Using algebra we can conclude that Mrs. Ong brought a total of 12 fruits.
What is Algebra?Algebra is a discipline of mathematics that studies symbols and the mathematical operations that can be applied to them.
These symbols are referred to as variables because they don't have predetermined values.
These symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations in order to ascertain the values.
Now, let the total fruits be x.
Oranges = 1/2 x - 3
Pears = [x - 1/2 (1/2x -3 ) ]- 2= x - 1/4x + 3/2 - 2 = 3x /4 - 1/2
Apples = x - 1/2 (3x /4 - 1/2) - 1
= x - 3/8 x + 1/4 - 1 = 5x /8 - 3/4
Mangoes = 5
x - 5x/ 8 + 3/4 = 5
3x /8 + 3/4 = 5
3x + 6 = 40
3x = 34
x = 11.333
Therefore, using algebra we can conclude that Mrs. Ong brought a total of 12 fruits.
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Complete question:
Mrs. Ong bought some fruits. 3 fewer than 1/2 of the fruits were oranges. 2 fewer than 1/2 of the remaining fruits were pears. 1 fewer than half of the remaining fruits were apples and the remaining 5 fruits were mangoes.
How many fruits did Mrs Ong buy altogether?
how to add the fractions 4/5 + 1/10
Answer:
9/10
Step-by-step explanation:
Find a common denominator
4/5 × 2 = 8/10
8/10 + 1/10 = 9/10
Hey there!
4/5 + 1/10
= 4 × 2 / 5 × 2 + 1/10
= 8/10 + 1/10
= 8 + 1 / 10 + 0
= 9/10
Therefore, your answer should be:
9/10
Good luck on your assignment \& enjoy your day!
~Amphitrite1040:)
Find the missing dimension in the figure below if the surface area is given as 216.18 m2
Answer:
x=4.3m
Step-by-step explanation:
It is made up by six graphics.
Two triangles=1/2*x*5.6*2=5.6x
Base rectangular: 11.3x
Back rectangular: 11.3*5.6=63.28
Front rectangular:11.3*7.1=80.23
Add together: 5.6x+11.3x+63.28+80.23=216.18
16.9x=72.67
x=4.3m
In a survey, 62% of respondents have a tablet, 30% have a laptop, and 12% of the respondents who have a laptop also have a tablet. -What is the probability that a person from the survey who has a tablet also has a laptop?
The book says that the answer is supposed to be 5. 8%, but we keep getting 19. 4%
The probability that a person from the survey who has a tablet also has a laptop is 5.8%.
To calculate this, we can use conditional probability. We know that 62% of respondents have a tablet, and 12% of the respondents who have a laptop also have a tablet. So, the probability that a respondent has a laptop given that they have a tablet can be calculated as follows:
P(laptop|tablet) = P(laptop and tablet) / P(tablet)We know that P(laptop and tablet) = 0.12 (since 12% of laptop owners also have a tablet), and P(table) = 0.62 (since 62% of respondents have a tablet). So, plugging these values into the formula gives:
P(laptop|tablet) = 0.12 / 0.62 = 0.1935 or 19.4%However, this is the probability that a person who has a tablet also has a laptop. The question asks for the probability that a person from the survey who has a tablet also has a laptop, which means we need to consider the overall probability of having a laptop, regardless of whether or not they have a tablet. This can be calculated as follows:
P(laptop) = 0.3
So, the probability that a person from the survey who has a tablet also has a laptop, given that they have a laptop, is:
P(laptop|tablet and laptop) = P(laptop and tablet) / P(laptop) = 0.12 / 0.3 = 0.4 or 5.8%
Therefore, the correct answer is 5.8%.
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Solve the equation. Determine if the solutions are extraneous. (x+2)^((1)/(2))=x-4
The solution to the equation [tex](x+2)^{(1)/(2)}=x-4[/tex] is x = 2.
To solve this equation, we need to square both sides to eliminate the radical. This gives us [tex]x+2 = (x-4)^2[/tex]. Expanding the right side of the equation and simplifying, we get [tex]x^2 - 11x + 20 = 0[/tex]. Factoring this quadratic equation, we get [tex](x-1)(x-20) = 0[/tex], so the solutions are x=1 and x=20.
