The green area of the figure above is grass.
The blue is a sidewalk. What is the area of the
sidewalk? Help ASAP
Answers
Answer:
80
Step-by-step explanation:
the answer is 80 because 8×10=80
Answer:
116
Step-by-step explanation:
Area of whole thing is 196 and area of grass is 80 so you do 196-80 to get 116.
Related Questions
A pentagon has angles that measure
80°, 90°, 130°, 120°, and t. What is t?
t=
Answers
Answer:
120°
Step-by-step explanation:
To find the measure of angle t in the pentagon, we can use the property that the sum of the interior angles of a pentagon is equal to 540 degrees.
So, we can set up an equation:
80° + 90° + 130° + 120° + t = 540°
Simplifying the equation:
420° + t = 540°
Next, we can isolate the variable t by subtracting 420° from both sides of the equation:
t = 540° - 420°
t = 120°
Therefore, the measure of angle t in the pentagon is 120 degrees.
Hope this helps!
Which situation would not be represented by the integer -5?
There is a temperature of 5°.
You owe $5.
There is a hole 5 feet deep.
There is a 5-yard penalty.
Answers
Answer:
A
Step-by-step explanation:
It is talking about a positive number
Answer:
a
Step-by-step explanation:
Let X be a Markov chain with transition probability matrix 0 1 2 0 0.7 0.2 0.1 P= 1 0.3 0.5 0.2 2 0 0 1 The Markov chain starts at time zero in state Xo = 0. Let T = min{n > 0: X = 2} be the first time that the process reaches state 2. Eventually, the process will reach and be absorbed into state 2. If in some experiment we observed such a process and noted that absorption had not yet taken place, we might be interested in the conditional probability that the process is in state 0 (or 1), given that absorption had not yet taken place. Determine P(X3 = 0 T > 3).
Answers
The conditional probability that the process is in state 0 (or 1), given that absorption had not yet taken place. Therefore, P(X3 = 0, T > 3) = 0.075.
The Markov chain starts at time zero in state Xo = 0 and the transition probability matrix of Markov Chain is as follows: 0 1 2 0 0.7 0.2 0.1 P= 1 0.3 0.5 0.2 2 0 0 1
P(X3 = 0, T > 3).We know that the probability of moving from state i to state j in two steps is given by P2(i, j).
Thus, the probability of moving from state i to state j in three steps is given by P3(i, j). We have P2(i, j) = P(i, ·)P(j, ·) = Σk P(i, k)P(k, j).
For a 3-step transition probability, we use the equation P3 = P2P = P2 (P2) and so on. Therefore,P3(1, 2) = P2(1, 1)P(1, 2) + P2(1, 2)P(2, 2) + P2(1, 3)P(3, 2) = (0.7)(0.3) + (0.2)(0.5) + (0.1)(0) = 0.235
Similarly,P3(1, 0) = P2(1, 0)P(0, 0) + P2(1, 1)P(1, 0) + P2(1, 2)P(2, 0) = (0)(0.7) + (0.3)(0) + (0.235)(0.2) = 0.047
Since we are interested in finding P(X3 = 0, T > 3), we need to find the probability that absorption had not yet taken place at time 3 and that the process is in state 0 at time 3, which can be expressed as:
P(X3 = 0, T > 3) = P(X3 = 0, X4 ≠ 2) = P(X3 = 0, X4 = 0) + P(X3 = 0, X4 = 1)
We know that T is the first time that the process reaches state 2 and the process will reach and be absorbed into state 2.
Thus, T is the absorption time for state 2 and it has a geometric distribution with parameter P2(2, 2) = 1, which implies that P(T = t) = (1 – 1)P2(2, 2) = 0 for all t < 1.
