B)
According to question
The amount of fruit juice she had = 12litre
Let x be 12litre
as we know that the ratio of fruit juice to the lemonade is 3 : 2
Thus,
Fruit juice = 3(x) = 12 litres
x = 12/3
x = 4
Fruit juice = 3 x 4 = 12litres
putting the value in lemonade
Lemonade = 2(4) = 8 litres
So, The extra punch she can make :
12 + 8 = 20litres
C)
The ratio for new second batch she added 4litres in lemonade
Then it become :
Fruit juice = 21 litres
Lemonade = 14 litres + 4litres = 18
The ratio will be
21:18
7:6
Thus, the ratio of fruit juice to lemonade in second batch will be 7:6
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there are grass in a farm and the same amount of grass is grown each day. it takes 10 days for 17 cows to eat all grass in the farm. it takes 12 days for 15 cows to eat all grass in the farm. how many days does it take for 7 cows to eat all grass in the farm?
7 cows will eat all the grass on the farm in 28 days
17 cows eat all the grass on the farm in 10 days
15 cows eat all the grass in the farm in 12 days
We can see a relationship between the number of cows and the number of days. A reduction of 2 cows leads to an increase in 2 days for the cows to eat the grass.
As a result, when we reduce the cows by 4 the increase in the number of days will be 8.
Hence, to reduce the cows from 15 to 7 we will reduce the number of cows by 8 resulting in an increase in 16 days for the grass on the farm to be eaten up
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Solve 7+y=5(2y-1)+3y. Y=
Answer: y = 1
Step-by-step explanation:
Given
7 + y = 5(2y - 1) + 3y ← distribute and simplify right side
7 + y = 10y - 5 + 3y
7 + y = 13y - 5 ( subtract y from both sides )
7 = 12y - 5 ( add 5 to both sides )
12 = 12y ( divide both sides by 12 )
1 = y
(Pythagorean Theorem MC)
Determine which set of side measurements could be used to form a right triangle.
O 9, 11, 13
O 6, 12, 17
O√8, 5, √17
O√√2. √7.9
Question 3(Multiple Choice Worth 2 points)
Previous Question
Question 1 (Answered)
>
pls helpppp
Using Pythagoras's theorem, the set of measurement that can be used to form a right triangle is √2, √7 and 9.
How to find the sides of a right triangle?A right angle triangle is a triangle that has one of its angles as 90 degrees.
The sides of a right angle triangle can be found using Pythagoras's theorem.
Therefore, Pythagoras's theorem is described as follows;
c² = a² + b²
where
a and b are the legs of the right trianglec is the hypotenuse of the right triangle.Therefore, the sides of the triangle that can be used to form a right triangle will have to obey the Pythagoras's theorem,
(√2)² + (√7)² = 9
2 + 7 = 9
Therefore, the last option can form a right triangle.
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Andrew earns 30% on all sales plus a retainer of £125 per week.
If he earns £893 in one week, find the value of his sales for that week.
If Andrew earns 30% on all sales plus a retainer of $125 per week and If he earns $893 in one week, then the value of the his sales for that week is $2560.
The total amount he earned = $893
The amount of retainer he earned = $125
The percentage he earned = 30%
30% of the sales = The total amount he earned - The amount of retainer he earned
Substitute the values in the equation
= 893-125
= $768
Consider the total sales as x
Then,
x × (30/100) = 768
0.3x = 768
x = 768/0.3
x = $2560
Hence, if Andrew earns 30% on all sales plus a retainer of $125 per week and If he earns $893 in one week, then the value of the his sales for that week is $2560.
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Find the requested angle supplement of 123^%A/246B/33C/57D/237
Given: The angle
[tex]123^0[/tex]To Determine: The supplement of the given angle
Solution
Please note that two angle are supplement if they are add up to 180⁰
So if the supplement is x. Therefore
[tex]\begin{gathered} x+123^0=180^0 \\ x=180^0-123^0 \\ x=57^0 \end{gathered}[/tex]Hence, the supplement of 123⁰ is 57⁰, OPTION C
Find the distance between the two points rounding to the nearest tenth (if necessary).
