On a piece of graph paper, plot the following points: A (3, 1), B (1, 5), C (9, 9), and D (11, 5). These coordinates will be the vertices of a quadrilateral. How would you use the distance formula and the slope formula to prove that this figure is actually a rectangle?
Answers
The given coordinates are actually a rectangle
How to determine the quadrilateral type?
The coordinates are given as:
A (3, 1), B (1, 5), C (9, 9), and D (11, 5).
Calculate the distance between the coordinates using:
[tex]d = \sqrt{(x_2 -x_1)^2 +(y_2 -y_1)^2[/tex]
So, we have:
[tex]AB = \sqrt{(3 -1)^2 +(1-5)^2} =\sqrt {20[/tex]
[tex]BC = \sqrt{(1 -9)^2 +(5-9)^2} =\sqrt {80[/tex]
[tex]CD = \sqrt{(9 -11)^2 +(9-5)^2} =\sqrt {20[/tex]
[tex]DA = \sqrt{(11 -3)^2 +(5-1)^2} =\sqrt {80[/tex]
The above shows that the opposite sides are congruent
Next, we calculate the slopes using:
m = (y2- y1)/(x2- x1)
So, we have:
AB = (1- 5)/(3-1) = -2
BC = (5- 9)/(1-9) = 1/2
CD = (9- 5)/(9-11) = -2
DA = (5- 1)/(11-3) = 1/2
The slopes of adjacent sides are opposite reciprocals.
This means that the sides are perpendicular
Hence, the given coordinates are actually a rectangle
Read more about quadrilateral at:
https://brainly.com/question/16691874
#SPJ1
Related Questions
The graph of a quartic function crosses the x-axis at -4 and -2 and touches it at 3. State the equation for the family.
Answers
The equation of the quartic function is f(x) = x⁴ - 18 · x² + 6 · x + 72.
How to find the possible equations for a quartic polynomial that passes through the x-axis at three points
Herein we must construct at least a polynomial that satisfies all conditions described in the statement. According to the fundamental theorem of algebra, quartic functions may have no real roots, two real roots or four real roots, which means that one of the roots must have a multiplicity of 2.
The root with a multiplicity of 2 is x = 3 and both x = - 4 and x = - 2 have only a multiplicity of 1, then we have the following expression by using the factor form of the definition of polynomials:
f(x) = (x - 3)² · (x + 4) · (x + 2)
Now we expand the expression to get the standard form:
f(x) = (x² - 6 · x + 9) · (x² + 6 · x + 8)
f(x) = x⁴ - 6 · x³ + 9 · x² + 6 · x³ - 36 · x² + 54 · x + 8 · x² - 48 · x + 72
f(x) = x⁴ - 18 · x² + 6 · x + 72
To learn more on quartic functions: https://brainly.com/question/25285042
#SPJ1
An exponential function f ( x ) = a b x f ( x ) = a b x passes through the points (0, 10000) and (3, 2160). What are the values of a and b ?
Answers
The values of a and b of the exponential function are 10000 and 0.6 respectively
How to solve exponential functions?We are given that the exponential function is expressed in general form as; f(x) = abˣ
where;
a is a non-zero real number called the initial value
b is any positive real number such that
b ≠ 1.
The domain of f is all real numbers.
The range of f is all positive real numbers if a > 0.
The range of f is all negative real numbers if a < 0.
The y-intercept is (0, a)
The horizontal asymptote is; y = 0.
We are told that this exponential function passes through the coordinate points (0, 10000) and (3, 2160).
At coordinate point (0, 10000), we have;
10000 = ab⁰
a = 10000
Now, at the coordinate point (3, 2160), we have;
2160 = 10000(b)³
2160/10000 = b³
0.216 = b³
b = ∛0.216
b = 0.6
Thus, we can conclude that the values of a and b of the given exponential function are respectively 10000 and 0.6.