Therefore, P(X3 = 0, X4 = 0) = P(X3 = 0, X4 = 0, T > 3)
= P(X3 = 0, T > 3)P(X4 = 0 | X3 = 0, T > 3) = P(X3 = 0, T > 3)P(0, 0) / P(X3 = 0, T > 3)P(0, 0) + P(X3 = 1, T > 3)
P(1, 0) + P(X3 = 2, T > 3)
P(2, 0) = (0.047)(1) / [(0.047)(1) + (0.235)(0.7) + (0)(0.2)]
= 0.067
Therefore, P(X3 = 0, T > 3) = P(X3 = 0, X4 = 0) + P(X3 = 0, X4 = 1) = (0.067)(0.7) + (0.235)(0.2) = 0.075.
Therefore, P(X3 = 0, T > 3) = 0.075.
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half of 1/2 cake = _____ cake.
Answers
Answer:
well
Step-by-step explanation:
please add the image
Applying the knowledge of fractions, half of 1/2 cake = 1/4 cake.
Multiplying FractionsWe are asked to find how many parts of are there in half of one-half of a cake.
Thus:
half = 1/2
1/2 of 1/2 cake = 1/2 × 1/2 (of = multiplication)
= (1 × 1)/(2 × 2)
= 1/4
Therefore, applying the knowledge of fractions, half of 1/2 cake = 1/4 cake.
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The base angles of an isosceles trapezoid are__________?
Answers
Answer:
Base angles of an isosceles triangle are always congruent.
2
When a set of data has an even member the median is found by:
(1 Point)
O adding the two middle numbers
finding the mean of the two numbers at the end.
O finding the mean of the middle members
O choosing the number in the middle
Answers
Answer:
It is 1, or A: adding the two middle numbers
Step-by-step explanation:
When you think of it, here's an example:
15
23
55
34
Add 23 and 55:
23+55=78
We've got this now:
15
78
34
Now, all you've gotta do is find the mediam.
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Which of the equations below could be the equation of this parabola?
(0,0)
Vertex
O A. y = 2x2
B. x = -2y2
O c. x = 2y2
O D. y= -2
Answers
Answer:
Y=2x^2
Step-by-step explanation:
The following equations represent a parabola with vertex (0,0): y=2x², x=-2y² and x=2y².
Quadratic functionThe quadratic function can be represented by a quadratic equation in the Standard form: ax²+bx+c=0 where: a, b and c are your respective coefficients. In the quadratic function the coefficient "a" must be different than zero (a≠0) and the degree of the function must be equal to 2.
A parabola also can be represented by a quadratic equation. The vertex of an up-down facing parabola of the form ax²+bx+c is [tex]x_v=\frac{-b}{2a}[/tex] . Knowing the x-coordinate of vertex, you can find the y-coordinate of vertex.
The another form for describing a parabola is [tex]4p\left(x-h\right)=\left(y-k\right)^2[/tex], where h and k are the vertex coordinates.
You should analyse each one of the options, considering the equations that can be represented a parabola.
Letter A - y=2x²
The coefficients of the quadratic equation are:
a=2, b=0, c=0
Then,
[tex]x_v=\frac{-b}{2a}\\ \\ x_v=\frac{-0}{2*2}=0[/tex].
If x-coordinate of vertex is equal to 0, from y=2x²you can:
[tex]y_v=2x^2\\ \\ y_v=2*0^2=0[/tex]
Therefore, the given equation ( y=2x²) represents a parabola with vertex (0,0).
Letter B - x=-2y²
From equation [tex]4p\left(x-h\right)=\left(y-k\right)^2[/tex], you can rewrite the given equation parabola for vertex (0,0) in:
[tex]4p(x-0)=(y-0)^2\\ \\ 4*\frac{-1}{8} x=y^2\\ \\ \frac{-1}{2} x=y^2\\ \\ x=-2y^2[/tex]
Therefore, the given equation ( x=-2y²) represents a parabola with vertex (0,0).
