Answer:
y''=-1.26
Step-by-step explanation:
We are given that [tex]2x^2+y^2=17[/tex]
We have evaluate the second order derivative of y w.r.t. x when x=2 and y=3.
Differentiate w.r.t x
Then , we get
[tex]4x+2yy'=0[/tex]
[tex]2x+yy'=0[/tex]
[tex]yy'=-2x[/tex]
[tex]y'=-\frac{2x}{y}[/tex]
Again differentiate w.r.t.x
Then , we get
[tex]2+(y')^2+yy''=0[/tex] [tex](u\cdot v)'=u'v+v'u)[/tex]
[tex]2+(y')^2+yy''=0[/tex]
Using value of y'
[tex]yy''=-2-(-\frac{2x}{y})^2[/tex]
[tex]y''=-\frac{2+(-\frac{2x}{y})^2}{y}[/tex]
Substitute x=2 and y=3
Then, we get [tex]y''=-\frac{2+(\frac{4}{3})^2}{3}[/tex]
[tex]y''=-\frac{18+16}{9\times 3}=-\frac{34}{27}[/tex]
Hence,y''=-1.26
Please check for me and if any are incorrect please explain!
Answer: (1) 14.4% (2) 3.78% (3) $222.48 (4) 21,176.47
(5) $425 (6) 14.3 cents per mile
Step-by-step explanation:
1) This is a calculator question. I = 0.144 --> 14.4%
[tex]A=P\bigg(1+\dfrac{r}{n}\bigg)^{nt}\\\\\bullet \text{A = Accrued amount (total amount paid)}\\\bullet \text{P = Principal (initial cost of the car)}\\\bullet \text{r = rate (interest rate in decimal form)}\\\bullet \text{n = number of times in a year (number of months)}\\\bullet \text{t = number of years}\\\\A = m \times nt\\\bullet \text{m = monthly payment amount}\\\bullet \text{nt = number of payments made}\\\implies m\times nt=P\bigg(1+\dfrac{r}{n}\bigg)^{nt}[/tex]
2) m = 93.33
nt = 36
P = 3000
n = 12
[tex]m\times nt=P\bigg(1+\dfrac{r}{n}\bigg)^{nt}\\\\\\93.33\times 36=3000\bigg(1+\dfrac{r}{12}\bigg)^{36}\\\\\\3359.88=3000\bigg(1+\dfrac{r}{12}\bigg)^{36}\\\\\\1.11996=\bigg(1+\dfrac{r}{12}\bigg)^{36}\\\\\\1.00315198=1+\dfrac{r}{12}\\\\\\0.00315198=\dfrac{r}{12}\\\\\\0.0378=r\\\\\\\large\boxed{3.78\%}=r[/tex]
3) same equation as #2 but deduct the down payment from the Principal
nt = 42
P - d = 5,555 - 555 = 5,000
r = 18% --> 0.18
n = 12
[tex]m\times nt=P\bigg(1+\dfrac{r}{n}\bigg)^{nt}\\\\\\m\times 42=(5,555-555)\bigg(1+\dfrac{.18}{12}\bigg)^{42}\\\\\\42m=5000(1.015)^{42}\\\\\\42m=5000(1.868847)\\\\\\42m=9344.23557\\\\\\m=\large\boxed{222.48}[/tex]
4) Gas + Oil = cost per mile × number of miles
1000 + 800 = 0.085x
1800 = 0.085x
[tex]\large\boxed{21,176.47}=x[/tex]
5) [tex]\text{Annual Depreciation}=\dfrac{\text{Cost of car - Trade-in value}}{\text{Years driven}}[/tex]
[tex].\qquad =\dfrac{3800-1250}{6}\\\\\\.\qquad =\dfrac{2550}{6}\\\\\\.\qquad = \large\boxed{425}[/tex]
6) [tex]\dfrac{\text{(Cost of car - Market value) + Gas & Oil + Parts, Maintenance + Insurance}}{\text{Miles driven}}[/tex]
[tex]=\dfrac{(5000-3800)+750+250+300}{17,500}\\\\\\=\dfrac{2500}{17,500}\\\\\\=0.142857\\\\=\large\boxed{14.3\ cents\ per\ mile}[/tex]
A hackberry tree has roots that reach a depth of 6 adn 5/12 meters. The top of the tree is 18.28 meters above the ground. Find the total height from the bottom of the roots to the top of the tree.
