Problem 3. (5 marks) Find the third order approximation of arctan(2) at r = 0. Use it to approximate arctan(0.1).
Answers
The third order approximation of arctan (2) at r = 0 gives us an approximation of arctan (0.1) as 0.1. To find the third order approximation of arctan (2) at r = 0, we can use the Taylor series expansion of arctan(x) around r = 0:
arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
Substituting x = 2, we get:
arctan(2) = 2 - 2^3/3 + 2^5/5 - 2^7/7 + ...
To find the third order approximation, we stop after the x^5 term:
arctan(x) ≈ x - x^3/3 + x^5/5
Substituting x = 2, we get:
arctan(2) ≈ 2 - 2^3/3 + 2^5/5
arctan(2) ≈ 1.1071
Now we can use this approximation to approximate arctan(0.1). We know that arctan(0.1) is close to arctan(0) = 0, so we can use a first-order approximation:
arctan(0.1) ≈ arctan(0) + arctan'(0)(0.1)
Taking the derivative of arctan(x), we get:
arctan'(x) = 1/(1+x^2)
Substituting x = 0, we get:
arctan'(0) = 1
So we have:
arctan(0.1) ≈ 0 + 1(0.1)
arctan(0.1) ≈ 0.1
Therefore, our third order approximation of arctan (2) at r = 0 gives us an approximation of arctan (0.1) as 0.1.
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Related Questions
Work in groups of two to three (If possible). For each of the activities described have the required number of people in your group (usually 1) perform the activity as the others observe. Following the performance and observation, record what was seen. Make sure that you understand the activity prior to performance. The observers should choose their positions carefully with reference to the planes of motion.
Laboratory Report - Reflexes
1. Long Jump - perform 2 trials of each of these jumps in turn. Record the relative length of each jump
(which jumps are longer than others).
Assign a rank order by distance:
_____a. Use a bobbing motion with the knees and arms.
_____b. Start the jump from a position of deep knee flexion, with no knee motion prior to the start of the jump.
_____c. Start the jump with the knees fully extended, using no knee motion.
Answers
a. Jump with a bobbing motion: Shortest jump b. Jump from deep knee flexion: Intermediate jump
c. Jump with fully extended knees: Longest jump ,In the long jump activity, three different jumping techniques were observed:
a) The bobbing motion involved bending and extending the knees and arms during the jump.
b) Starting from deep knee flexion meant initiating the jump from a position with knees bent and no motion prior to the jump.
c) Starting with fully extended knees involved jumping with the knees straight and no motion before the jump. During the performance, the lengths of each jump were measured. Trial results consistently indicated that the jump with the bobbing motion had the shortest length, while the jump starting from deep knee flexion had an intermediate length. The jump with fully extended knees consistently achieved the longest distance.
The observations reveal that starting the jump with fully extended knees and no knee motion results in the longest jump. The bobbing motion with knee and arm movements leads to the shortest jump. Deep knee flexion position falls between the two in terms of jump length.
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PLS PLS PLS HELPP
Translate the figure 7 units to the left and 6 units up, then reflect it across the line X=1
Is the resulting figure similar or congruent to the original figure ? Explain why.
Answers
Answer:
Step-by-step explanation:
Figures are congruent, because corresponding angles and sides are congruent.
you have a sample of 50 women recently diagnosed with endometrial cancer; 10 of them report that their partner smokes cigarettes. you have another sample of 50 women without endometrial cancer; 5 of them report that their partner smokes cigarettes. based upon this information, what is the probability (to one decimal place) that a woman who has endometrial cancer has a partner who smokes?
Answers
The probability that a woman has endometrial cancer has a partner who smokes is 0.2
Probability: -Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event. For an experiment having 'n' number of outcomes, the number of favorable outcomes can be denoted by x. The formula to calculate the probability of an event is as follows.
Probability(Event) = Favorable Outcomes/Total Outcomes = x/n
OR
P(A) = n(A)/n(S)
Where,
P(A) is the probability of an event “A”
n(A) is the number of favorable outcomes
n(S) is the total number of events in the sample space.
According to the question,
50 women has endometrial cancer because 5 of them reports that their partner smokes cigerate.