Letter C - x=2y²
From equation [tex]4p\left(x-h\right)=\left(y-k\right)^2[/tex], you can rewrite the given equation parabola for vertex (0,0) in:
[tex]4p(x-0)=(y-0)^2\\ \\ 4*\frac{1}{8} x=y^2\\ \\ \frac{1}{2} x=y^2\\ \\ x=2y^2[/tex]
Therefore, the given equation ( x=2y²) represents a parabola with vertex (0,0).
Letter D - y=-2
The degree of equation is not equal 2. Therefore, it does not represent a parabola.
Only the equations of letter A, B and C represent a parabola with vertex (0,0). See the attached image.
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A company rents out 18 food booths and 26 game booths at the county fair. The fee for a food booth is $100 plus $10 per day. The fee for a game booth is $95 plus $5 per day. The fair lasts for d days, and all the booths are rented for the entire time. Enter a simplified expression for the amount, in dollars, that the company is paid.
Answers
Answer:
4,270 + 310d
Step-by-step explanation:
Let
d = number of days
Each food booth = 100 + 10d
18 food booth = 18(100 + 10d)
= 1,800 + 180d
Each game booth = 95 + 5d
26 game booth = 26(95 + 5d)
= 2,470 + 130d
Total amount the company is paid = total cost of food booth + total cost of game booth
= (1,800 + 180d) + (2,470 + 130d)
= 1,800 + 180d + 2,470 + 130d
= 1,800 + 2,470 + 180d + 130d
= 4,270 + 310d
Total amount the company is paid = 4,270 + 310d
Jen has 3 bags of pears. Each Bag has 5 pears. If Jen gives the same number of pears to 4 friends, how many pears will each friend get?
Answers
3bags x 5 pearls per bag=15 bags
15 / (4 friends+you)=5
15/5=3 pearls per friend.
Answer:3 pearls per person.
Answer:
It is 3x5 is 15 the you divide 15 by 4 which is 3.75 per person
Step-by-step explanation:
3x5=15
15\4=3.75
I need help pls is due today
Answers
Answer:
a) 1/2, 0.5, 50% b)3/4, 0.75, 75% c) 1/12, 0.08333, 8.333% d) 5/12, 41.6667%, 0.416
Step-by-step explanation:
Answer:
a] 1/2 b] 5/6 c] 1/6 d] 5/4
Step-by-step explanation:
no of even no present/total numbers and so on
try it later
Which of the following is not determined in the case of multiple linear regression?
Question 4 options:
1)Potential confounding effects from "lurking" variables included in the analysis
2)Observed relationship between predictor and outcome variables adjusting for confounding relationships
3)Causal relationships between predictor and outcome variables adjusting for confounding relationships
4) Potential observed interaction effects between variables Save
Answers
It does not determine the observed interaction effects between variables.
In the case of multiple linear regression, the option that is not determined is:
Variable interactions that have been observed to have effects.
Option 1: Multiple linear regression can estimate the observed relationship between predictor and outcome variables after adjusting for confounding relationships and option 2: it can determine the potential confounding effects from lurking variables. Furthermore, numerous direct relapse can evaluate the causal connections among indicator and result factors adapting to jumbling connections (choice 3).
However, the potential observed interaction effects between variables cannot be determined by multiple linear regression on its own. Association impacts happen when the connection between two indicators and the result variable isn't added substance however relies upon the mix of the indicator values. In order to evaluate the effects of interactions, additional analysis methods like including interaction terms and carrying out specific interaction tests are required.
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PROCESS A: "Driftless" geometric Brownian motion (GBM). "Driftless" means no "dt" term. So it's our familiar process: ds = o S dw with S(O) = 1. o is the volatility. PROCESS B: ds = a S2 dw for some constant a, with S(0) = 1 As we've said in class, for any process the instantaneous return is the random variable: dS/S = (S(t + dt) - S(t)/S(t) = [1] Explain why, for PROCESS A, the variance of this instantaneous return (VAR[ds/S]) is constant (per unit time). Hint: What's the variance of dw? The rest of this problem involves PROCESS B. [2] For PROCESS B, the statement in [1] is not true. Explain why PROCESS B's variance of the instantaneous return (per unit time) depends on the value s(t).