Answer:
24 209/300 m ≈ 24.69667 m ≈ 24.70 m
Step-by-step explanation:
The total height is the sum of the above-ground height and the below-ground depth:
6 5/12 m + 18.28 m = (6 + 18) m + (5/12 + 7/25) m
= 24 m + (125+84)/300 m
= 24 209/300 m ≈ 24.6966666... m ≈ 24.70 m
_____
The problem statement does not say what rounding is required. The least-resolution number is 18.28, with resolution of 1 cm, so the answer might rightly be rounded to that resolution: 24.70 m. If we take the numbers to be exact, their sum can only be represented exactly as the ratio of integers or a mixed number or a repeating decimal.
Ted brought a cooler containg 7.5 liters of water to a picnic.If 500 milliters of water milliliters of water are served to each person ,how many people can get water before the cooler is empty?
Answer:
Step-by-step explanation:
15 people can be served because a liter equals to 1000 milliliters and that means that 7 times two equals 14 plus the other 500 milliliters then that's 15.
15 people can get water before the cooler is empty.
In order to find the number of people who would be able to get water before the cooler empties, you need to divide the total quantity of water in the cooler by the quantity given to each person.
To do that, both quantities have to have the same unit. The liters should therefore be converted to milliliters.
1 liter = 1,000 milliliters
7.5 liters = 7.5 x 1,000
= 7,500 milliliters
The number of people that would get a drink is:
= Quantity of water in cooler / quantity of water per person
= 7,500 / 500
= 15 people
Find out more at https://brainly.com/question/17614855.
A farmer produces both beans and corn on her farm. If she must give up 16 bushels of corn to be able to get 6 bushels of beans, then her opportunity cost of 1 bushel of beans is ______________.
Answer:
2.67 bushels of corn
Step-by-step explanation:
We can see it in the picture.
In a study of factors that might affect memory, research participants were assigned to drink either an alcoholic or a nonalcoholic beverage prior to completing a memory test. Those who drank the nonalcoholic beverage were assigned to the group. A) survey B) control C) experimental D) correlational
Answer:
Answer is B) control group
Step-by-step explanation:
Survey is when you ask questions to a group of people. Has nothing to do in this case. Correlational does not exist but correlation means the association between two groups.
In an experiment, you apply a treatment, or a conditions that differentiates groups and measure an outcome . In this case the treatment is drinking alcohol to measure memory. The group that receives the treatment is the experimental group and the one that not receives the treatment is the control because, we are comparing or controlling alcohol consumption in this group to measure differences in memory with the experimental group.
Jane has been growing two bacteria farms. Bacteria farm Rod has a starting population of 2 bacteria, while Bacteria farm Sphere has a starting population of 8 bacteria. However, Jane starts growing Rod five hours before she starts growing Sphere. At 8 p.m., Jane checks her farms and finds that they have the exact same population. If the population of Rod doubles every hour, but the population of Sphere is quadrupled every hour, how many hours ago did she start growing Sphere?
Answer: Jane started growing Sphere 3 hours ago
Step-by-step explanation:
Farm Rod starting population (Rsp) = 2
Farm Sphere starting population (Ssp) = 8
Let´s name "Rh" the quantity of hours since Rod started growing, and
"Sh" the quantity of hours since Sphere started growing.
And, let´s name "R" the population of farm Rod at 8 p.m. and "S" the population of farm Sphere at 8 p.m.
Population of Rod doubles every hour, therefore:
R = [tex]Rsp * 2^{Rh}[/tex]
R = [tex]2(2^{Rh})[/tex]
Population of Sphere is quadrupled every hour, therefore:
S = [tex]Ssp * 4^{Rh}[/tex]
S = [tex]8(4^{Rh})[/tex]
At 8 p.m. Jane found that R = S
Therefore, at 8 p.m:
[tex]2(2^{Rh})[/tex] = [tex]8(4^{Sh})[/tex]
dividing both sides by 2
[tex]2^{Rh} =4(4^{Sh})[/tex]
adding exponents
[tex]2^{Rh} = 4^{Sh+1}[/tex]
[tex]2^{Rh} =2^{2^{Sh+1} }[/tex]
the bases are the same; exponents must be the same
Rh = 2Sh + 2 (equation 1)
And we also know that Jane started growing Rod five hours before Sphere:
Rh = Sh + 5 (equation 2)
Replacing equation 2 into equation 1:
(Sh + 5) = 2Sh + 2
5 - 2 = 2Sh - Sh
3 = Sh, or
Sh = 3
Jane started growing Sphere 3 hours ago.