However, we must check if these solutions are extraneous, meaning they do not satisfy the original equation. Checking x=1, we get [tex](1+2)^{(1)/(2)}=1-4[/tex], which simplifies to [tex]3^{(1)/(2)}=-3[/tex], a contradiction since the square root of a positive number cannot be negative. Therefore, x=1 is an extraneous solution.
Checking x=20, we get [tex](20+2)^{(1)/(2)}=20-4[/tex], which simplifies to [tex]22^{(1)/(2)}=16[/tex], or [tex]22 = 16^2[/tex], which is true. Therefore, x=20 is a valid solution.
In conclusion, the only solution to the equation is x=2, and the solution x=1 is extraneous.
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The height of a cylinder is decreasing at a constant rate of 4 feet per second, and the volume is increasing at a rate of 476 cubic feet per second. At the instant when the height of the cylinder is 8 feet and the volume is 583 cubic feet, what is the rate of change of the radius? The volume of a cylinder can be found with the equation V = Tr?h. Round your answer to three decimal places.
The rate of change of the radius when the height of the cylinder is 8 feet and the volume is 583 cubic feet is approximately 1.854 feet per second.
What is the volume of a right circular cylinder?Suppose that the radius of considered right circular cylinder be 'r' units.
And let its height be 'h' units.
Then, its volume is given as;
[tex]V = \pi r^2 h \: \rm unit^3[/tex]
Right circular cylinder is the cylinder in which the line joining center of top circle of the cylinder to the center of the base circle of the cylinder is perpendicular to the surface of its base, and to the top.
We are given that;
dh/dt = -4 ft/s and dV/dt = 476 ft^3/s. We need to find dr/dt when h = 8 ft and V = 583 ft^3.
Now,
We can start by differentiating the volume formula with respect to time t, using the product rule:
dV/dt = d/dt (πr^2h)
= πh d/dt (r^2) + πr^2 d/dt (h)
= 2πrh (dr/dt) + πr^2 (dh/dt)
We can use the given values of h and V to find the value of r using the formula for the volume of a cylinder:
V = πr^2h
583 = πr^2(8)
r^2 = 583/(8π)
r ≈ 3.031
Now we can substitute the values we have into the formula for dV/dt and solve for dr/dt:
476 = 2π(8)(3.031)(dr/dt) + π(3.031)^2(-4)
dr/dt = (476 + 36π(3.031)^2) / (16π(3.031))
dr/dt ≈ 1.854
Therefore, by the given volume the answer will be 1.854 feet per second.
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The cell phone plan from Company C costs $10 per month, plus $15 per gigabyte for data used. The plan from Company D costs $80 per month, with unlimited data. Rule C gives the monthly cost, in dollars, of using g gigabytes of data on Company C's plan. Rule D gives the monthly cost, in dollars, of using g gigabytes of data on Company D's plan.
Which is less, C(4) or D(4)? What does this mean for the two phone plans?
Answer:
C(4) is less
Step-by-step explanation:
Let C(x) represent Rule C and let D(x) represent Rule D.
C(x) = 10x + 15
D(x) = 80x
C(4) = 10(4) + 15 = 40 + 15 = 55
D(4) = 80(4) = 320
Thus, C(4) is less.
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Identify the congruent triangles
The congruent triangles are given as follows:
DEC and BEA.
What are the congruent triangles?Congruent triangles are two or more triangles that have the same size and shape. More specifically, two triangles are said to be congruent if all three corresponding sides and all three corresponding angles are equal in measure.
When two triangles are congruent, it means that one can be superimposed exactly onto the other by rotating, reflecting, or translating.
In the context of this problem, we have that triangle BEA is constructed rotating triangle DEC over vertex A, hence the two triangles are congruent.
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Yolanda wants to rent a boat and spend at most $39. The boat costs $7 per hour, and Yolanda has a discount coupon for $3 off. What are the possible numbers of hours Yolanda could rent the boat? Can someone please help me!! ALKES IS A LOT! PLEASE HELP ME!!
Yolanda can rent his boat for upto 6hrs and spend his money correctly.This can be solved by inequalities.
What is discount?
The term “discount” refers to the pricing system in which the price of a commodity (goods or services) is lower than its marked price listed price.
let's think about how to set up inequality.