Answers
In Process A, which is a driftless geometric Brownian motion, the variance of the instantaneous return (VAR[ds/S]) is constant per unit time. However, in Process B, where ds = aS^2dw, the variance of the instantaneous return depends on the value of S(t).
In Process A, since there is no drift term (dt), the random variable dw follows a standard normal distribution with a constant variance of dt. When calculating the instantaneous return dS/S, we can see that the dt terms cancel out, resulting in a constant variance of the instantaneous return (VAR[ds/S]) per unit time. This is because the volatility o remains constant, and the random variable dw has a constant variance.
In Process B, the equation ds = aS^2dw suggests that the variance of the instantaneous return is proportional to the value of S(t). As S(t) increases, the magnitude of ds also increases, leading to a larger variance of the instantaneous return. In other words, the volatility in Process B depends on the level of the underlying process S(t). This is different from Process A, where the variance of the instantaneous return is constant regardless of the value of S(t). Hence, the variance of the instantaneous return in Process B is not constant per unit time, but rather dependent on the value of S(t).
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Find the Inverse of the following function.
Answers
Answer: A
Step-by-step explanation:
Can somebody plz help answer these questions correctly (only if u know how to do them) thx sm! :3
WILL MARK BRAINLIEST WHOEVER ANASWERS FIRST :DDDD
Answers
Answer:
s= 100
r= 130
x= 38
y= 30
Step-by-step explanation:
Which of the following is a solution to the differential equation xy′−3y=6xy′−3y=6 ?
Answers
A solution to the differential equation xy' - 3y = 6 is y = -2/x. This is a particular solution that satisfies the given differential equation. Therefore, y = -2/x is a solution to the differential equation xy' - 3y = 6.
To find a solution to the differential equation xy' - 3y = 6, we need to solve the equation and find a function that satisfies it. We can begin by rearranging the equation:
xy' - 3y = 6
To solve this linear first-order ordinary differential equation, we can use the method of integrating factors. The integrating factor is given by the exponential of the integral of the coefficient of y, which in this case is -3:
IF = e^(-3x)
Multiplying both sides of the equation by the integrating factor, we have:
e^(-3x)xy' - 3e^(-3x)y = 6e^(-3x)
This can be rewritten as:
(d/dx)(e^(-3x)y) = 6e^(-3x)
Integrating both sides with respect to x, we get:
e^(-3x)y = ∫(6e^(-3x))dx
Simplifying the integral and applying the constant of integration, we have:
e^(-3x)y = -2e^(-3x) + C
Dividing both sides by e^(-3x), we obtain:
y = -2 + Ce^(3x)
The constant C can take any value. By choosing C = 0, we have y = -2/x, which is a particular solution to the given differential equation. Therefore, y = -2/x is a solution to the differential equation xy' - 3y = 6.
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Point S is on line segment RT. Given ST=3 and RT=20, determine the length RS.
Answers
Answer:
RS = 17
Step-by-step explanation:
Given S lies on RT , then
RS + ST = RT , that is
RS + 3 = 20 ( subtract 3 from both sides )
RS = 17
Please help! This is my last question and I can’t get it... I gave the most points I could. (I will also mark you the brainliest if it works)
Answers
Answer:
18.
Step-by-step explanation:
Which measure is equivalent to 65 kilograms?
659
6509
65000
65,0000
Answers
Answer:
65,000 grams
Step-by-step explanation:
Because 1000 grams is in 1 kilogram, you would multiply the 65 kilograms by 1,000 to convert it to grams.