You are helping Bobby find the length and width of his garden. He knows that the area of the garden is LaTeX: x^2+8x+15x 2 + 8 x + 15 square feet. The length of his garden is LaTeX: \left(x+5\right)( x + 5 )square feet.
Part 1) What is the width of the garden? Explain how you came to this conclusion and what method you used (algebra tiles, the x-method, another method, etc. )
Part 2) If LaTeX: xx is 3 feet, what would the area of the garden be? Show your work.
Answer:
If the length of the garden is (x + 5) then the width will be (x + 3)
The area of the garden is 48 sq feet
Step-by-step explanation:
The garden is x^2 + 8x + 15 sq feet
Use the factorization method and break the middle term:
x^2+3x+5x+15
Group the first two terms and last two terms:
(x^2+3x)+(5x+15)
Take out the common from each pair.
x(x+3) +5(x+3)
(x+5)(x+3)
We have already been given that x+5 is the length. Then this shows that x+5 is the length and x+3 is the width
If the length of the garden is (x + 5) then the width will be (x + 3)
The method I used is the F.O.I.L. method
The letters FOIL stand for First, Outer, Inner, Last. First means multiply the terms which occur first in each binomial. Then Outer means multiply the outermost terms in the product.
(x + 5)(x + 3) = x^2 + 3x + 5x + 15
Solve the like terms:
= x^2 + 8x + 15
b) x= 3 feet
(x + 5)(x + 3)
Substitute the value in the expression
(3 + 3)(5 + 3)= (8)(6)= 48 square feet
Area of the garden is 48 sq feet.
Simplify. Write the answer in scientific notation. HELP ASAP!!
Answer:
The answer to your question is: the last option 3.0 x 10⁻¹⁰
Step-by-step explanation:
First divide 3.3 by 1.1
3.3 / 1.1 = 3
Then subtract -8 - (2) = -8 -2 = -10
Finally use scientific notation where the power will be -10
result : 3 x 10 ⁻¹⁰
On Tuesday at 2 p.m., the ocean’s surface at the beach was at an elevation of 2.2 feet. Winston’s house is at an elevation of 12.1 feet. The elevation of his friend Tammy’s house is 3 1/2 times the elevation of Winston’s house.
a. What is the elevation of Tammy’s house?
b. How many times greater is the elevation of Tammy’s house than the elevation of the ocean’s surface at 2 p.m.?
c. On Wednesday at 9 a.m., Winston went diving. Near the beach, the ocean’s surface was at an elevation of -2.5 feet. During his deepest dive, Winston reached an elevation that was times the elevation of the ocean’s surface. What elevation did Winston reach during his deepest dive?
d. While diving, Winston stopped at a point located halfway between his deepest dive and the ocean’s surface. At what elevation was Winston when he stopped halfway?
Caleb will be going to college next year. He would like to save some money for living expenses. Select the goal that would be the most helpful for him.
I will earn a college degree in business from the University of Vermont by the time I am twenty-two years old.
I will save $2,600 in one year by depositing $50 dollars per week in a savings account.
I will save $100 per paycheck.
I will attend college.
Answer:
save 100$ each check.
Step-by-step explanation:
Answer:
I will save $2,600 in one year by depositing $50 dollars per week in a savings account.
Step-by-step explanation:
When you set a goal, it should clearly state what you want to accomplish, it has to be measurable and relevant according to what you want to do. Also, it has to establish a deadline and it should be something that you can achieve.
According to this and the options given, the goal that would be most helpful for Caleb is I will save $2,600 in one year by depositing $50 dollars per week in a savings account because it is well defined, it is measurable and establishes a deadline as he will save $50 dollars per week to get $2600 in a year. Additionally, the goal can be accomplished and it is relevant to what he wants to do that is to save money for living expenses because he is going to college next year.
The function f(x) = -X2 - 2x + 15 is shown on the graph.
What are the domain and range of the function?