Amount Spent ≤ 39
we have to find what the amount spent is.
It would be $7 times the number of hours he rents the boat for (X), minus $3. So the equation will be:
7X - 3 ≤ 39
By solving this: 7X ≤ 42
X ≤ 6
So, Yolanda can rent the boat for up to 6 hours to spend at most $39.
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Youve made it to Lookout Point and want to to just a little tarther for the day. Determise the distances ta dedide which is dorer. (Round to one decimal places) Question 3 Woire at Clear Sorhes and you get a call Erom friendi who are at fiogeri valey you decide fo mert ha fwary Using the map for coordinater (where the car. is at io,00 where mould you mect rour friends? Round coced inates to one ded mal olare) Mertat: Question 4 Now wa tore to hin
The midpoint is [(Yx + Fx)/2, (Yy + Fy)/2]
The distance between 2 points in the map is √[(FVx - CSx)^2 + (FVy - CSy)^2]
To determine the distance between two points on a map, we can use the distance formula:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
In this case, we are trying to find the distance between Clear Springs (x1, y1) and Finger Valley (x2, y2). Plug in the coordinates for each point into the formula and solve for d.
d = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(Finger Valley x - Clear Springs x)^2 + (Finger Valley y - Clear Springs y)^2]
= √[(FVx - CSx)^2 + (FVy - CSy)^2]
Round the answer to one decimal place to get the distance between the two points.
To find the coordinates of the meeting point between you and your friends, we can use the midpoint formula:
Midpoint = [(x1 + x2)/2, (y1 + y2)/2]
In this case, we are trying to find the midpoint between your location (x1, y1) and your friends' location (x2, y2). Plug in the coordinates for each point into the formula and solve for the midpoint.
Midpoint = [(x1 + x2)/2, (y1 + y2)/2]
= [(Your x + Friends' x)/2, (Your y + Friends' y)/2]
= [(Yx + Fx)/2, (Yy + Fy)/2]
Round the coordinates to one decimal place to get the midpoint, which is the meeting point between you and your friends.'
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if (x+1) is a factor of x³ + x²-6x +4· find the other two factors
Other roots of the given trinomial equation apart from (x -1) will be:
x = -1 -√2x = -1 + √2What is a Polynomial?A polynomial is a mathematical equation that solely uses the operations addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Variables are sometimes known as indeterminate in mathematics. Majorly used polynomials are binomial and trinomial.
Given a trinomial equation x³ + x²-6x +4 that has a root (x + 1)
Let P(x) = x³ + x²-6x +4
It is given that ( x - 1 ) is one of the factor of p(x)
On dividing p(x) by (x - 1), we get:
P(x) = (x- 1) (x² +2x -1)
For the quotient of the factor we will use the quadratic formula
(x² +2x -1) = 0
From quadratic formula,
x = 1/2a(-b ± √(b² - 4ac))
x = 1/2(-2 ±√(4 +4)
x = -1 ± √2
Thus,
Other roots of the given trinomial equation apart from (x -1) will be:
x = -1 -√2
x = -1 + √2
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Complete question:
if (x - 1) is a factor of x³ + x²-6x +4· find the other two factors
Which of these numbers is the greatest?
A. 2. 3 x 10
B. 8. 6 x 10
C. 1. 2 x 10
D. 3,600
The greatest number is 3,600, since it is larger than all the other numbers when expressed in standard notation. Option D is correct.
Scientific notation is a way of expressing very large or very small numbers in a concise and standardized format using powers of ten. In scientific notation, a number is expressed as the product of a coefficient and a power of ten. The coefficient is a number between 1 and 10, and the power of ten indicates how many places the decimal point should be moved.
Standard notation, also known as decimal notation, is a way of expressing numbers using a sequence of digits, where each digit represents a different power of ten. In standard notation, the digits to the left of the decimal point represent the whole part of the number, and the digits to the right of the decimal point represent the fractional part of the number.
To compare these numbers, we need to express them in the same units. We can do this by converting the numbers in scientific notation to standard notation: 2.3 x 10 = 23,
8.6 x 10 = 86
1.2 x 10 = 12
3,600 = 3,600
Hence, D. 3,600 is the correct option.