Read more about Exponential Functions at; https://brainly.com/question/11464095
#SPJ1
Use the information provided to write the general conic form equation of the circle: Ends of a diameter: (11, -2) and (9, 4)
Answers
The equation for a circle is:
[tex](x-a)^{2} + (y-b)^{2} + = r^{2}[/tex]
Where (a,b) is the circle's center and r is the circle's radius.
First, we can find the center point of the circle. Because the two points from the problem are the endpoints of a diameter, the midpoint of the line segment is the center point of the circle.
The formula to find the mid-point of a line segment giving the two endpoints is:
M = [tex](\frac{x_{1} + x_{2} }{2}[/tex], [tex]\frac{y_{1} + y_{2} }{y} )[/tex]
Where M is the midpoint and the given points are:
[tex](x_{1}, y_{1} )[/tex] and [tex](x_{2} , y_{2})[/tex]
Substituting the values from the two points in the problem gives:
[tex]M = (\frac{11+9}{2}[/tex],[tex]\frac{-2+4}{2} )[/tex]
[tex]M = (\frac{11+9}{2}[/tex],[tex]\frac{2-4}{2})[/tex]
[tex]M = (\frac{20}{2} , \frac{2}{2})[/tex]
[tex]M = (10,1)[/tex]
The Rangers won five of the first six games. how many of the next 30 games must the rangers wins have twice as many wins as losses?
Answers
Using proportions, it is found that they must win 19 of their next 30 games.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
We want a win-to-loss proportion of 2:1, which is equivalent to a win-to-total proportion of 2:3. We have that:
The team will have won 5 + x games.The team will have played 36 games.Hence:
[tex]\frac{5 + x}{36} = \frac{2}{3}[/tex]
3(5 + x) = 72
15 + 3x = 72
3x = 57
x = 19.
The Rangers must win 19 of their next 30 games.
More can be learned about proportions at https://brainly.com/question/24372153
#SPJ1
A university is interested in whether there's a difference between students who live on
campus and students who live off campus with respect to absenteeism. Over one
semester, researchers take random samples of on-campus and off-campus students and
record the following number of missed classes over a semester:
On-campus: (3, 4, 0, 6, 2, 1, 3, 3, 5, 2, 4, 4, 6, 5, 2)
Off-campus: (6, 5, 2, 6, 2, 0, 7, 8, 1, 7, 2, 6, 5, 3, 2)
A. Would we use a t confidence interval or a z confidence interval to determine
whether there's a significant difference between the two groups? What are
the conditions for using this kind of confidence interval? Do these data meet
the necessary conditions? Use sketches of modified box-and-whisker plots to
support your decision. (2 points)
B. What are the degrees of freedom (k) for this test using the conservative
method? (Hint: Don't pool and don't use your calculator.) (1 point)
C. What are the sample statistics for this test? Consider on-campus students to
be sample one and off-campus students to be sample two. (2 points)
D. Compute a 95% confidence interval for the difference between the number of
classes missed by each group of students. (2 points)
E. Compute a 90% confidence interval for the difference between the number of
classes missed by each group of students. (2 points)
F. Based on the two confidence intervals you computed in parts d and e, draw a conclusion about the differences between the means of the two groups.
Answers
It is to be noted that the determination of whether or not there is a significant difference between the two groups will be done using a t test.
What is a t test?
A t-test is a statistical test that juxtaposes two samples' means. It is used in hypothesis testing, using a null hypothesis that the variance in group means is zero and an alternative hypothesis that the difference is not zero.