(6,7) and (4,−1)
Answer:
8.2 units (nearest tenth)
Step-by-step explanation:
Distance between two points
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
where (x₁, y₁) and (x₂, y₂) are the two points.
Given points:
(x₁, y₁) = (6, 7)(x₂, y₂) = (4, -1)Substitute the given points into the distance formula and solve for d:
[tex]\implies d=\sqrt{(4-6)^2+(-1-7)^2}[/tex]
[tex]\implies d=\sqrt{(-2)^2+(-8)^2}[/tex]
[tex]\implies d=\sqrt{4+64}[/tex]
[tex]\implies d=\sqrt{68}[/tex]
[tex]\implies d=8.2 \; \sf (nearest\;tenth)[/tex]
Therefore, the distance between the two given points is 8.2 units to the nearest tenth.
This is an online ACT prep guide problem that I’m having trouble on.Below, is the answer options to this problem.A. -4B. 1C. 6D. The limit does not exist.
Step 1
Given; A graph
Required; To find f(-4)
Step 2
The graph at x=-4 has a removable discontinuity. Therefore we can then conclude that f(-4) is undefined.
Hence, f(-4) is undefined
A removable discontinuity is a point on the graph that is undefined or does not fit into the rest of the graph.
what is the rule for this pattern?
17, 20, 19, 22, 21
Answer:
+3, -1
Step-by-step explanation:
17 + 3 = 20
20 - 1 = 19
19 + 3 = 22
22 - 1 = 21
Please mark me brainliest if this helps!
The senior class sells hamburgers and hot dogs at a football game and makes a profit of $1.75 on each hamburger and $1.25 on each hot dog. The class would like a profit of at least $280. Let x represent the number of hamburgers and y represent the number of hot dogs sold
Answer:
1.75x + 1.25y >= 280
Step-by-step explanation:
1.75x + 1.25y >= 280
Which of the following options results in a graph that shows exponential decay?
Group of answer choices
f(x) = 0.6(2)x
f(x) = 3(0.7)x
f(x) = 0.4(1.6)x
f(x) = 20(3)x
The equation of a function that results in a graph that shows exponential decay is f(x) = 3(0.7)ˣ
How to determine the function that represents an exponential decay?An exponential function is represented as
f(x) = abˣ
Where
Variable a represents the initial valueVariable b represents the rateA function that represents an exponential decay would have a rate less than 1
This is represented as b <1
From the list of options, we have
f(x) = 3(0.7)ˣ
By comparing f(x) = 3(0.7)ˣ and f(x) = abˣ, we have
b = 0.7
0.7 is less than 1
Hence, the exponential decay function is f(x) = 3(0.7)ˣ
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A balcony at an apartment can hold up to 1,816 kilograms. How many 160-pound people can the balcony hold?
The balcony can hold 25 people of 160-pound weight.
According to the question,
We have the following information:
The maximum weight balcony can hold = 1816 kg
Weight of 1 person = 160 pound
Now, we will convert this weight into kilogram so that both the measurements have same units.
1 pound = 0.454 kg
160 pound = 160*0.454 kg
160 pound = 72.64 kg
Now, let's take the number of people balcony can hold to be x.
So, we have:
72.64x = 1816
x = 1816/72.64
x = 25
Hence, the number of people the given balcony can hold is 25.
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◄) Solve for w.
29 + 17w = 624
W =
Helppp
Answer:
w = 35
Step-by-step explanation:
29 + 17w = 624
subtract 29 from both side
17 w = 595
divide both sides by 17
w = 35
an environmentalist wishes to conduct a hypothesis test on the percentage of cars driven in the city that are hybrids. is it sufficient for him to use a simple random sample of 172172 cars if hybrids currently account for 6%6% of the car sales in the country and he claims that the percentage of hybrids in the city is higher than that?
It is enough for him to use a simple random sample of 172 cars.
It is required that there are at least 10 successes and 10 failures to test a hypothesis that involves a proportion p in a sample of size n.
Following conditions are required for it:
1) np ≥ 10
2) n (1-p) ≥ 10
p = 6% = 0.06
n = 172
np = 172x0.06 = 10.322 ≥ 10172 (1-0.06) = 161.68 ≥ 10So, there are at least 10 successes and 10 failures in the 172 cars sample, it is enough for him to use a simple random sample of 172 cars.