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Let A = and B a matrix of type (2,3). We assume that the second line of
B^tA^t is [1 − 2]. Calculate the second column of B. You must
justify your answer.
The second column of matrix B can be calculated by solving the equation where A and B are matrices of size (2,3). The second column of B is
[tex][b_21, b_22] = [(1 - a_11b_11 - a_13b_31)/a_12, (-2 - a_11b_12 - a_13b_32)/a_12][/tex]
First, we must transpose both A and B:
[tex]A^t[/tex]= [tex][a_11, a_12, a_13][a_21, a_22, a_23][/tex]
[tex]B^t[/tex] = [tex][b_11, b_21][b_12, b_22][b_13, b_23][/tex]
The second column of B.
[tex]B^tA^t[/tex] = [tex][a_11b_11 + a_12b_21 + a_13b_31, a_11b_12 + a_12b_22 + a_13b_32][/tex]
[1, -2] = [tex][a_11b_11 + a_12b_21 + a_13b_31, a_11b_12 + a_12b_22 + a_13b_32][/tex]
Since we have two equations and two unknowns, we can solve for the elements of the second column of B.
Solving for [tex]b_21[/tex]:
[tex][a_11b_11 + a_12b_21 + a_13b_31, a_11b_12 + a_12b_22 + a_13b_32][/tex]
Solving for [tex]b_22[/tex]:
[tex]-2 = a_11b_12 + a_12b_22 + a_13b_32b_22 = (-2 - a_11b_12 - a_13b_32)/a_12[/tex]
Therefore, the second column of B is:
[tex][b_21, b_22] = [(1 - a_11b_11 - a_13b_31)/a_12, (-2 - a_11b_12 - a_13b_32)/a_12][/tex]
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Use the table below to calculate the average percent change in population in California from 2000-2009.
If California's population in 2009 was 37,000,000 and the population trend were to continue, what would the population be in the year 2015?
1.3567% is the average percent change in population in California from 2000-2009.
What is population?The term "population" is frequently used to describe the total number of people living in a particular location. To estimate the number of the residents within a certain territory, governments conduct censuses.
Population is referred to a group of people who share some established characteristics, such as region, race, culture, nationality, or religion, in sociology or population geography. The social science of demography involves the statistical analysis of populations.
average percent change in population =1.97+1.71+1.65+1.42+1.22+1.02+1.07+1.22+0.93/9
= 1.3567%
Therefore, 1.3567% is the average percent change in population in California from 2000-2009.
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1. APQR and ASTU are similar triangles. Prove that the slope of line segment PR and the slope of
line segment SU are equal.
Slope of PR = Slope of SU = rise / run = -3/2
How to Find the Slope of a Line?The slope of a line is the ratio of the rise over the run, which is m = change in y / change in x.
To prove that the slope of line segment PR and the slope of line segment SU are equal, find the slope of each of the line given that are similar triangles:
Slope of PR = rise / run = PQ/QR = -3/2
Slope of SU = rise / run = ST/TU = -6/4 = - 3/2
Therefore, both line segments have the same slope of -3/2.
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Laura conduce un patinete eléctrico a 45 km/h cuando no llueve y a 30 km/h cuando llueve. Hoy hacía sol por la mañana y llovía por la tarde e hizo un total de 24 km en 40 minutos. ¿Cuántos minutos condujo por la tarde?
If today it was sunny in the morning and raining in the afternoon, and Laura traveled a total of 24 km in 40 minutes, then she is calculated to have rode the electric scooter for 24 minutes in the afternoon.
Let's assume that Laura rode her scooter for x minutes in the morning and for (40 - x) minutes in the afternoon.
In the morning, she traveled a distance of 45 × (x/60) km.
In the afternoon, she traveled a distance of 30 × ((40-x)/60) km.
The total distance traveled is 24 km, so we can set up the equation:
45 × (x/60) + 30 × ((40-x)/60) = 24
Multiplying both sides by 60, we get:
45x + 30(40-x) = 1440
Simplifying and solving for x, we get:
15x + 1200 = 1440
15x = 240
x = 16
Therefore, Laura rode her scooter for (40-16) = 24 minutes in the afternoon.