What are the conditions for using this kind of confidence interval?The conditions to use the t test are:
The sample must be independentThe mean of the population and variance must be unknown.The Box plot is attached.What are the degrees of freedom (k) for this test using the conservative method?The degrees of freedom (k) to be utilized for this text will be derived using the conservative method given below:
df = [(s₁²/n₁) + (s₂²/n²)/[((s₁²/n₁)²/((n₁-1)) + (s₂²/n₂)²/((n₂-1))]
= [(3.0952/15) + (6.4095/15)]² / [((3.0952/15)²/14) + ((6.4095/15)²/14)]
= 24.965
Hence,
df ≈ 24 (if approximated to the floor)
What are the sample statistics for this test?Recall the the standard deviation of the population are unequal and unknown. This thus requires that we utilize the two-sample unpooled t-test.
Here, H₀ is given as;
[tex]t = \frac{\bar{x_{1} -\bar{x_{2}}}}{\sqrt{\frac{s_{1}^{2} }{n_{1} } + \frac{S_{2}^{2} }{n_{2}} } } \sim t_{df}[/tex]
t = [(3.33333 - 4.13333)]/√[(3.0952/15) + (6.4095/15)]
= - 0.8/√0.6337
t = - 1.005
What is the 95% confidence interval for the difference between the number of classes missed by each group of students?
The 95% confidence interval is computed using the following formula:
[tex](\bar x_{2} - \bar x_{1}) \pm t_\alpha_/_2_,_df \left({\sqrt{\frac{S_{1}^{2} }{n_{1} } + \frac{S_{2}^{2} }{n_{2}} } } \right)[/tex]
= - 0.8 ± t₀.₀₂₅,₂₄ (√0.6337)
= - 0.8 ± 2.064 (√0.6337)
= -2.4429, 0.8429
What is the a 90% confidence interval for the difference between the number of classes missed by each group of students?To derive the 90% interval, we state:
[tex](\bar x_{2} - \bar x_{1}) \pm t_\alpha_/_2_,_df \left({\sqrt{\frac{S_{1}^{2} }{n_{1} } + \frac{S_{2}^{2} }{n_{2}} } } \right)[/tex]
= - 0.8 ± t₀.₀₅₀,₂₄ (√0.6337)
= - 0.8 ± 0.685 (√0.6337)
= -2.162, 0.562
Based on the two confidence intervals computed in parts d and e, what is the conclusion about the differences between the means of the two groups?
From the intervals computed, we must fail to reject H₀
H₀ : μ₁ = μ₂
It is clear from the above intervals computed from that the differences between the mean of both groups is significant. This is because, zero is included on the two intervals.
Learn more about t-test at;
https://brainly.com/question/6589776
#SPJ1
If 4x+2=5y+3 then y =
Answers
Answer:4x/5-1/5 = y
Step-by-step explanation:
We need to get y by itself
4x+2=5y+3
subtract 3 from both sides
4x-1=5y
divide both sides by 5
4x/5-1/5 = y
Answer:
[tex]4x + 2 = 5y + 3[/tex]
[tex]5y + 3 = 4x + 2[/tex]
[tex]5y = 5x + 2 - 3[/tex]
[tex]5y = 4x - 1[/tex]
[tex] \frac{5y}{5} = \frac{4x - 1}{5} [/tex]
[tex]y = \frac{4x - 1}{5} [/tex]
Scores on the math portion of the SAT are believed to be normally distributed and range from 200 to 800. A researcher from the admissions department at the University of New Hampshire is interested in estimating the mean math SAT scores of the incoming class with 95% confidence. How large a sample should she take to ensure that the margin of error is below 29?
Answers
Using the z-distribution, it is found that she should take a sample of 46 students.
What is a z-distribution confidence interval?
The confidence interval is:
[tex]\overline{x} \pm z\frac{\sigma}{\sqrt{n}}[/tex]
The margin of error is:
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
In which:
[tex]\overline{x}[/tex] is the sample mean.z is the critical value.n is the sample size.[tex]\sigma[/tex] is the standard deviation for the population.In this problem, we have a 95% confidence level, hence[tex]\alpha = 0.95[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so the critical value is z = 1.96.