According to the probability formula:
P(A) = n(A)/ n(s)
= 10/50
= 0.2
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HELP HELP PLS!!! 9th Grade Math!!! WILL MARK BRAINLIST!!!
Answers
Answer:
x=9
Step-by-step explanation:
These angles are equal to each other because they are alternate exterior angles. Thus, we can solve the for the value of x by creating the equation
6x+5=7x-4
From here, we solve it like a normal algebraic equation, isolating the variables and making sure it is on the same side of the other. From there we subtract 7x from both sides, and subtract 5 from both sides, leaving -x = -4-5. Summing up -4-5, we get -x=-9. Now we make them positive on both sides, leaving the answer that x = 9
Assume you deposit $1,000 every three months at a 3 percent
annual rate, compounded quarterly. How
much will you have at the end of 5 years?
Answers
The future value of the deposits at the end of 5 years is approximately $6,138.74.
To calculate the future value of the deposits, we can use the formula for compound interest:
FV = P(1 + r/n)^(nt)
Where: FV is the future value
P is the principal amount (initial deposit)
r is the annual interest rate (3% in this case)
n is the number of times interest is compounded per year (quarterly in this case)
t is the number of years
In this case, the principal amount is $1,000, the annual interest rate is 3% (or 0.03), the number of times interest is compounded per year is 4 (quarterly), and the number of years is 5.
Plugging in these values into the formula, we get: FV = $1,000(1 + 0.03/4)^(4*5)
Calculating this, the future value of the deposits at the end of 5 years is approximately $6,138.74.
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A summary of two stocks is shown.
Name of Stock Symbol Closing Price Day 1 Closing Price Day 2 Closing Price Day 3
Metroplis, Ltd MTP 17.95 18.28 18.25
Suburbia, Inc SBR 5.63 5.88 4.98
Suppose you purchase 30 shares of Metropolis stock and 55 shares of Suburbia stock on Day 1 at the closing price. Which day, during the following two days, would be the best to sell both stocks?
Day 2 is the best by $26.75.
Day 3 is the best by $26.75.
Day 2 is the best by $23.65.
Day 3 is the best by $23.65.
Answers
The best day to sell both stocks is Day 2, since the total value of both stocks is highest on that day, option C is correct.
On Day 1, the total value of the 30 shares of Metropolis stock is:
30 shares x $17.95 per share = $538.50
And the total value of the 55 shares of Suburbia stock is:
55 shares x $5.63 per share = $309.65
So the total value of both stocks on Day 1 is:
$538.50 + $309.65 = $848.15
On Day 2, the total value of the shares of Metropolis stock is:
30 shares x $18.28 per share = $548.40
And the total value of the shares of Suburbia stock is:
55 shares x $5.88 per share = $323.40
The total value of both stocks on Day 2 is: $871.8
for Day 3, total value of both stocks on Day 3 is $$821.40
Therefore, we can see that the best day to sell both stocks is Day 2, since the total value of both stocks is highest on that day:
$871.80 - $848.15 = $23.65
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Find the mean median and mode for 44, 4, 27, 5
Answers
Answer:
Mean: 20
Median: 16
Mode: No mode
Step-by-step explanation:
For the mean:
44 + 4 + 27 + 5 = 80
80/4 = 20
For the median:
5 + 27 = 32
32/2 = 16
For the mode:
None of the numbers are repeated.
5^-8 x 5^4 what is the answer to this equation?
Answers
Answer:
0.0016
Step-by-step explanation:
calculator was used
Answer: not simplified -4 simplified 1/5^4
Step-by-step explanation:
The common stock of Dayton Rapur sells for $48 49 a shame. The stock is inxpected to pay $2.17 per share next year when the annual dividend is distributed. The company increases its dividends by 2.56 percent annually What is the market rate of retum on this stock? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, eg-32.16.)
Answers
The market rate of return on the Dayton Rapur stock is approximately 4.59%.
To calculate the market rate of return on the Dayton Rapur stock, we need to use the dividend discount model (DDM). The DDM calculates the present value of expected future dividends and divides it by the current stock price.