So to convert 65 kilograms to grams you would do:
65 * 1,000 = 65,000 grams
A researcher interested in the age at which women have their first child surveyed a simple random sample of 250 women who have one child and found an approximately normal distribution with a mean age of 27.3 and a standard deviation of 5.4. According to the Empirical rule, approximately 95% of women had their first child between the ages of a. 11.1 years and 43.5 years b. 16.5 years and 38.1 years C. 21.9 years and 32.7 years d. 25.0 years and 29.6 years
Answers
According to the Empirical rule, approximately 95% of women had their first child between the ages of 16.5 years and 38.1 years. The correct option is b. 16.5 years and 38.1 years.
The given question states that a researcher interested in the age at which women have their first child surveyed a simple random sample of 250 women who have one child and found an approximately normal distribution with a mean age of 27.3 and a standard deviation of 5.4. We are to find the range of age at which approximately 95% of women had their first child, according to the empirical rule.
There are three ranges for the empirical rule as follows:
Approximately 68% of the observations fall within the first standard deviation from the mean.
Approximately 95% of the observations fall within the first two standard deviations from the mean.
Approximately 99.7% of the observations fall within the first three standard deviations from the mean.
Now, we will apply the empirical rule to find the age range at which approximately 95% of women had their first child. The mean age is 27.3 years and the standard deviation is 5.4 years, hence:
First, find the age at which 2.5% of women had their first child:
µ - 2σ = 27.3 - (2 × 5.4) = 16.5
Then, find the age at which 97.5% of women had their first child:
µ + 2σ = 27.3 + (2 × 5.4) = 38.1
Therefore, approximately 95% of women had their first child between the ages of 16.5 years and 38.1 years. Hence, the correct answer is option b. 16.5 years and 38.1 years.
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According to the Chronicles of Contrived Statistics (March, 2016), the probability that Elon Musk will offer you a free trip into space in the next month is 41%. The probability that scientists from Area 51 will start selling pet aliens next month is 27%. Finally, the probability that either Elon Musk will offer you a free trip to space or that scientists will begin sales of pet aliens next month is 24%. What is the probability that both Elon Musk will offer you a free trip into space and scientists from Area 51 will start selling pet aliens in the next month?
Answers
Answer:
[tex]P(A\ and\ B) = 44\%[/tex]
Step-by-step explanation:
Given
Represents the event as follows:
A = [tex]Elon\ Musk[/tex] will offer you a [tex]free\ trip[/tex] into space in the next month\
B = Scientists from [tex]Area\ 51[/tex] will [tex]start\ selling[/tex] pet aliens next month
So, we have:
[tex]P(A) = 41\%[/tex]
[tex]P(B) = 27\%[/tex]
[tex]P(A\ or\ B) = 24\%[/tex]
Required
Determine [tex]P(A\ and\ B)[/tex]
This is calculated as:
[tex]P(A\ and\ B) = P(A) + P(B) - P(A\ or\ B)[/tex]
So, we have:
[tex]P(A\ and\ B) = 41\% + 27\% - 24\%[/tex]
[tex]P(A\ and\ B) = 44\%[/tex]
A study at the University of Illinois found that young men who drank two pints of beer first were better able to solve certain word puzzles than sober men. Design an experiment that could attempt to verify this result. Describe the population, how you’d collect your sample, how you’d execute the experiment, and what data you’d collect .
Answers
Experiment to verify the result of the study:A study at the University of Illinois found that young men who drank two pints of beer first were better able to solve certain word puzzles than sober men. An experiment to verify this result should be designed with the following steps:
Population: The population in this experiment would be young men who are eligible to consume beer legally.
Sampling: The sampling method will be convenient sampling. In this type of sampling, participants will be selected based on their availability to participate. Any participant that is within the age range of eligibility and is willing to participate can be considered for the study.
The participants will be divided into two groups, one group will drink two pints of beer while the other group will not drink any beer.
Executing the experiment: Both groups will be given word puzzles to solve after the beer is consumed by the test group and given to the control group directly.
The participants will not be given any hints on how to solve the puzzle to keep it fair. Data Collection: Both groups will be timed to solve the puzzle.