Scores on the math portion of the SAT are believed to be normally distributed and range from 200 to 800, hence, by the Empirical Rule the standard deviation is found as follows:
[tex]6\sigma = 800 - 200[/tex]
[tex]6\sigma = 600[/tex]
[tex]\sigma = 100[/tex]
The sample size is n when M = 29, hence:
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
[tex]29 = 1.96\frac{100}{\sqrt{n}}[/tex]
[tex]29\sqrt{n} = 196[/tex]
[tex]\sqrt{n} = \frac{196}{29}[/tex]
[tex](\sqrt{n})^2 = \left(\frac{196}{29}\right)^2[/tex]
n = 45.67.
Rounding up, a sample of 46 students should be taken.
More can be learned about the z-distribution at https://brainly.com/question/25890103
#SPJ1
Last PreCalc Question, Need help with writing piecewise functions with graphs. Giving brainliest!
Answers
The piece-wise linear functions can be written as follows:
[tex]f(x) = x, x \leq -2[/tex].[tex]f(x) = -x - 7, -2 < x \leq 1[/tex].[tex]f(x) = 2x - 9, x > 1[/tex].What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.For x equal or less than -2, the line passes through (-3,-3) and (-2,-2), hence the rule is:
[tex]f(x) = x, x \leq -2[/tex].
For x greater than -2 up to 1, the y-intercept is of -7, and the line also passes through (1,-8), hence the rule is:
[tex]f(x) = -x - 7, -2 < x \leq 1[/tex].
For x greater than 1, the function goes through (2,-5) and (3,-3), hence the slope is:
m = (-3 - (-5))(3 - 2) = 2.
The rule is:
y = 2x + b.
When x = 2, y = -5, hence:
-5 = 2(2) + b
b = -9.
Hence:
[tex]f(x) = 2x - 9, x > 1[/tex].
More can be learned about linear functions at https://brainly.com/question/24808124
#SPJ1
A dry Waller wants to order core board for a 15-story elevator shaft. Each floor uses 18.5 sheets.Calculate the number of boards needed. Add 10%of the total for waste.
Answers
Answer:
305.25
Step-by-step explanation:
18.5 for one floor, multiply by 15 to get the amount for 15 floors: 18.5*15 = 277.5 sheets
We need to add 10% for waste, meaning 277.5 plus the waste: 1.1*277.5 = 305.25
The drywaller needs to order 306 core boards for the 15-story elevator shaft, considering the 10% waste.
How to determine the number of boards needed. Add 10%of the total for wastecalculate the number of core boards needed for the 15-story elevator shaft, we can follow these
Calculate the total number of sheets for all floors:
Total sheets = Number of floors × Sheets per floor
Total sheets = 15 floors × 18.5 sheets per floor
Total sheets = 277.5 sheets
Add 10% of the total for waste:
Waste percentage = 10% = 0.10 (in decimal form)
Waste sheets = Total sheets × Waste percentage
Waste sheets = 277.5 sheets × 0.10
Waste sheets = 27.75 sheets
Calculate the final number of boards needed (including waste):
Number of boards = Total sheets + Waste sheets
Number of boards = 277.5 sheets + 27.75 sheets
Number of boards = 305.25 sheets
Since you can't have a fraction of a sheet, we need to round up to the nearest whole number.
Therefore, the drywaller needs to order 306 core boards for the 15-story elevator shaft, considering the 10% waste.
Learn more about expression at https://brainly.com/question/22688504
#SPJ3
: A native wolf species has been reintroduced into a national forest. Originally 210 wolves were transplanted. After 10 years, the population had grown to 410 wolves. If the population grows exponentially, write an explicit formula for the number of wolves, where w is the number of wolves and t is the number of years.
Answers
The explicit formula that can be used to determine the number of wolves is w = 210 (1.0692^t) .
What is the explicit formula?The first step is to determine the growth rate of the wolf species.