First, let's calculate the expected dividend for the next year. The annual dividend is $2.17 per share, and it increases by 2.56% annually. So the expected dividend for the next year is:
Expected Dividend = Annual Dividend * (1 + Annual Dividend Growth Rate)
Expected Dividend = $2.17 * (1 + 0.0256)
Expected Dividend = $2.23
Now, we can calculate the market rate of return using the DDM:
Market Rate of Return = Expected Dividend / Stock Price
Market Rate of Return = $2.23 / $48.49
Market Rate of Return ≈ 0.0459
Finally, we convert this to a percentage:
Market Rate of Return ≈ 0.0459 * 100 ≈ 4.59%
Therefore, the market rate of return on the Dayton Rapur stock is approximately 4.59%.
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Which angle refers to the same angle as ∠EDC
Answers
Answer:
Point D does
Step-by-step explanation:
I took the quiz
find the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y) = sin(xy), (9, 0)
Answers
To find the maximum rate of change of the function \(\(f(x, y) = \sin(xy)\)\) at the point (9, 0) and the direction in which it occurs, we can calculate the gradient vector and evaluate it at the given point.
The gradient vector of \(\(f\)\) is given by:
\(\(\nabla f(x, y) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)\)\)
Let's calculate the partial derivatives:
\(\(\frac{\partial f}{\partial x} = y\cos(xy)\)\)
\(\(\frac{\partial f}{\partial y} = x\cos(xy)\)\)
Now, we can evaluate the gradient vector at the point (9, 0):
\(\(\nabla f(9, 0) = \left(0\cos(0\cdot 9), 9\cos(9\cdot 0)\right) = \left(0, 9\right)\)\)
The magnitude of the gradient vector gives us the maximum rate of change of the function at the point (9, 0). The magnitude of \(\(\nabla f(9, 0)\)\) is:
\(\(|\nabla f(9, 0)| = \sqrt{0^2 + 9^2} = \sqrt{81} = 9\)\)
So, the maximum rate of change of \(\(f\)\) at the point (9, 0) is 9.
To determine the direction in which this maximum rate of change occurs, we normalize the gradient vector by dividing it by its magnitude:
\(\(\frac{\nabla f(9, 0)}{|\nabla f(9, 0)|} = \frac{(0, 9)}{9} = (0, 1)\)\)
Therefore, the maximum rate of change of \(\(f\)\) at the point (9, 0) occurs in the direction of the vector (0, 1), which is the positive y-direction.
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Rex wanted to estimate the mean finishing time for the approximately 50 , 000 50,00050, comma, 000 finishers at the New York City Marathon. He took a random sample of 10 1010 finishers' times, which were skewed to the right with a sample mean of x ˉ = 271. 5 x ˉ =271. 5x, with, \bar, on top, equals, 271, point, 5 minutes. He's considering using his data to make a confidence interval for the mean finishing time. Which conditions for constructing a t tt interval have been met? Choose all answers that apply: Choose all answers that apply: (Choice A) A The data is a random sample from the population of interest. (Choice B) B The sampling distribution of x ˉ x ˉ x, with, \bar, on top is approximately normal. (Choice C) C Individual observations can be considered independent
Answers
Answer:
The data is a random sample from the population of interest.
Individual observations can be considered independent.
Step-by-step explanation:
From Khan Academy
The random condition:
The problem reads, "He took a random sample of 10 finishers' times," so the random condition is met.
The normal condition:
The times in the sample were skewed, and the sample size is small (n=10<30)(n=10<30)left parenthesis, n, equals, 10, is less than, 30, right parenthesis, so the normal condition has not been met.
The independence condition:
The independence condition says that when sampling without replacement, we can still treat each observation in the sample as independent as long as we sample less than 10%, percent of the population. Since the sample of 10 finishers' times is less than 10%, percent of the approximately 50,000 times in the population, the independence condition has been met.
The answers:
The data is a random sample from the population of interest.
Individual observations can be considered independent.