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8
(b) Then determine whether the above relation is a function.
A) Function
B Not a function
The above relation is not a function
What is a function?
A function from either a set X to just a set Y allocates precisely one element of Y to each element of X. The set X is known as the function's domain, while the set Y is known as the function's codomain. The Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi are credited with the first known attempt to the concept of function. Originally, functions were the idealization of how a variable quantity depended on another quantity. A planet's location, for example, is a function of time. Historically, the notion was developed with the infinitesimal calculus at the end of the 17th century, and the functions investigated were differentiable until the 19th century.
The above relation is not a function
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9y = -2x is this true or false
this is a little hard for me help me out please
The simplified form of the expression as a fraction is -43/390.
Solving expressions and simplifying fractionsFractions are expression written as a ratio of two integers. For instance, a/b are known as fractions.
Given the expression below:
[2-|-2/3-2(-1/5)|]÷(-13)
In order to solve the expression, we will use the PEMDAS rule to have:
[2-|-2/3-2(-1/5)|]÷(-13)
[2-|-2/3+1/10)|]÷(-13)
Find the LCM
[2-|-20+3/30|]÷(-13)
[2-|-17/30|]÷(-13)
[2-(17/30)]÷(-13)
[60-17/30]÷(-13)
43/30÷(-13)
Change the division sign to multiplication
43/30÷(-13)
43/30 * -1/13
-43/390
This gives the simplified form of the expression.
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multistage cluster sampling with stratification, systematic sampling, simple random sampling and disproportionate stratified sampling are examples of:
Multistage cluster sampling with stratification, systematic sampling, simple random sampling and disproportionate stratified sampling are examples of probability sampling.Probability sampling is the process of selecting a sample from a population when the selection is based on the randomization principle, often known as chance or random selection.
What does stratified multistage cluster sampling mean?Multistage sampling, also known as multistage cluster sampling, involves taking a sample from a population in successively smaller groupings. In national surveys, for instance, this technique is frequently employed to collect data from a sizable, geographically dispersed population.For instance, a researcher might be interested in the various eating customs throughout western Europe. It is essentially impossible to gather information from every home. The researcher will first pick the target nations. He or she selects the states or regions to survey from among these nations.
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A rectangles width (W) is 5 inches more than a third the length (L). If the perimeter is 42 inches what are the dimensions of the rectangle ?
The length and width of the rectangle are 5.5 inches and 10.5 inches.
What do we mean by rectangles?A rectangle is a quadrilateral with four right angles in Euclidean plane geometry.It can also be explained in terms of a parallelogram with a right angle or an equiangular quadrilateral—a quadrilateral whose angles are all equal.A square is an asymmetrical shape with four sides of equal length.So, the dimensions of the rectangle:
Let the length of the rectangle be 'x' and the width be 'x + 5'.The perimeter is 42 inches.Perimeter formula: 2(l + b)Substitute the values in the formula as follows:
P = 2(l + b)42 = 2(x + x + 5)42 = 2(2x + 5)42 = 4x + 204x = 42 - 204x = 22x = 22/4x = 5.5 inches (Length)x + 5 = 5.5 + 5 = 10.5 inches (Width )Therefore, the length and width of the rectangle are 5.5 inches and 10.5 inches.
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3. What is true for an image and a preimage in a reflection? (1 point)
The image is larger than the preimage.
The image is smaller than the preimage.
The image and the preimage have the same orientation.
The image and the preimage have different orientations.
Answer: The image and the preimage have the same orientation.
Answer:
The image and the preimage have the different orientation.
Step-by-step explanation:
i
took
the
test
Point (2,11.9),(4,10.5)
(2,8) , (4,7.5)
(2,6) , (4,1.5)
(2,0.5) , (4.-1)
(2,-1) , (4,-1)
(-2.8,-5.0) , (-3.6,-5.9)
rise over run so what's the answer
The slopes formed by each of the pairs of points are given as follows:
(2,11.9),(4,10.5): -0.7.(2,8) , (4,7.5): -0.25.(2,6) , (4,1.5): -2.25.(2,0.5) , (4.-1): -0.75.(2,-1) , (4,-1): 0.(-2.8,-5.0) , (-3.6,-5.9): 0.89.SlopeThe slope is equivalent to the rate of change, hence, given two points, the slope is calculated as the change in y, also called rise, of these two points divided by the change in x, also called run.