Growth rate = [(future population / present population)^(1/number of years)] - 1
[(410 / 210)^(1/10)] - 1 = 6.92
The exponential function would have the form:
FV = P (1 + r)^t
FV = Future population P = Present populationR = rate of growtht = number of yearsw = 210 (1.0692^t)
To learn more about exponential functions, please check: https://brainly.com/question/26331578
#SPJ1
What are the length and width of a rectangle if the length is
3 inches longer than twice the width and the area of the
rectangle is 5 in2?
Answers
The length and width of the rectangle are 5 inches and 1 inches respectively.
How to find the length and width of a rectangle?The length and width of the rectangle can be found as follows;
l = 3 + 2w
area of a rectangle = lw
where
l = lengthw = widthTherefore,
5 = lw
5 = (3 + 2w)w
5 = 3w + 2w²
2w² + 3w - 5 = 0
Hence,
2w² + 3w - 5 = 0
Therefore,
w = 1 and w = - 5 / 2
width = 1 inches
length = 3 + 2(1) = 5 inches
learn more on rectangle here: https://brainly.com/question/14026521
#SPJ1
Compute the monthly payments for the add on interest loan. The amount of the loan is $8,276.17. The annual interest rate is 5.7%. The term of the loan is 5.5 years.
Answers
The monthly payments for this add on interest loan are of $164.71.
Given Information and Formula Used
It is given that for an add on interest loan,
Principal Amount, p = $8,276.17
Annual Interest Rate, r = 5.7%
Term of the loan, T = 5.5 years
The formula for simple interest is given as follows,
I = (p)(r)(t)/100 ............... (1)
The formula for total amount of add on interest is given by,
A = p + I ....................... (2)
Computing the Interest
Substitute the given values of p, r, and t in the formula (1) of interest to get,
I = (8276.17)(5.7)(5.5)/100
I = 259457.9295/100
I = $2,594.58
Computing the Monthly Payment for Add-on Interest Loan
Substituting the values of p and I in the formula (2), we obtain the total amount as,
A = $ (8276.17 + 2594.58)
A = $ 10,870.75
Monthly payment for the add on interest loan = A/t(in months)
= $ (10,870.75/66)
= $164.71
Therefore, monthly payments of $164.71 are to be made for the add on interest loan.
Learn more about interest here:
https://brainly.com/question/26457073
#SPJ1
which of the following functions is graphed below?
Answers
The graphed piecewise function is the one in option B.
Which of the given functions is the graphed one?
On the graph, we can see that on the lowest part we have a closed dot at x = 1
And the above part has a open dot at x = 1.
Then the piecewise function is of the form:
y = f(x) for x ≤ 1y = g(x) for x > 1Such that the above part seems to be quadratic, and the part below seems to have a larger degree.
With that, we can conclude that the correct option is:
y = x^2 + 4, for x > 1y = x^3 - 2, for x ≤ 1Which is option b.
If you want to learn more about piecewise functions:
https://brainly.com/question/27262465
#SPJ1
Assume a = b and c ≠ 0
Which of the following sentences is not true?
1) a+c=b+c
2) a-c=b-c
3) ac = bc
4) a/c = b/c
5) none of these
Answers
Answer:
None of these is not true
36-u=261 solve for u
Answers
Answer:
u = -225
Step-by-step explanation:
36 - u = 261
-u = 261 - 36
-u = 225
u = -225
The perimeter of a regular hexagon is 72 inches. Find the length of each of its sides.
Answers
12 inches
Step-by-step explanation:A regular polygon is any 2-D shape where all the sides and angles are congruent.
Regular Hexagon
As denoted by the prefix "hexa-", hexagons have 6 sides and internal angles. The perimeter is equal to all 6 sides added together. However, we know that these sides must be congruent because it is a regular hexagon.
Solving for Perimeter
We can create an equation that describes the perimeter of this shape, with "x" representing the length of one side.