O
The domain is all real numbers. The range is {yly < 16).
The domain is all real numbers. The range is {yly > 16).
The domain is xxl-5
The domain is {xl-5 5x53). The range is {yly s 16).
Answer:
Step-by-step explanation:
i can't see graph.
[tex]f(x)=-x^2-2x+15=-(x^2+2x+1-1)+15=-(x-1)^2+1+15=-(x-1)^2+16\\domain=all~real~numbers.\\[/tex]
Range≤16
so A
The domain is all real numbers. The range is y less then or equal to 16.
Quadratic functionQuadratic function is a polynomial of degree 2. It is in the form:
f(x) = ax² + bx + c
where a, b, c are constants
The domain of a function is the set of input variables while the range of a function is the set of output variables.
From the image attached, The domain is all real numbers. The range is y less then or equal to 16.
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For school photos, 1/5 of the students choose to have a blue background. 2/5 of the students chose to have purple and 1/5 chose to have a gray background. What portions of the students choose to have another background
Answer:
1/5 is the answer
Step-by-step explanation:
1/5 had blue
2/5 had purple
1/5 had gray
add those together you get 4/5
leaving you with 1/5 who chose different backgrounds.
What is the simplified form of StartRoot 64 x Superscript 16 Baseline EndRoot?
Answer:
[tex]8x^8[/tex] (Please see my interpretation of the problem.)
The problem:
Simplify [tex]\sqrt{64x^{16}}[/tex] (Please tell me if I did or didn't interpret your problem correctly. Thank you!)
Step-by-step explanation:
[tex]\sqrt{64x^{16}}[/tex] (Given)
[tex]\sqrt{64}\sqrt{x^{16}}[/tex] (if u and v are positive then [tex]\sqrt{uv}=\sqrt{u}\sqrt{v}[/tex])
[tex]8\sqrt[2]{x^{16}}[/tex] ([tex]\sqrt{ }=\sqrt[2]{}[/tex])
[tex]8x^{\frac{16}{2}}[/tex] ([tex]\sqrt[n]{x^m}=x^\frac{m}{n}[/tex])
[tex]8x^8[/tex] (simplified/reduced the fractional-exponent)
Answer:
8x8
Step-by-step explanation:
so B
Professor Green is interested in determining the average SAT score for the entire population of individuals who took the SAT. She wants to know how her class compares to the population of students who took the SAT. She finds that the average SAT score for the population is 1000. Is this score an example of a descriptive or an inferential statistic?
Answer:
Descriptive statistics
Explanation:
The population average is a descriptive statistic. It informs about how the population looks like, in this case, the population looks like having a average of 1000.
If, using her class's average where to infer the population average that is inferential statistic. But that is not the case
Paolo buys a used car for $3,500 and wants to fill it up with gasoline. His gas tank holds 15 gallons and is currently empty. Paolo has $100. One gas station sells gas for $2.50 per gallon. The station across the street sells gas for $2.25 per gallon. What is the cheapest cost to fill up Paolo's tank?
Round your answer to the nearest hundredth. Leave the dollar sign off your answer, so for example if your answer is $10.25, write 10.25.
The cheapest cost to fill the Paolo's tank = $33.75
We have Paolo who bought a used car for $3,500 and wants to fill it up with gasoline. His gas tank holds 15 gallons and is currently empty. Paolo has $100. One gas station sells gas for $2.50 per gallon. The station across the street sells gas for $2.25 per gallon.
We have to determine the cheapest cost to fill up Paolo's tank.
What is mathematics important in Price Estimation ?The price estimation is important in mathematics as it helps us to create mathematical relations among the various factors on which the price depends in order to extract accurate results.
According to question, we have -
Cost of Car = $3500.
Car's gasoline capacity = 15 gallons.
Amount of money with Paolo = $100.
Cost of gas at station A = $2.25.
Cost of gas at station B = $2.50.
Case 1 - If Paolo goes to Station A
Total cost to fill the tank = 2.25 x 15 = $33.75
Case 2 - If Paolo goes to Station B
Total cost to fill the tank = 2.50 x 15 = $37.50
Hence, the cheapest cost to fill the Paolo's tank = $33.75
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What is an algebraic expression to find the number of squares in the border of a 75x75 grid?