The group that solves the puzzle faster will be regarded as the winner. The number of people in each group that solve the puzzle will be recorded.
A correlation test would be performed to determine if the solution time is related to the consumption of beer.
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Data will be collected and analyzed using statistical tools such as the t-test to determine if there is a significant difference in performance between the two groups.
The experiment is designed to verify whether young men who drank two pints of beer first were better able to solve certain word puzzles than sober men. This question requires a well-planned experimental design. The experiment requires a hypothesis and a null hypothesis.
Hypothesis
Drinking two pints of beer can improve the performance of young men in word puzzles than sober men.
Null Hypothesis
Drinking two pints of beer cannot improve the performance of young men in word puzzles than sober men.
Population
The target population of the study is young men aged between 18 to 30 years.
Sample collection
To collect the sample, we will identify potential participants based on the age range of 18-30 years. The study will recruit volunteers who drink alcohol regularly and those who don't. Participants who have consumed alcohol before the study will be required to take a breathalyzer test to ensure they are within the recommended limits. Only those with a blood alcohol concentration of 0.08% and below will be included in the study. Participants will also be required to sign informed consent to participate in the study.
Execute the experiment
Participants will be randomly assigned into two groups: the control group and the experimental group. The control group will be given water to drink while the experimental group will be given two pints of beer. Participants will then be given a set of word puzzles to solve, and their performance will be recorded. Each group will be given an equal time limit to solve the word puzzles.
Data Collection
The data collected will include the number of word puzzles solved by each group, the time taken to solve the word puzzles, and the number of incorrect answers. The data collected will be analyzed using statistical tools such as the t-test to determine if the difference in performance between the two groups is statistically significant. ConclusionThe experiment is designed to verify if drinking two pints of beer can improve the performance of young men in solving certain word puzzles than sober men. The experiment involves a sample size of young men aged 18-30 years who will be randomly assigned to two groups; the experimental group and the control group. Data will be collected and analyzed using statistical tools such as the t-test to determine if there is a significant difference in performance between the two groups.
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Can someone plz help me with #2 and #6 plz thank oyu
Answers
Answer:
njbrdjbbgdbjbfjfj
Step-by-step explanation:
The following is a scatter plot of the heights and weights of the students in Mr. Baldwin’s PE class.
A graph titled Heights and Weights has weight in pounds on the x-axis, and height in inches on the y-axis. A line goes through (146, 47.5).
Given the line of best fit, estimate the height of a student who weighs 146 pounds.
a.
About 67.5 inches
b.
About 65 inches
c.
About 72 inches
d.
About 72.5 inches
Please select the best answer from the choices provided
A
B
C
D
Answers
the answer is b ............
HELPP PLSSSS NO BOTS OR I WILL REPORT!!!
Answers
Answer:
B. false
Step-by-step explanation:
If today is Wednesday what is it like hood tomorrow will be Saturday?
Answers
Answer:
zero
Step-by-step explanation:
If today is Wednesday, the probability of tomorrow being Saturday is zero.
If m∠4 = 35°, find m∠2 and m∠3.
There are two parallel lines. Point A and B are on the upper line and point C and D are same are A and B on the lower line. The segment AC is perpendicular to both parallel lines. There is another segment CB. The angle CAB is denoted by 1. The angle ABC is denoted by 2. The angle ACB is denoted by 3. The complementary angle of angle 3 is denoted by 4.
m∠2 =
°
m∠3 =
°
Answers
Answer:
Step-by-step explanation:
yes
The ∠3 is an alternate angle of ∠4 so it is 35° while ∠3 is a complimentary angle of ∠4 so it will be 55°.
What is an angle?An angle is a geometry in plane geometry that is created by 2 rays or lines that have an identical terminus.
Since we lack a measurement for angular rotation, the angle is a valuable tool for measuring angular distance. For instance, a meter or an inch is a unit of measurement for linear motion.