6x = 72Since all the sides are equal we can use multiplication instead of addition. To solve this equation, divide both sides by 6.
x = 12This means that all sides must be 12 inches.
tiff basic insurance cost is $613.50 per year. His insurance company offers a typical "safe-driver discount." What would be his rate if he went three years without an accident? Annual premium?
Answers
The rate if he went three years without an accident based on the information about the insurance is $552.15.
How to calculate the insurance?From the information given, his basic insurance cost is $613.50 per year and his insurance company offers a typical "safe-driver discount.
The amount will be:
= 613.50 - (20.45 × 3)
= 552.15
In conclusion, the rate is $552.15.
Learn more about insurance on:
brainly.com/question/25855858
#SPJ1
A pitcher originally contains a juice drink with 20 percent cranberry juice. After 4 ounces of cranberry juice is added, the new drink is one-fourth cranberry juice. How many total ounces of juice drink are in the pitcher after the addition of the cranberry juice?
Answers
20%=0.2
1/4=0.25
0.2x+4=0.25x
-0.2x -0.2x
4=0.05x
4/0.05=x
80=x
80 total ounces
Landscape artists frequently hand-draw their landscape layouts (blueprints) because this allows them more creativity and precision over their plans. Although done by hand, the layouts must be extremely accurate in terms of angles and distances.
a. A landscape artist has drawn the outline of a house. Describe three different ways to make sure the corners of the house are right angles.
b. A bench needs to be placed in the exact middle of two trees. Describe three different methods the designer can use to find out where to place the bench.
c. The landscape designer has drawn an angle on one side of the yard and wants to draw the same angle on the other side. Describe three different ways he can do this.
Answers
The information given about the landscape is illustrated below:
How to explain the landscape?a. You'll need a measuring tape, two range poles, pegs, and three people.
The first person holds the zero mark between their thumbs and fingers, the second person holds the 3m mark on the tape between their thumbs and fingers, and the third person holds the 8m. When all sides of the rope are stretched, a triangle is created, and an angle near one is a right angle.
Wrap one loop of the rope around peg A with a peg through the other loop, draw a circle on the ground, place pegs B and C where the circle crosses the base line, and place peg D half way between pegs B and C, allowing pegs D and A to form lines perpendicular to the base line, forming a right angle.
B. Method 1; Take a measurement of the distance between both trees and divide by 2 to get the midpoint.
Method 2; By use of a compass to bisect the distance between both points.
Method 3; By using a protractor to mark the 90° vertical point when placed between the two endpoints.
C The three different ways he can do this.
1) by using a protractor
2) by finding a point
3) by extending the line of the angle until it reaches the other side of the field.
Read more about landscape on;
https://brainly.com/question/16825239
#SPJ1
In 2016, Alberta had about 4.2 million people.
Assuming they follow the same population
growth rate, it is predicted they will have 6.65
million people in 20 years. At what rate is the
province's population growing?
Answers
The province's population is growing at the rate of 58.34 %
Rate of change is used to mathematically describe the percentage change in value over a defined period of time, and it represents the momentum of a variable. The calculation for ROC is simple in that it takes the current value of a stock or index and divides it by the value from an earlier period.