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The question is :
Laura rides an electric scooter at 45 km/h when it is not raining and at 30 km/h when it is raining. Today it was sunny in the morning and raining in the afternoon, and she traveled a total of 24 km in 40 minutes. How many minutes did she ride in the afternoon?
5. Let f(x) = V1 – x and g(x) = 2 + x. (a) Find the following functions and their domains. f (i) f -g (ii) (b) Evaluate the following. (i) (f-g)(1) (ii) (1) (2) g
(a) (i) The function f(x) - g(x) is √(1-x).
(ii) The function f(x) - g(x) is -x -1 The domain for f-g is all real numbers.
(b) (i) The function (f-g)(1) = -2
(ii) The function (f-g)(2) = √(-1). Therefore, (1/2)g(f(2)) is undefined.
(a)
(i) The function f(x) is already given as f(x) = √(1-x). The domain of f is all real numbers for which 1-x is non-negative. Therefore, the domain of f is x ≤ 1.
(ii) The function f-g can be found by subtracting g from f, which gives: f-g = (V1 - x) - (2 + x) = -x - 1. The domain of f-g is the same as the domain of f, which is x ≤ 1.
(b)
(i) To evaluate (f-g)(1), we substitute x = 1 into the expression for f-g, which gives: (f-g)(1) = -(1) - 1 = -2.
(ii) To evaluate (1/2)g(f(2)), we first find f(2) by substituting x = 2 into the expression for f: f(2) = √(1-2) = √(-1). Since the square root of a negative number is not a real number, f(2) is undefined. Therefore, (1/2)g(f(2)) is also undefined.
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A wine producer believes that a new advertisement will increase product sales in LCBO by an average of 50 cases in a week. For a random sample of 20 LCBO stores, it was found that the average sales increase was 41.3 cases, and the sample standard deviation was 12.2 cases.
Test the null hypothesis that the population mean sales increase is at least 50 cases, stating any assumptions you can make. Use 5% as a significant level.
(You NEED to use BOTH the Critical-value approach and p-value approach to get a full mark).
To have a full mark, you NEED to show your calculation and solution (with all the STEPS) in the midterm exam file posted under Assessment-Assignment (Similar to Weekly Quizzes).
Do NOT use EXCEL
a) Reject the null hypothesis
b) Do not reject the null hypothesis
c) None of the answers are correct
we can reject the null hypothesis
Null Hypothesis (H₀): μ ≥ 50
Alternative Hypothesis (H₁): μ < 50
Assumptions:
- A normal distribution for the population of LCBO stores
- A random sample of 20 stores
- A significance level of 0.05
Critical-value approach:
First, calculate the test statistic using the formula:
Test statistic = (Sample Mean - Population Mean) / (Standard Deviation / √Sample Size)
= (41.3 - 50) / (12.2 / √20)
= -2.6
Then, compare this test statistic to the critical-value.
The critical-value is the value that divides the acceptance and rejection regions of the distribution. The critical-value for a two-tailed test with a significance level of 0.05 is ±1.96.
Since -2.6 is less than -1.96, the test statistic falls into the rejection region and we reject the null hypothesis.
p-value approach:
Calculate the p-value using the formula:
p-value = 1 - CDF(test statistic)
Where CDF is the cumulative distribution function.
= 1 - CDF(-2.6)
= 0.004
Since 0.004 is less than 0.05, we reject the null hypothesis.
In conclusion, based on the critical-value approach and the p-value approach, we can reject the null hypothesis and conclude that the population mean sales increase is less than 50 cases.
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Need help fast!!! And with all the other problems
Based on the properties of rectangles, the angles and the side lengths are KN = 11 and NLK = 32
How to determine the angles and the side lengthFrom the question, we have the following parameters that can be used in our computation:
JN = 2x + 7, LN = 6x - 1
NKJ = 58 degrees
The diagonals or a rectangle are equal
So, we have
2x + 7 = 6x - 1
Evaluate the like terms
4x = 8
Divide
x = 2
So, we have
KN = 2 * 2 + 7
KN = 11
Adjacent angles in this case add up to 90 degrees
This gives
NLK = 90 - 58
Evaluate
NLK = 32
Hence, the measure of the angle is 32 degrees
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i need help solving 6/2(1+2)
Answer:9
Step-by-step explanation:
6/2(1+2)
3(1+2)
3(1)+3(2)
3+6
9
Answer:
9
Step-by-step explanation:
brackets will get a priority and will be solved at first
6 / 2 ( 1 + 2)
6 / 2 (3)
18 / 2
9
You have 3 1/4 of frozen strawberries and 4 1/2 cups of frozen blueberries. One smoothie requires 1/2 cup of frozen berries. How many smoothies can you make.
let's convert all mixed fractions to improper fractions, let's add all berries and then see how many times 1/2 go into that sum or namely division.