Answer:
Well, if the grip is 75 x 75, you can rule out that the grid is 75 squares in each border.
Step-by-step explanation:
75 x 75 = a x b
a= horizontal side lengths
b= vertical side lengths
Rule that a= 75 squares and b= 75 squares.
A species of fish was added to a lake. The population size P(t) of this species can be modeled by the following exponential function:P(t)=1500/(1+7e^-0.4t)where t is the number of years from the time the species was added to the lake.Find the initial population size of the species and the population size after 9 years. Round your answers to the nearest whole number as necessary.
Initial population size is 188 and after 9 years we get 1259 fishes.
We have given that the function
[tex]P(t)=\frac{1500}{1+7e^(-0.04t)}[/tex]
Where t is the number of years from the time the species was added to the lake.
We have to find the initial population size of the species and the population size after 9 years.
where p(0) is the population size of species and p(9) is the population size after 9 years.
Therefore we have the given function is at t=0
[tex]P(0)=\frac{1500}{1+7e^(-0.04(0)))}=\frac{1500}{1+07e^0} \\=\frac{1500}{1+7} \\=\frac{1500}{8} \\=187.5\\=188[/tex]
at t=9 we have
[tex]P(9)=\frac{1500}{1+7ex^{-0.4(9)} }\\=\frac{1500}{1+7e^{-3.6} } \\=\frac{1500}{1.191}\\ =1259.16\\=1259 fishes[/tex]
Therefore after 9 years we get 1259 fishes.
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The initial population size of the species is approximately 188. After 9 years, the population size is approximately 1260.
To find the initial population size of the species, we need to evaluate the population function ( P(t) ) at ( t = 0 ).
Given the population function:
[tex]\[ P(t) = \frac{1500}{1 + 7e^{-0.4t}} \][/tex]
1. **Initial population size (( P(0) )):**
Substitute ( t = 0 ) into the function:
[tex]\[ P(0) = \frac{1500}{1 + 7e^{-0.4 \cdot 0}} \][/tex]
[tex]\[ P(0) = \frac{1500}{1 + 7e^0} \][/tex]
[tex]\[ P(0) = \frac{1500}{1 + 7 \cdot 1} \][/tex]
[tex]\[ P(0) = \frac{1500}{1 + 7} \][/tex]
[tex]\[ P(0) = \frac{1500}{8} \][/tex]
[tex]\[ P(0) = 187.5 \][/tex]
Round to the nearest whole number:
[tex]\[ P(0) \approx 188 \][/tex]
So, the initial population size is[tex]\( \boxed{188} \).[/tex]
2. **Population size after 9 years (( P(9) )):**
Substitute ( t = 9 ) into the function:
[tex]\[ P(9) = \frac{1500}{1 + 7e^{-0.4 \cdot 9}} \][/tex]
[tex]\[ P(9) = \frac{1500}{1 + 7e^{-3.6}} \][/tex]
Calculate [tex]\( e^{-3.6} \):[/tex]
[tex]\[ e^{-3.6} \approx 0.02732 \][/tex]
Substitute this value back into the equation:
[tex]\[ P(9) = \frac{1500}{1 + 7 \cdot 0.02732} \][/tex]
[tex]\[ P(9) = \frac{1500}{1 + 0.19124} \][/tex]
[tex]\[ P(9) = \frac{1500}{1.19124} \][/tex]
[tex]\[ P(9) \approx 1260 \][/tex]
Round to the nearest whole number:
[tex]\[ P(9) \approx \boxed{1260} \][/tex]
So, the population size after 9 years is [tex]\( \boxed{1260} \).[/tex]
Solve for y.
byl – 19 =-15
If there is more than one solution, separate them with commas,
If there is no solution, click on "No solution".
Answer:
Solution: -4, 4
Step-by-step explanation:
You need to do 2 cases:
Case 1: [tex]y\geq 0[/tex]
[tex]When\ y\geq 0 => | y | = y[/tex]
y - 19 = -15
y = 19 - 15
y = 4
Case 2: [tex]y< 0[/tex]
[tex]When\ y < 0 => |y| = -y[/tex]
-y -19 = -15
-19 + 15 = y
-4 = y
Both are solutions of the equation.
Answer:
-4, 4 would be your answer
Step-by-step explanation:
Anyone know this Geometry problem?