Given:
Initial Population = 4.2 million
Population after 20 years = 6.65 million
Change in population = 6.65 - 4.2 = 2.45 million
Rate at which the province's population growing is
= [tex]\frac{Change in population}{Inital Population}[/tex] x100 %
= [tex]\frac{2.45}{4.2}\\[/tex] x 100%
= 58.34 %
Thus the province's population is growing at the rate of 58.34 %
Learn more about Rate here :
https://brainly.com/question/13103052
#SPJ1
ABCD is a rectangle, with M the midpoint or BC and N the midpoint of CD. If CM=4 and NC=5, what percent of the area of the rectangle is shaded
Answers
Answer:
87.5%
Step-by-step explanation:
Given:
⇒ ABCD is a rectangle, with M the midpoint of BC and N the midpoint of CD.⇒ CM = 4 and NC = 5.Area of triangle NCM
⇒ 1/2 × base × height
⇒ 1/2 × CN × CM
⇒ 1/2 × 5 × 4
⇒ 10cm²
Length of rectangle ⇒ 10
Breadth of the rectangle ⇒ 8
↓
Area of rectangle ⇒ Length × Breadth
Area of rectangle ⇒ 10 × 8 = 80cm²
Area of shaded region ⇒ Area of rectangle - Area of triangle
Area of shaded region ⇒ 80 - 10 = 80cm²
Percentage of the shaded region = Shaded/complete rectangle × 100
⇒ 70/80 × 100 = 87.5
⇒ The percent of the area of the rectangle shaded is therefore 87.5%.
Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options. x < 5 –6x – 5 < 10 – x –6x + 15 < 10 – 5x A number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right. A number line from negative 3 to 3 in increments of 1. An open circle is at negative 5 and a bold line starts at negative 5 and is pointing to the left.
Answers
The solution to the inequality expression is x>5 and the correct representation is a number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.
Inequality expressionInequality expressions are expression not separated by an equal sign
Given the following inequality expression
–3(2x – 5) < 5(2 – x)
Expand
-6x + 15 < 10 - 5x
Collect the like terms
-6x + 5x < 10 -15
-x < -5
x > 5
Hence the solution to the inequality expression is x>5 and the correct representation is a number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.
Learn more on inequality here: https://brainly.com/question/24372553
#SPJ1
I need help with my work
Answers
The area of the interior above the polar axis is -0.858 square units
The area bounded by a polar curveThe area bounded by a polar curve between θ = θ₁ and θ = θ₂ is given by
[tex]A = \int\limits^{\theta_{2} }_{\theta_{1} } {\frac{1}{2}r^{2} } \, d\theta[/tex]
Now, since we have the curve r = 1 - sinθ and we want to find the area of the interior above the polar axis, we integrate from θ = 0 to θ = π, since this is the region above the polar axis.
So, [tex]A = \int\limits^{\theta_{2} }_{\theta_{1} } {\frac{1}{2}r^{2} } \, d\theta\\= \int\limits^{\pi}_{0} {\frac{1}{2}(1 - sin\theta)^{2} } \, d\theta\\= \int\limits^{\pi}_{0} {\frac{1}{2}[1 - 2sin\theta + (sin\theta)^{2}] } \, d\theta\\= \int\limits^{\pi}_{0} {\frac{1}{2}\, d\theta - \int\limits^{\pi}_{0} 2sin\theta \, d\theta+ \int\limits^{\pi}_{0} (sin\theta)^{2} } \, d\theta\\[/tex]
[tex]A = \int\limits^{\pi}_{0} {\frac{1}{2}\, d\theta - 2\int\limits^{\pi}_{0} sin\theta \, d\theta+ \int\limits^{\pi}_{0} \frac{(1 - cos2\theta)}{2} } \, d\theta\\= \int\limits^{\pi}_{0} {\frac{1}{2}\, d\theta - 2\int\limits^{\pi}_{0} sin\theta \, d\theta+ \int\limits^{\pi}_{0} \frac{1}{2} } \, d\theta -\int\limits^{\pi}_{0} \frac{cos2\theta}{2} } \, d\theta\\[/tex]
[tex]A = \int\limits^{\pi}_{0} \, d\theta - 2\int\limits^{\pi}_{0} sin\theta \, d\theta -\int\limits^{\pi}_{0} \frac{cos2\theta}{2} } \, d\theta\\= [\theta]_{0}^{\pi} - 2[-cos\theta]^{\pi} _{0} - [\frac{sin2\theta}{4}]_{0}^{\pi} \\= [\theta]_{0}^{\pi} + 2[cos\theta]^{\pi} _{0} - [\frac{sin2\theta}{4}]_{0}^{\pi}\\= [\pi - 0] + 2[cos\pi - cos0] - \frac{ [sin2\pi - sin0]}{4}\\= \pi + 2[-1 - 1] - \frac{ [0 - 0]}{4}\\= \pi + 2[-2] - \frac{ [0]}{4}\\= \pi - 4 - 0\\= \pi - 4\\= 3.142 - 4\\= -0.858[/tex]
So, the area of the interior above the polar axis is -0.858 square units
Learn more about area bounded by polar curve here:
https://brainly.com/question/27624501
#SPJ1
Two-thirds of a number x is at least 10. Find the smallest possible prime number x.