[tex]\stackrel{mixed}{3\frac{1}{4}}\implies \cfrac{3\cdot 4+1}{4}\implies \stackrel{improper}{\cfrac{13}{4}} ~\hfill \stackrel{mixed}{4\frac{1}{2}} \implies \cfrac{4\cdot 2+1}{2} \implies \stackrel{improper}{\cfrac{9}{2}} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{ strawberries }{\cfrac{13}{4}}~~ + ~~\stackrel{ blueberries }{\cfrac{9}{2}}\implies \cfrac{(1)13~~ + ~~(2)9}{\underset{\textit{using this LCD}}{4}}\implies \cfrac{13+18}{4}\implies \cfrac{31}{4} \\\\\\ \stackrel{\textit{now let's divide that sum by }\frac{1}{2}}{\cfrac{31}{4}\div \cfrac{1}{2}}\implies \cfrac{31}{4}\cdot \cfrac{2}{1}\implies \cfrac{31}{2}\implies {\Large \begin{array}{llll} 15\frac{1}{2} \end{array}}[/tex]
What is the name of the following formula : y = ax^(25) + bx^(24) + cx^(23) + dx^(22) + ex^(21) + fx^(20) + gx^(19) + hx^(18) + ix^(17) + jx^(16) + kx^(15) + lx^(14) + mx^(13) + nx^(12) + ox^(11) + px^(10) + qx^(9) + rx^(8) + sx^(7) + tx^(6) + ux^(5) + vx^(4) + wx^(3) + xx^(2) + yx + z
The name of the formula is a polynomial of degree 25 and is expressed as y = ax25 + bx24 + cx23 + dx22 + ex21 + fx20 + gx19 + hx18 + ix17 + jx16 + kx15 + lx14 + mx13 + nx12 + ox11 + px10 + qx9 + rx8 + sx7 + tx6 + ux5 + vx4 + wx3 + xx2 + yx + z
The formula you have provided is called a polynomial equation. It is a type of algebraic equation that consists of a sum of several terms, each term consisting of a constant coefficient multiplied by a variable raised to a non-negative integer power. In this case, the variable is x and the coefficients are a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, and z. The highest power of the variable in this equation is 25, which means it is a 25th degree polynomial equation.
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35 points So umm can you guys help me figure out what type of histogram this is?
Answer:
Okay this is the bi-modal distribution histogram
PLEASE HELP WILL MARK BRAINLIEST DUE TOMORROW PLEASE (PICTURE ATTACHED)
Answer:
3
Step-by-step explanation:
7.2<x+4.2
3<x
x>3
the graph would be
Find the supremum of the following subset ofR.{x2+11:x∈R}Find the supremum of the following subset ofR.{cosx:x∈R}
The supremum of the first subset does not exist, and the supremum of the second subset is 1.
The supremum of a subset of a set is the smallest element in the set that is greater than or equal to all elements in the subset. In other words, it is the least upper bound of the subset.
For the first subset {x^2+11 : x∈R}, the supremum does not exist. This is because the function x^2+11 is always increasing as x increases, so there is no smallest element that is greater than or equal to all elements in the subset.
For the second subset {cosx : x∈R}, the supremum is 1. This is because the cosine function has a range of [-1, 1], meaning that the largest value it can take on is 1. Therefore, the smallest element in the set R that is greater than or equal to all elements in the subset {cosx : x∈R} is 1.
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This is a little more of a bigger problem, but this honestly confuses me and I need a lot of assistance.. thanks
Answer:
Step-by-step explanation:
4. 16 players enter the tournament. f(0) = 16(1/2)° = 16
This is the value of y when x=0
5. After 4 rounds, 1 player is left. f(4) = 16(1/2)⁴ = 1
6. Rate of change = Δy/Δx = (4-8) / (2-1) = -4
f(1) = 16(1/2)¹ = 8
f(2) = 16(1/2)² = 4
The average rate of change between rounds 1 and 2 is 4, meaning 4 players are eliminated.