(-3)+(-10)+(-17)+....... as far as the 11th term
Answers
Answer: The answer is y=-7x. yeah
I go out star gazing again, but this time the chances of seeing a shooting star are 80% in any given hour. Assuming that the probability of seeing a shooting star is uniform for the entire hour, what is the probability that I will see one during the first 15 minutes of the hour I am out
Answers
The probability of seeing a shooting star during the first 15 minutes of the hour is approximately 19.95%
To solve this problem, we need to use conditional probability. We want to find the probability of seeing a shooting star during the first 15 minutes of the hour given that the chances of seeing a shooting star in any given hour are 80%.
First, we need to determine the probability of seeing a shooting star in the first 15 minutes of the hour. Since the probability is uniform for the entire hour, we can assume that the probability of seeing a shooting star in any given minute is 80% divided by 60 (the number of minutes in an hour), which is approximately 1.33%.
To find the probability of seeing a shooting star during the first 15 minutes of the hour, we need to add up the probabilities of seeing a shooting star in each of the 15 minutes. This can be calculated as follows:
P(seeing a shooting star in the first 15 minutes) = P(seeing a shooting star in minute 1) + P(seeing a shooting star in minute 2) + ... + P(seeing a shooting star in minute 15)
= 15 x 1.33%
= 19.95%
Therefore, the probability of seeing a shooting star during the first 15 minutes of the hour is approximately 19.95%. This means that if you go out star gazing again and the chances of seeing a shooting star are 80%, there is a 19.95% chance that you will see one during the first 15 minutes of the hour you are out.
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What is the total pressure of a mixture of two gases?
The total pressure of a mixture of gases can be defined as the sum of the pressures of each individual gas: Ptotal=P1+P2+… +Pn. + P n . The partial pressure of an individual gas is equal to the total pressure multiplied by the mole fraction of that gas.
Answers
The total pressure of a mixture of two gases is the finalized value of the summation of the total pressure of each of the gases.
The total pressure of a mixture of two gases can be expressed in the form of an equation, as below,
Total Pressure = Pressure of Gas 1 + Pressure of Gas 2
It is noteworthy to be mentioned that the partial pressure of an individual gas is completely in contrast to that of the individual pressure of a gas. It can be expressed in the form of an equation, as below,
Partial Pressure = Total Pressure of Gas x Mole Fraction of Gas
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Help! I DONT GET THIS AT ALL?!
Whoever answers I give points.
Solving Two step inequalities
Which inequality statement below is false? Explain.
(1). 6>6 (3). -4 < 15
(2). 10<10 (4). 3 < 7/2
Please help! And if you do thank you!
Answers
Answer:
Number 3 and 4 are correct, but I have no clue about 1 or 2.
Step-by-step explanation:
I'm just gonna start with number 4
if you put 7/2 into decimals you get 3.5 7/2 is greater than 3
number 3. -4 is in the negative zone, so it is less than 15 which is positive
if I were you, I would guess that number 1 is false. but i cant be sure
1. You decide to buy some new clothes. The subtotal comes to $123.79, but
you have a 15% off coupon. What is the total you will spend? *
Answers
Answer:
$105.22
Step-by-step explanation:
simplify the expression 6(a-4+2b)
Answers
Answer:
6a + 12b - 24
Step-by-step explanation:
What is the special angle pair relationship between ∠1 and ∠3?
Answers
Answer:
see explanation
Step-by-step explanation:
∠ 1 and ∠ 3 are corresponding angles
each is in the same position ( the upper left hand corner) in its group of 4 angles.
find the value of x, round to nearest tenth
Answers
Answer:
x ≈ +15.3
Step-by-step explanation:
Because this is a right triangle, the Pythagorean Theorem applies:
8² + 13² = x² = 233, so that x = +√233, or approximately x ≈ +15.3
find the distance between the points (-3,-2) and (1,-5)
Answers
In how many ways can the letters in the word spoon be arranged?
a. 24
b. 30
c. 60
d. 120
Answers
Choose the missing step in the given solution to the inequality −6x − 10 > 14 + 2x. (1 point)
−6x − 10 > 14 + 2x
_______________
-8x>24
x 14
B.) -8x -10 >14
C.)−4x − 10 < 14
D.) −8x − 10 < 14
Answers
Answer:
- 8x > 24
Step-by-step explanation:
Given
- 6x - 10 > 14 + 2x ( subtract 2x from both sides )
- 8x - 10 > 14 ( add 14 to both sides )
- 8x > 24 ← required missing step
Divide both sides by - 8, reversing the symbol as a result of dividing by a negative quantity.