Answer:
ST = 20
Step-by-step explanation:
RT is the sum of RS and ST
Replacing with length you get:
17 + x + 6 = 3x - 56
17 + 6 + 5 = 3x - x
28 = 2x
14 = x
ST = x + 6 = 14 + 6 = 20
Suppose you need to deliver 40 terabytes of data to your co-workers in Atlanta (200 km). You have an available 100 Mbps dedicated link for data transfer. Alternatively, your friend raises homing pigeons and would be willing to let you borrow one. You have a single 1 TB SD card you can strap to the pigeon. Assume pigeons can fly 1000 km per day and that it takes no time to copy to/from the SD card.
Answer: All data will be sent in 89.39 hours
Step-by-step explanation:
Data transfer rate: 100 Mbps=100 [tex]\frac{Mb}{s}[/tex] ×[tex]\frac{1Tb}{1024^{2} Mb}[/tex]×[tex]\frac{3600s}{h}[/tex]=0.3433 [tex]\frac{Tb}{h}[/tex]
The pigeon can fly 1000 km/day and it needs to fly 400 km (round trip).
Pigeon rate: [tex]\frac{1000km}{24h}[/tex]=41.67 [tex]\frac{km}{h}[/tex]
Time of round trip: 400 km÷[tex]\frac{1000km}{24h}[/tex]=9.6 h
So the pigeon can send 1 Tb every 9.6 hours.
If we sum the rates we can get the time for sending all the data:
40 Tb= (0.3433 Tb/h + 1 Tb/9.6h)×t
T= 89.39 hours
In a basic sine curve, where can the zeros NOT be found?
Answer:
At the maxium point
Step-by-step explanation:
A maxium point is a point where the function gets the higher value. For example, in the basic sine curve, we can find a maxium point when [tex]x=\frac{\pi }{2}[/tex]. We won't find any maxium point when x=0.
As you can see on the graph I attached, the zeros of a basic sine curve (or, said in other words, the points were x=0), are at the beginning, ath the middle and at the end of the function's graphic.
Two parallel lines are crossed by a transversal. Horizontal and parallel lines p and q are cut by transversal m. At the intersection of lines p and m, the bottom left angle is b degrees. At the intersection of lines q and m, the top right angle is 128 degrees. What is the value of b? B = 32 b = 52 b = 118 b = 128
Answer: Last option.
Step-by-step explanation:
Observe the figure attached.
In order to find the value of "b", you need to remember the definition of "Alternate interior angles". These are the pairs of angles located in the interior of the parallel lines and on opposite side of the transversal. They are congruent.
Based on this definition, you can conclude that the angle "b" and the angle that measures 128° are Alternate interior angles; therefore they are congruent.
This means that the value of "b" is:
[tex]b=128\°[/tex]
Answer:
The last answer!
Step-by-step explanation:
edge 2020
A truck can be rented from Company A for $130 a day plus $0.20 per mile. Company B charges $70 a day plus $0.30 per mile to rent the same truck. Find the number of miles in a day at which the rental costs for Company A and Company B are the same.
130+x/5=70+3x/10 x=400
Mr.Nolan's code for his ATM card is a four digit number. The digits of the code are the prime factors of 84 listed from least to greatest. What is the code for Mr.Nolan's ATM card?
Answer:
2237
Step-by-step explanation:
84 = 2·2·3·7 . . . . . . prime factorization
A certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn?
(A) 4
(B) 6
(C) 8
(D) 10
(E) 12
Answer:
The answer is (C) 8
Step-by-step explanation:
First, let's calculate the length of the side of the square.
[tex]A_{square}=a^2[/tex], where [tex]a[/tex] is the length of the side. Now, let's try to build the square. First we need to find a point which distance from (0, 0) is 10. For this, we can use the distance formula in the plane:
[tex]d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}[/tex] which for [tex]x_1=0[/tex] and [tex]y_1 = 0[/tex] transforms as [tex]d=\sqrt{(x_2)^2 + (y_2)^2}[/tex]. The first point we are looking for is connected to the origin and therefore, its components will form a right triangle in which, the Pythagoras theorem holds, see the first attached figure. Then, [tex]x_2[/tex], [tex]y_2[/tex] and 10 are a Pythagorean triple. From this, [tex]x_2= 6[/tex] or [tex]x_2=8[/tex] while [tex]y_2= 6[/tex] or [tex]y_2=8[/tex]. This leads us with the set of coordinates:
[tex](\pm 6, \pm 8)[/tex] and [tex](\pm 8, \pm 6)[/tex]. (A)
The next step is to find the coordinates of points that lie on lines which are perpendicular to the lines that joins the origin of the coordinate system with the set of points given in (A):
Let's do this for the point (6, 8).