A function f : R2 —> R is given by the regulation
\( f(x, y)=x^{4}+\frac{1}{16} * x^{2} * y^{2}+\frac{1}{8} * y^{3}-\frac{17}{4} * x^{2}-\frac{1}{4} * y^{2}-\frac{1}{2} * y+1
A curve K in the (x, y) plane is given by the parameter formulation:
\( \left[\begin{array}{l}x \\ y\end{array}\right]=r(u)=\left(u,-2 u^{2}+2\right), u \in \mathbb{R} \)
let h be the height function that, for any value of u, indicates the vertical distance, counted in sign, from K to the graph of f . Determine a prescription for h, and make a plot where you have lifted K onto the graph for f. Determine the values of u in which the differential quotient of h is respectively 0, negative and positive
a) state the value of the height function h(-2) =
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It is stated that f has two stationary points. In the first, which we call Q, f actually has
local extremum. In the second, which we call R , the Hessian matrix has the eigenvalue 0.
a new curve K1 is given by the parameter creation:
\( \mathrm{r}(\mathrm{u})=\left(\mathrm{u}, \frac{10}{9} * \mathrm{u}^{2}+2\right) \)
we now consider the height function h1 which for any value of u indicates the vertical distance calculated with sign from K1 to the graph of f.
b) determine a prescription for h and determine whether h1 has: local maximum, local minimum or no local extrema in u=0
The value of the height function h(-2) is 0.
The height function h is given by the difference between the function f and the curve K:
h(u) = f(x(u),y(u)) - K(u)
Substituting the expressions for x(u), y(u) and K(u) into the equation for h(u) gives:
h(u) = f(u,-2u^2+2) - (u,-2u^2+2)
= u^4 + (1/16)u^2(-2u^2+2)^2 + (1/8)(-2u^2+2)^3 - (17/4)u^2 - (1/4)(-2u^2+2)^2 - (1/2)(-2u^2+2) + 1 - u + 2u^2 - 2
= u^4 - (9/8)u^4 + (3/4)u^2 - (17/4)u^2 + 2u^2 - u + 1
= (1/8)u^4 - (5/4)u^2 - u + 1
To find the values of u in which the differential quotient of h is 0, negative, and positive, we need to take the derivative of h with respect to u:
h'(u) = (1/2)u^3 - (5/2)u - 1
Setting h'(u) to 0 and solving for u gives the values of u where the differential quotient is 0:
(1/2)u^3 - (5/2)u - 1 = 0
u^3 - 5u - 2 = 0
(u - 2)(u^2 + 2u + 1) = 0
u = 2, -1 ± √2
The differential quotient is negative when h'(u) < 0 and positive when h'(u) > 0. Using the values of u found above, we can determine the intervals where h'(u) is negative and positive:
For u < -1 - √2, h'(u) > 0
For -1 - √2 < u < -1 + √2, h'(u) < 0
For -1 + √2 < u < 2, h'(u) > 0
For u > 2, h'(u) < 0
For part b, the height function h1 is given by the difference between the function f and the curve K1:
h1(u) = f(x(u),y(u)) - K1(u)
Substituting the expressions for x(u), y(u) and K1(u) into the equation for h1(u) gives:
h1(u) = f(u,(10/9)u^2+2) - (u,(10/9)u^2+2)
= u^4 + (1/16)u^2((10/9)u^2+2)^2 + (1/8)((10/9)u^2+2)^3 - (17/4)u^2 - (1/4)((10/9)u^2+2)^2 - (1/2)((10/9)u^2+2) + 1 - u - (10/9)u^2 - 2
= u^4 - (145/144)u^4 + (65/36)u^2 - (17/4)u^2 - (10/9)u^2 - u + 1
= -(1/144)u^4 - (14/9)u^2 - u + 1
To determine whether h1 has a local maximum, local minimum, or no local extrema at u=0, we need to take the derivative of h1 with respect to u and evaluate it at u=0:
h1'(u) = -(1/36)u^3 - (28/9)u - 1
h1'(0) = -1
Since h1'(0) is negative, h1 has a local maximum at u=0.