x < - 3 ← is the solution
Kate has 3 quarts of a 5% sugar solution she wants to mix this with 2478% sugar solution and X quarts of a 12% sugar solution to make a 10% sugar solution how many quarts of a 12% sure solution to Katie‘s
Answers
Answer:
The volume of the 12% sugar solution required, 'X' for a given volume of the 8% sugar solution, 'Y' is;
X = (2·Y + 15)/2
When the volume of the 8% sugar solution, Y = 2 quartz, the volume of the 12% sugar solution required, X = 9.5 gallons
Step-by-step explanation:
The data of the sugar solutions are;
The volume of 5% sugar solution Kate has = 3 quarts
The volume of the 12% sugar solution = X
Let 'Y' represent the volume of the 8% sugar solution, we have;
The volume of the resultant solution, Z = X + Y + 3
The concentration of the resultant solution = 10%
0.05 × 3 + 0.08·Y + 0.12·X = 0.1·(X + Y + 3)
Solving gives the relationship between the volume of the 12% sugar solution, 'X', and the 8% sugar solution, 'Y' in quartz, as follows;
X = (2·Y + 15)/2
When the volume of the 8% sugar solution, Y = 2 quartz, we have;
X = (2 × 2 + 15)/2 = 9.5
Therefore, the volume of the 12% sugar solution Katie requires to make the 10% sugar solution is 9.5 quartz.
You and a group of friends wish to start a company. You have an idea, and you are comparing startup incubators to apply to. (Start up incubators hold classes and help startups to contactventure capitalists and network with one another) Assume funding is normally distributed. Incubator A has a 70% success ratio getting companies to survive at least 4 years from inception. The average venture funding of the 57 companies reaching that 4 year mark,is 1.3 million dollars with a standard deviation of 0.6 million Incubator B has a 39% success ratio getting companies to survive at least 4 years from inception. The average venture funding of the 40 companies reaching that 4 year mark,is 1.9 million dollars with a standard deviation of 0.55 millionAre the success ratios significantly different?
Answers
To determine if the success ratios of Incubator A and Incubator B are significantly different, we can perform a hypothesis test.
Let's set up the null and alternative hypotheses as follows:
Null Hypothesis (H0): The success ratios of Incubator A and Incubator B are not significantly different.
Alternative Hypothesis (H1): The success ratios of Incubator A and Incubator B are significantly different.
We can use a significance level (α) of 0.05, which is a common choice.
To test the hypothesis, we can use a two-proportion z-test since we are comparing the success ratios of two groups. The formula for the test statistic is:
z = (p1 - p2) / sqrt(p_hat * (1 - p_hat) * (1/n1 + 1/n2))
where:
p1 and p2 are the success ratios of Incubator A and Incubator B, respectively,
p_hat is the pooled sample proportion,
n1 and n2 are the sample sizes of Incubator A and Incubator B, respectively.
Let's calculate the test statistic and compare it to the critical value to make a decision.
Given:
Success ratio of Incubator A (p1) = 0.70
Sample size of Incubator A (n1) = 57
Success ratio of Incubator B (p2) = 0.39
Sample size of Incubator B (n2) = 40
First, calculate the pooled sample proportion (p_hat):
p_hat = (x1 + x2) / (n1 + n2)
where x1 is the number of successes in Incubator A and x2 is the number of successes in Incubator B. Since we are not given the actual counts, we cannot calculate the exact value of p_hat.
Next, calculate the test statistic (z) using the formula above.
Once we have the test statistic, we can compare it to the critical value from the standard normal distribution at the specified significance level (α) to make a decision.
If the test statistic falls within the rejection region (i.e., it is beyond the critical value), we reject the null hypothesis. If it falls within the acceptance region (i.e., it is within the critical value), we fail to reject the null hypothesis.
Without knowing the actual counts of successes in Incubator A and Incubator B, we cannot perform the calculations to determine if the success ratios are significantly different.
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Samantha has decided to donate $501,000 to a university. If the
endowment will earn a return of 9%, how much can be spent each year
while ensuring the funds last forever?