For the first case, the points are:
(2, 11.9) and (4, 10.5)
Hence:
The change in x is of 4 - 2 = 2.The change in y is of 10.5 - 11.9 = -1.4.The slope is:
m = -1.4/2 = -0.7.
The other slopes are calculated as follows:
(2,8) , (4,7.5): m = (7.5 - 8)/(4 - 2) = -0.25.(2,6) , (4,1.5): m = (1.5 - 6)/(4 - 2) = -2.25.(2,0.5) , (4.-1): m = (-1 - 0.5)/(4 - 2) = -0.75.(2,-1) , (4,-1): m = (-1 - (-1))/(4 - 2) = 0. (no change in the output).(-2.8,-5.0) , (-3.6,-5.9): m = (-3.6 - (-2.8))/(-5.9 - (-5)) = 0.89.More can be learned about slopes at https://brainly.com/question/28954211
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suppose that the antenna lengths of woodlice are approximately normally distributed with a mean of inches and a standard deviation of inches. what proportion of woodlice have antenna lengths that are more than inches? round your answer to at least four decimal places.
Proportion = 0.2119
P (X is less than or equal to 0.18) = P [(X-μ)/ sigma is less than or equal to (0.18-0.22)/ 0.05] = P(Z is less than or equal to -0.80). Using z-table proportion = 0.2119
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consider the experiment of a worker assembling a product. a. define a random variable that represents the time in minutes required to assemble the product. b. what values may the random variable assume? c. is the random variable discrete or continuous?
(a) Let "X" be the random variable that represents the time in minutes required to assemble the product.
(b) The random variable may assume the values in the interval (0, ∞).
(c) The random variable "X" is continuous in nature.
A random variable is a mathematical representation of a number or object that is affected by random occurrences. It is a mapping or function between probable outcomes in a sample space and a measured space, which is frequently real numbers.A random variable is a variable with an unknown value or a function that gives values to each of the results of an experiment. A random variable might be discrete (with fixed values) or continuous (any value in a continuous range).To learn more about variables, visit :
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find the 2 points if the (x,-1) which are 4 units from the pooint (3,2)
The possible coordinates of the other points are (3 + √7, -1) and (3 - √7, -1)
How to calculate the coordinates of the two points?From the question, we have
Points = (3, 2) and (x, -1)Distance = 4 unitsWhere (x, -1) represents the other points
The distance between the points is the number of units between them
It is calculated using the following distance formula
d = √[(x₁ - x₂)²+ (y₁ - y₂)²]
Where x and y represent the coordinates of the given points
Substitute the known values in d = √[(x₁ - x₂)²+ (y₁ - y₂)²]
So, we have
d = √[(3 - x)²+ (2 + 1)²]
Evaluate the expression
d = √[(3 - x)²+ 9]
Recall that d = 4
So, we have
√[(3 - x)²+ 9] = 4
Square both sides
(3 - x)²+ 9 = 16
This gives
(3 - x)² = 7
So, we have
3 - x = ±√7
Solve for x
x = 3 ± √7
Hence, the coordinates are (3 + √7, -1) and (3 - √7, -1)
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what is the arithmetic mean of all of the positive two-digit integers with the property that the integer is equal to the sum of its first digit plus its second digit plus the product of its two digits?
The arithmetic mean is 59.
What is arithmetic mean?
It is the sum of collection of numbers divided by the count of the numbers.
Conider AB is the nuber satisfying the condition. Hence,
[tex]10A+B=A+B+A\times B\\9A=A\times B\\[/tex]
Since AB is a two digit number hence, [tex]A\neq 0\\[/tex]. Hence, divide both sides by [tex]A[/tex].
[tex]9=B[/tex]
Hence, B is 9 and A can take any value from 1 to 9.
Hence, numbers are 19, 29, 39, 49, 59, 69, 79, 89,99.
Now, calculate arithmetic mean as follows:
[tex]AM=\frac{Sum \ of \ numbes}{Count \ of \ numbers}\\=\frac{19+29+39+49+59+69+79+89+99}{9}\\=59[/tex]
Hence, arithmetic mean of numbers is 59.