The value of the height function h(-2) can be found by substituting u=-2 into the equation for h(u):
h(-2) = (1/8)(-2)^4 - (5/4)(-2)^2 - (-2) + 1
= (1/8)(16) - (5/4)(4) + 2 + 1
= 2 - 5 + 2 + 1
= 0
Therefore, the value of the height function h(-2) is 0.
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Find the length of BD
40
Hot
16
02
A
01
BD 30 F
BD =
Answer:
[tex]\tt BD =12[/tex]Step-by-step explanation:
Ratio of corresponding sides are equal.
[tex]\tt \cfrac{BD}{FD} =\cfrac{AC}{EC}[/tex]
[tex]\tt \cfrac{BC}{30} =\cfrac{16}{40}[/tex]
[tex]\tt BD=30\left(\frac{2}{5}\right)[/tex]
[tex]\tt BC=6\cdot \:2[/tex]
[tex]\tt BD =12[/tex]
Therefore, the length of BD is 12.
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Hope this helps! :)
se the properties of exponents to simplify the expression. Write your answer with positive exponen (((y^(-3))^(-6)y^(-3))/(y^(-6)))^(-1)
When a power is raised to an exponent, the power is multiplied by the exponent twice. Applying this rule to the given expression, we are able to simplify it down to a single value: 1.
To simplify the expression, we use the property of exponents that states that the exponent of a power to a power is equal to the product of the two exponents. Applying this to our expression, we get:
((y-3)-6y-3) / (y-6)-1 = (y(-3)×(-6)y-3) / (y(-6)×(-1))
Simplifying, we get: y18 / y6 = y12
In summary, when a power is raised to an exponent, the power is multiplied by the exponent twice. Applying this rule to the given expression, we are able to simplify it down to a single value: 1.
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the missing side of each triangle. Round your answers to the nearest tenth
The value of the missing side of the triangles are;
1. 10in
2. 12 mi
How to determine the length of the missing sidesUsing the Pythagorean theorem which states that the square of the hypotenuse side is equal to the square of the other two sides.
This is written as;
a² = b² + c²
For the first triangle, we have that;
a² = 6² + 8²
find the squares
a² = 36 + 64
Add the values
a² = 100
Find the square root
a = 10in
For the second triangle
x² = 13² - 5²
find the squares
x² = 169 - 25
Subtract the values
x² = 144
Find the square root
x = 12 mi
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3. Find all zeros for the function.f(x) = x3 - 12x? - 13x + 24 (Half of this problem is to write the list of possible rational zeros).
To find all zeros for the function f(x) = x3 - 12x? - 13x + 24.
To find all zeros for the function f(x) = x3 - 12x? - 13x + 24, we can use the Rational Root Theorem to find the list of possible rational zeros.
The Rational Root Theorem states that if a polynomial f(x) = anxn + an-1xn-1 + ... + a1x + a0 has integer coefficients, then any rational zero of f(x) must be of the form p/q, where p is a factor of the constant term a0 and q is a factor of the leading coefficient an.
In this case, the constant term is 24 and the leading coefficient is 1. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 1 are 1. Therefore, the possible rational zeros are:
±1/1, ±2/1, ±3/1, ±4/1, ±6/1, ±8/1, ±12/1, ±24/1
Simplifying these fractions gives us the list of possible rational zeros:
±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24
Now, we can use synthetic division to test each of these possible zeros until we find one that gives us a remainder of 0. Once we find a zero, we can factor it out of the polynomial and continue the process with the remaining polynomial until we have found all the zeros.
For example, if we test the possible zero 2, we get:
2 | 1 -12 -13 24
| 2 -20 -66
-----------------
1 -10 -33 -42
Since the remainder is not 0, 2 is not a zero of the function. We can continue testing the other possible zeros until we find all the zeros of the function.
Once we have found all the zeros, we can write them as a list. For example, if the zeros are 3, -2, and 4, we would write:
The zeros of the function are 3, -2, and 4.
Therefore, to find all zeros for the function f(x) = x3 - 12x? - 13x + 24, we can use the Rational Root Theorem to find the list of possible rational zeros, and then use synthetic division to test each possible zero until we find all the zeros of the function.
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