The equation of the line that join the point (6, 8) with the origin (0, 0) has the equation [tex]y = mx +n[/tex], however, we only need to find its slope in order to find a perpendicular line to it. Thus,
[tex]m = \frac{y_2-y_1}{x_2-x_1} \\m = \frac{8-0}{6-0} \\m = 8/6[/tex]
Then, a perpendicular line has an slope [tex]m_{\bot} = -\frac{1}{m} = -\frac{6}{8}[/tex] (perpendicularity condition of two lines). With the equation of the slope of the perpendicular line and the given point (6, 8), together with the equation of the distance we can form a system of equations to find the coordinates of two points that lie on this perpendicular line.
[tex]m_{\bot}=\frac{6}{8} = \frac{8-y}{6-x}\\ 6(6-x)+8(8-y)=0[/tex] (1)
[tex]d^2 = \sqrt{(y_o-y)^2+(x_o-x)^2} \\(10)^2=\sqrt{(8-y)^2+(6-x)^2}\\100 = \sqrt{(8-y)^2+(6-x)^2}[/tex] (2)
This system has solutions in the coordinates (-2, 14) and (14, 2). Until here, we have three vertices of the square. Let's now find the fourth one in the same way we found the third one using the point (14,2). A line perpendicular to the line that joins the point (6, 8) and (14, 2) has an slope [tex]m = 8/6[/tex] based on the perpendicularity condition. Thus, we can form the system:
[tex]\frac{8}{6} =\frac{2-y}{14-x} \\8(14-x) - 6(2-y) = 0[/tex] (1)
[tex]100 = \sqrt{(14-x)^2+(2-y)^2}[/tex] (2)
with solution the coordinates (8, -6) and (20, 10). If you draw a line joining the coordinates (0, 0), (6, 8), (14, 2) and (8, -6) you will get one of the squares that fulfill the conditions of the problem. By repeating this process with the coordinates in (A), the following squares are found:
(0, 0), (6, 8), (14, 2), (8, -6)(0, 0), (8, 6), (14, -2), (6, -8)(0, 0), (-6, 8), (-14, 2), (-8, -6)(0, 0), (-8, 6), (-14, -2), (-6, -8)Now, notice that the equation of distance between the two points separated a distance of 10 has the trivial solution [tex](\pm10, 0)[/tex] and [tex](0, \pm10)[/tex]. By combining this points we get the following squares:
(0, 0), (10, 0), (10, 10), (0, 10)(0, 0), (0, 10), (-10, 10), (-10, 0)(0, 0), (-10, 0), (-10, -10), (0, -10)(0, 0), (0, -10), (-10, -10), (10, 0)See the attached second attached figure. Therefore, 8 squares can be drawn
Enter your answer and show all the steps that you use to solve this problem in the space provided.
x y
7 11
8 13
9 15
10 17
Determine whether y varies directly with x. If so, find the constant of variation k and write the equation.
Answer:
y DOES NOT vary directly with x
Step-by-step explanation:
If variation is direct, the ratio of y to x is a constant. Here, we have ...
11/7 ≠ 13/8
so variation is not direct.
___
It appears that y = 2x-3, rather than y = kx for some k. The added constant (-3) means variation is not direct.
To determine if y varies directly with x, we calculate the ratios between the corresponding values. The ratios are approximately constant, so y varies directly with x. The equation representing the direct variation is y = 1.57x.
Explanation:To determine whether y varies directly with x, we need to check if the ratio between the corresponding values of y and x is constant. Let's calculate the ratio for each pair of values:
For x = 7, y = 11, the ratio is 11/7 = 1.57For x = 8, y = 13, the ratio is 13/8 = 1.63For x = 9, y = 15, the ratio is 15/9 ≈ 1.67For x = 10, y = 17, the ratio is 17/10 = 1.7Since the ratios are approximately constant, we can conclude that y varies directly with x. To find the constant of variation (k), we can choose any of the ratios. Let's take the ratio from the first pair: 11/7 ≈ 1.57. Therefore, the equation that represents the direct variation is y = 1.57x.