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Help on this question
Answer: Yes
Step-by-step explanation: 250-250
An outlier is when some of the data is spread farther away from the rest of the data. It is far off from other data
Q1 A ball is thrown upwards with some initial speed. It goes up to a height of 19.6m and then returns. Find (a) The initial speed (b) The time taken in reaching the highest point (c) The velocity of the ball one second before and one second after it reaches the maximum height (d) The time taken by the ball to return to its original position
Answer:
(a) 19.6 ms⁻¹
(b) 2 s
(c) 9.8 ms⁻¹
(d) 4 s
Step-by-step explanation:
Constant Acceleration Equations (SUVAT)
[tex]\boxed{\begin{array}{c}\begin{aligned}v&=u+at\\\\s&=ut+\dfrac{1}{2}at^2\\\\ s&=\left(\dfrac{u+v}{2}\right)t\\\\v^2&=u^2+2as\\\\s&=vt-\dfrac{1}{2}at^2\end{aligned}\end{array}} \quad \boxed{\begin{minipage}{4.6 cm}$s$ = displacement in m\\\\$u$ = initial velocity in ms$^{-1}$\\\\$v$ = final velocity in ms$^{-1}$\\\\$a$ = acceleration in ms$^{-2}$\\\\$t$ = time in s (seconds)\end{minipage}}[/tex]
When using SUVAT, assume the object is modeled as a particle and that acceleration is constant.
Acceleration due to gravity = 9.8 ms⁻².
Part (a)When the ball reaches its maximum height, its velocity will momentarily be zero.
Given values (taking up as positive):
[tex]s=19.6 \quad v=0 \quad a=-9.8[/tex]
[tex]\begin{aligned}\textsf{Using} \quad v^2&=u^2+2as\\\\\textsf{Substitute the given values:}\\0^2&=u^2+2(-9.8)(19.6)\\0&=u^2-384.16\\u^2&=384.16\\u&=\sqrt{384.16}\\\implies u&=19.6\; \sf ms^{-1}\end{aligned}[/tex]
Therefore, the initial speed is 19.6 ms⁻¹.
Part (b)Using the same values as for part (a):
[tex]\begin{aligned}\textsf{Using} \quad s&=vt-\dfrac{1}{2}at^2\\\\\textsf{Substitute the given values:}\\19.6&=0(t)-\dfrac{1}{2}(-9.8)t^2\\19.6&=4.9t^2\\t^2&=\dfrac{19.6}{4.9}\\t^2&=4\\t&=\sqrt{4}\\\implies t&=2\; \sf s\end{aligned}[/tex]
Therefore, the time taken to reach the highest point is 2 seconds.
Part (c)As the ball reaches its maximum height at 2 seconds, one second before this time is 1 s.
Given values (taking up as positive):
[tex]u=19.6 \quad a=-9.8 \quad t=1[/tex]
[tex]\begin{aligned}\textsf{Using} \quad v&=u+at\\\\\textsf{Substitute the given values:}\\v&=19.6+(-9.8)(1)\\v&=19.6-9.8\\\implies v&=9.8\; \sf ms^{-1}\end{aligned}[/tex]
The velocity of the ball one second before it reaches its maximum height is the same as the velocity one second after.
Proof
When the ball reaches its maximum height, its velocity is zero.
Therefore, the values for the downwards journey (from when it reaches its maximum height):
[tex]u=0 \quad a=9.8 \quad t=1[/tex]
(acceleration is now positive as we are taking ↓ as positive).
[tex]\begin{aligned}\textsf{Using} \quad v&=u+at\\\\\textsf{Substitute the given values:}\\v&=0+9.8(1)\\\implies v&=9.8\; \sf ms^{-1}\end{aligned}[/tex]
Therefore, the velocity of the ball one second before and one second after it reaches the maximum height is 9.8 ms⁻¹.
Part (d)From part (a) we know that the time taken to reach the highest point is 2 seconds. Therefore, the time taken by the ball to travel from the highest point to its original position will also be 2 seconds.
Therefore, the total time taken by the ball to return to its original position after it is thrown upwards is 4 seconds.