Answers
Samantha can spend approximately $45,090 each year from the endowment to ensure the funds last forever, assuming a return rate of 9%.
To determine how much can be spent each year while ensuring the funds last forever, we can use the concept of a perpetuity. A perpetuity is a series of equal payments that continues indefinitely.
The amount that can be spent each year from an endowment can be calculated using the following formula:
Annual Spending = Endowment Amount * Return Rate
In this case, Samantha has decided to donate $501,000 to the university, and the endowment is expected to earn a return of 9%.
Annual Spending = $501,000 * 0.09
Annual Spending = $45,090
Therefore, Samantha can spend approximately $45,090 each year from the endowment to ensure the funds last forever, assuming a return rate of 9%.
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Write a function rule to represent the situation: The volume, V, remaining im a 253ft^3 pile of gravel decreases by .2ft^3 with each shovelful, s, of gravel spread in a walkway. (Brainliest, no bots)
Answers
Answer:
V=243 ft^3 - .2 ft^3s
Step-by-step explanation:
A church window consisting of a rectangle topped by a semicircle is to have perimeter p. Find the radius of the semicircle if the area of the window is to be maximum. Use pi to represent.
Answers
Answer:Function assigns the value of each element of one set to another element of another set. At point, , function A will be maximum.
What is a Function?
A function assigns the value of each element of one set to the other specific element of another set.
Assume that the radius of the semicircle is r, while the length of the rectangle is L. therefore, 2r or the diameter of the width of the rectangle.
Now, the perimeter of the window consists of two lengths, one width, and length of the semicircle, therefore,
The length of the window can be written as,
The area of the window can be written as,
Substituting the value of the length of the window,
Finding the derivative of the Area, A with respect to the radius, r.
Equating the derivative with 0,
Thus, if then A'>0 therefore, the function A is increasing, and when then A'<0 therefore, the function A is decreasing.
Hence, this means that at the point the function A will be maximum.
Function assigns the value of each element of one set to another element of another set. At point, , function A will be maximum.
What is a Function?
A function assigns the value of each element of one set to the other specific element of another set.
Assume that the radius of the semicircle is r, while the length of the rectangle is L. therefore, 2r or the diameter of the width of the rectangle.
Now, the perimeter of the window consists of two lengths, one width, and length of the semicircle, therefore,
The length of the window can be written as,
The area of the window can be written as,
Substituting the value of the length of the window,
Finding the derivative of the Area, A with respect to the radius, r.
Equating the derivative with 0,
Thus, if then A'>0 therefore, the function A is increasing, and when then A'<0 therefore, the function A is decreasing.
Hence, this means that at the point the function A will be maximum.
Function assigns the value of each element of one set to another element of another set. At point, , function A will be maximum.
What is a Function?
A function assigns the value of each element of one set to the other specific element of another set.
Assume that the radius of the semicircle is r, while the length of the rectangle is L. therefore, 2r or the diameter of the width of the rectangle.
Now, the perimeter of the window consists of two lengths, one width, and length of the semicircle, therefore,
The length of the window can be written as,
The area of the window can be written as,
Substituting the value of the length of the window,
Finding the derivative of the Area, A with respect to the radius, r.
Equating the derivative with 0,
Thus, if then A'>0 therefore, the function A is increasing, and when then A'<0 therefore, the function A is decreasing.
Hence, this means that at the point the function A will be maximum.
Step-by-step explanation:
Simply (5+1)^2-(11+3^2)/4
Answers
Answer:
31
Step-by-step explanation:
\((5+1)^2-(11+3^2)/4\)
\((5+1)^{2} \\= 25+5+5+1\\= 36\)
36 - (11+9)/4
= 36 - 20/4
= 36 - 5
= 31
Answer:
Step-by-step explanation:
(5+1)²-(11+3²)÷4
=(6)²-(11+9)÷4
=36-(20)÷4(we use DMAS rule here)
=36-5
=31
The y -intercept of the line whose equation is 3 x + 2 y = 7:
A. 7/2
B. 7/3
C. -3/2
Answers
Answer:
A)
Step-by-step explanation:
Answer:
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