Answers
x>15
Smallest prime number of x is 17
The smallest possible prime number x is 17.
What is Fraction?An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a correct fraction is smaller than the denominator.
given, Two-thirds of a number x is at least 10.
So,
2/3 x ≥ 10
x ≥ 15
First prime number that is bigger than 15 is 17.
therefore, The smallest possible prime number x is 17.
Learn more about fraction here:
https://brainly.com/question/10354322
#SPJ2
What is the current yield on a 3 year bond with 10% annual coupons, a par value of 100, and a current price of 107.87?
Answers
The current yield of the bond is 9.27%.
What is current yield?
Current yield is the bond rate of return computed as the annual coupon divided by the current price of the bond
The annual coupon is the coupon rate of 10% multiplied by the bond's par value of 100
annual coupon=10%*100
annual coupon=10
current bond price=107.87
current yield=annual coupon/current bond price
current yield=10/107.87
current yield=9.27%
Find more on current yield on:https://brainly.com/question/12909555
#SPJ1
f(x)=2x. If g(x) is a vertical stretch, compression, and or reflection of f(x) followed by a, what is the equation of g(x)?
Answers
The function g(x) is g(x)= (3x)^2
How to solve for g(x)?
The complete question is in the image
From the graph in the image, we have:
f(x) = x^2
The function f(x) is stretched by a factor of 3 to form g(x).
This means that:
g(x) = f(3x)
So, we have:
g(x)= (3x)^2
Hence, the function g(x) is g(x)= (3x)^2
Read more about function transformation at:
brainly.com/question/10222182
#SPJ1
46.Find the quotient and remainder
Answers
Answer:
30624; 1514,6; a1,6.
The inverse of G(x) is a function.
• A. True
• B. False
Answers
Answer: False
Step-by-step explanation:
G(x) is not a one-to-one function (since it fails the horizontal line test). Therefore, it doesn't have an inverse.
Cuál es el número que no pertenece a esta sucesión 99 koma 105 koma 111 koma 117 koma 123 koma 129 koma 135 koma 141 koma 147 koma 153 Cuál es el número que no pertenece esta sucesión 99coma 105 koma 111 koma 117 koma 123 koma 129 koma 135 koma 141 koma 147 koma 153
Answers
Based on the calculations, all of the numbers belong to this arithmetic sequence.
How to calculate an arithmetic sequence?Mathematically, the nth term of an arithmetic sequence can be calculated by using this expression:
[tex]a_n = a_1 +(n-1)d[/tex]
Where:
d is the common difference.a₁ is the first term of an arithmetic sequence.n is the total number of terms.Next, we would determine the common difference as follows:
d = a₂ - a₁
d = 105 - 99 = 6.
d = a₃ - a₂
d = 111 - 105 = 6.
d = a₄ - a₃
d = 117 - 111 = 6.
Based on the calculations, all of the numbers belong to this arithmetic sequence.
Read more on arithmetic sequence here: brainly.com/question/12630565
#SPJ1
Complete Question:
What is the number that does not belong to this sequence 99, 105, 111, 117, 123, 129, 135, 141, 147, 153
x minus 2 is equal to 7
Answers
Add both values 7 + 2 = 9, which is x.
9 - 2 = 7.