Given that AB and CD are parallel and AC is perpendicular to both.
So,
∠ACD = 90°
∠3 + ∠4 = 90°
∠3 = 90 - 35 = 55°.
Now,
Since ∠4 and ∠2 are alternative angles so both will be the same.
So,
∠2 = 35°
Hence "The ∠3 is an alternate angle of ∠4 so it is 35° while ∠3 is a complimentary angle of ∠4 so it will be 55°".
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A rectangle’s length is three times its width, w. Its area is 243 square units. Which equation can be used to find the width of the rectangle?
a. w2 = 3(243)
b. 3w2 = 243
c. 4w2 = 243
d. 3w2 = 3(243)
please im on a timed test
Answers
Answer:
b) 3w² = 243
Step-by-step explanation:
area = L x w
L = 3w
substitute for L:
243 = 3w x w = 3w²
3w² = 243
A cylindrical canister has a radius of 7 cm and a height of 16 cm. Find the volume of
the canister.
pls show your work thx pls help me will give brainly.
Answers
Answer:
7/16 = 0.4375
Step-by-step explanation:
sry
hey hottie pls help me! thank u ;)
Select the context(s) that could be modeled by a linear function.
A)The amount Ms. Ji Woo and Mrs. Rose pay to rent a car is $50 per day.
B) Ms. Jisoo adds $20 to a savings account each month.
C)The value of Principal Kai's car decreases by 15.5% each year.
D)The price of a stock on Robinhood each year is 110% of its price from the previous year.
E)Mrs. Manoban pays $1000 for car insurance the first year and pays an additional $25 per year
Answers
Answer: A,B, and E for sure. I'm not sure about C though.
Step-by-step explanation:
use the chain rule to find dz/dt. z = xy7 − x2y, x = t2 1, y = t2 − 1
Answers
If z = xy⁷ − x²y, x = t² + 1, y = t² − 1, then using chain rule the value of dz/dt = 2t(y⁷ + xy⁶(dy/dx) - 2xy - x²(dy/dx)) + 2t(xy⁷ + 7xy⁶ - x²)
To find dz/dt using the chain rule, we need to differentiate z with respect to t while considering the relationship between z, x, y, and t.
Given:
z = xy⁷ − x²y
x = t² + 1
y = t² − 1
We can start by finding dz/dx and dz/dy separately and then use the chain rule to combine them to find dz/dt.
First, let's find dz/dx:
Differentiating z with respect to x, keeping y constant:
dz/dx = (d/dx)(xy⁷) - (d/dx)(x²y)
Using the product rule for differentiation:
dz/dx = y⁷ + xy⁶(dy/dx) - 2xy - x²(dy/dx)
Next, let's find dz/dy:
Differentiating z with respect to y, keeping x constant:
dz/dy = (d/dy)(xy⁷) - (d/dy)(x²y)
Using the product rule again:
dz/dy = x(y⁷ + 7y⁶(dy/dy)) - x²
Since dy/dy = 1, we can simplify dz/dy as:
dz/dy = xy⁷ + 7xy⁶ - x²
Now, let's use the chain rule to find dz/dt:
dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt)
Substituting the expressions we found earlier for dz/dx and dz/dy, and using dx/dt = 2t and dy/dt = 2t:
dz/dt = (y⁷ + xy⁶(dy/dx) - 2xy - x²(dy/dx))(2t) + (xy⁷ + 7xy⁶ - x²)(2t)
Simplifying this expression gives us dz/dt in terms of t, x, and y:
dz/dt = 2t(y⁷ + xy⁶(dy/dx) - 2xy - x²(dy/dx)) + 2t(xy⁷ + 7xy⁶ - x²)
This is the complete calculation for finding dz/dt using the chain rule.
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Complete question is:
Use the chain rule to find dz/dt. z = xy⁷ − x²y, x = t² + 1, y = t² − 1