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Freedonia has 50 senators. Each senator is either honest or corrupt. Suppose you know that at least one of the Freedonian senators is honest and that, given any two Freedonian senators, at least one is corrupt. Based on these facts, you can determine how many Freedonian senators are honest and how many are corrupt.
Answer:
There are 49 corrupt senators and 1 honest senator.
Step-by-step explanation:
Freedonia has 50 senators. Each senator is either honest or corrupt. At least one of the Freedonian senators is honest.
As we can see that there are no two senators where both of them are honest. So either there is one senerator who is honest or none.
As given that at least one of the Freedonian senators is honest.
Hence, we can conclude there is one and only one honest senator.
Final answer:
In Freedonia's senate, there must be one honest senator and forty-nine corrupt senators to satisfy the given conditions.
Explanation:
To solve the problem presented, we must work under the conditions that in Freedonia's senate of 50 senators, which implies that there are 50 senators in total, each is either honest or corrupt, at least one senator is honest, and in any pair of senators, at least one is corrupt. This means it's impossible to have two honest senators together, as this would contradict the condition that in any pair of senators, there must be at least one corrupt senator.
Given that there is at least one honest senator, we can use logic to deduce the following: If we were to assume that there are two honest senators, we would have a pair of senators where both are honest, which violates the given condition. Therefore, there can be only one honest senator in Freedonia's senate to meet all the conditions laid out in the problem.
Since the total number of senators is 50, and we have established that there is only one honest senator, it follows that the remaining 49 senators must be corrupt. Hence, the Freedonia's senate consists of one honest senator and forty-nine corrupt senators.
Dr. Cyril conducts a simple random sample of 500 men who became fathers for the first time in the past year. He finds that 23% of them report being unsure of their ability to be good fathers, plus or minus 4%. What is another term for the 4% value?
Answer:
The another term for the 4% value is : Margin of error
Step-by-step explanation:
Dr. Cyril conducts a simple random sample of 500 men.
He finds that 23% of them report being unsure of their ability to be good fathers, plus or minus 4%.
Now, the another term for the 4% value is : Margin of error
Means the estimate of father being unsure about his ability would be between 19% and 27%.
When sample size is increased, the margin of error becomes smaller.
Evaluate the expression uv^2 + 5uv + u^2 for u = 3 and v = 4. HELP PLEASE!!
A 84
B 96
C 117
D 112
Answer:
The answer to your question is letter C.117
Step-by-step explanation:
The original expression is: uv² +5uv + u²
and u= 3 and v = 4
Then: (3)(4)² + 5(3)(4) + (3)² Substitution
(3)(16) + 5(12) + 9 Expanding
48 + 50 + 9 Simplify
117 Result
The value of the algebraic expression [tex]uv^2 + 5uv + u^2[/tex] at u = 3, v = 4 is 117
Third option is correct
What is algebraic expression?
Algebraic expression consist of variables and numbers connected with addition, subtraction, multiplication and division.
For example - in 2x + 3y -7, the numbers are 2, 3, 7, variables are x, y and operations are addition, subtraction and multiplication.
Here,
[tex]uv^2 + 5uv + u^2[/tex]
Here, u = 3, v = 4
[tex]3(4)^2 + 5(3)(4) + (3)^2\\48 + 60 + 9\\117[/tex]
Third option is correct.
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Which property justifies this statement? If x=3, then x−3=0. Subtraction Property of Equality Division Property of Equality Reflexive Property of Equality Multiplication Property of Equality
The statement 'If x=3, then x−3=0' is justified by the Subtraction Property of Equality. This property allows you to subtract the same value from both sides of an equation while maintaining equality.
Explanation:The statement 'If x=3, then x−3=0' is justified by the Subtraction Property of Equality. This property states that if you subtract the same number from both sides of an equation, the equation remains equal. So, in this case, you start with x=3 and then you subtract 3 from both sides to get x-3=0. It's also important to note that the other properties mentioned, Division, Reflexive, and Multiplication, aren't appropriate in this scenario because no division, reflexion or multiplication operation is being performed.
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