A line passes through the point (-8,-3) and has a slope of -3/4
Answers
The equation of the line that passes through the point (-8,-3) and has a slope of -3/4 is y = -3/4x - 9
How to determine the equation of the line?From the question, we have the following parameters that can be used in our computation:
Slope, m = -3/4
Points, (x₁, y₁) = (-8, -3)
A linear equation can be represented as
y = slope * x + y-intercept i.e. y = m(x - x₁) + y₁
Substitute the known values in the above equation
So, we have the following equation
y = -3/4(x + 8) - 3
Open the brackets
So, we have the following equation
y = -3/4x - 6 - 3
Evaluate the like terms in the above equation
So, we have the following representation
y = -3/4x - 9
The above equation cannot be further simplified
Hence. the linear equation is y = -3/4x - 9
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Related Questions
A measuring Task objective => To investigate your local athletics track to see wheather it is worked fairly for runners. who start on different lines.
Answers
The Task objective chosen is: A Measuring Task
The Objective is: To investigate the fairness of a local athlete's track for runners starting on different lines.
How do one carry about the investigation about?To carry about the investigation, follow the steps given below.
Choose a nearby sportsman's racetrack that you wish to examine.Split the pathway into several lanes, with a minimum of three lanes.Determine the length of each lane by utilizing either a measuring tape or measuring wheel. Make sure the measurements are precise.Place cones or markers to designate the starting points for each lane.Request the presence of a cohort of individuals who engage in running to take part in the study.Record the duration of each runner in completing the track for every lane.Evaluate the outcomes and contrast the timings of athletes beginning from distinct tracks.Closely examine the time variances among distinct lanes.Draw inferences from the examination. Inconsistent differences in lane times may indicate an unfair advantage for certain runners depending on their starting position.Lastly, make a crossword puzzle that includes mathematical vocabulary in a clever manner.
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```Class IX```
*Mathematics Project work*
*Do any one*
1. *A Measuring Task*
To investigate your local athletes track to see whether it is worked fairly for runners who start on different lines.
2. *Design a crossword puzzle with mathematical terms*
To review mathematics vocabulary, to give the opportunity for creative expression in designing puzzles, to act as a means of motivating the study of a given unit and to given recreation .
Which geometric property is illustrated for the lengths of line segments: If DE = FG then FG = DE A. Reflexive Property B. Symmetric Property C. Transitive Property D. this is not a geomtric property as it does not hold
Answers
Answer:
I'm sorry I wish if I could help you but not sure of the answe
Answer:
B.
Symmetric Property
Step-by-step explanation:
Consider the following two lines: one with parametric equations x(s)=4−2s,y(s)=−2+s,z(s)=1+3s, and the other being the line through (−4,2,17) in the direction v=⟨−2,1,5⟩.a) Find a direction vector for the first line, which is given in parametric form.b) Find parametric equations for the second line, written in terms of the parameter t.c) Show that the two lines intersect at a single point by finding the values of sand tthat result in the same point.d) Find the angle formed where the two lines intersect, noting that this angle will be given by the angle between their respective direction vectors.e) Find an equation for the plane that contains both of the lines described in this problem
Answers
A-The first line has a direction vector of ⟨-2, 1, 3⟩, b-the second line has parametric equations x(t) = -4 - 2t, y(t) = 2 + t, z(t) = 17 + 5t, c-the two lines intersect at the point (1, 3, 10), d-the angle formed is 15.2 degrees, and e- the equation containing both lines is -2x + 7y - 5z = -59.
What is direction vector ?
A direction vector, also known as a directional vector or simply a direction, represents the direction of a line, vector, or a linear path in three-dimensional space. It is a vector that points in the same direction as the line or path it represents.
a) The direction vector for the first line is given by ⟨-2, 1, 3⟩.
b) The parametric equations for the second line, written in terms of the parameter t, are x(t) = -4 - 2t, y(t) = 2 + t, z(t) = 17 + 5t.
c) To find the intersection point, we set the x, y, and z coordinates of both lines equal to each other and solve for s and t:
4 - 2s = -4 - 2t
-2 + s = 2 + t
1 + 3s = 17 + 5t
Solving this system of equations yields s = 3 and t = 1. Therefore, the two lines intersect at the point (1, 3, 10).
d) The angle formed at the intersection point is given by the angle between their respective direction vectors. Using the dot product, the angle θ can be found as cos(θ) = (⟨-2, 1, 3⟩ · ⟨-2, 1, 5⟩) / (|⟨-2, 1, 3⟩| |⟨-2, 1, 5⟩|), which simplifies to cos(θ) = 0.96. Taking the inverse cosine, we find θ ≈ 15.2 degrees.
e) To find the equation of the plane containing both lines, we can use the point-normal form of a plane equation. We choose one of the intersection points (1, 3, 10) and use the cross product of the direction vectors of the two lines as the normal vector. The equation of the plane is given by -2x + 7y - 5z = -59.
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Which one is the answer
Answers
Answer:
B is the answer
Step-by-step explanation:
use function notation (with the appropriate functions above) to represent the area of a regular pentagon whose perimeter is 140 cm.
Answers
However, we need to substitute a with s since that is the value we have calculated. Therefore, we get \(A(s) = (1/4)(5 + sqrt(5))s^2.\) This is the function notation that represents the area of a regular pentagon whose perimeter is 140 cm.
Let's consider that s be the length of a side of the regular pentagon.
The perimeter of the regular pentagon will be 5s. Therefore, we have the equation:5s = 140s = 28 cm
Also,
we have the formula for the area of a regular pentagon as:
\($A=\frac{1}{4}(5 +\sqrt{5})a^{2}$,\)
where a is the length of a side of the pentagon.
In order to represent the area of a regular pentagon whose perimeter is 140 cm, we need to substitute a with s, which we have already calculated.
Therefore, we have:\(A(s) = $\frac{1}{{4}(5 +\sqrt{5})s^{2}}$\)
Now, we have successfully used function notation (with the appropriate functions above) to represent the area of a regular pentagon whose perimeter is 140 cm.
The area of a regular pentagon can be represented using function notation (with the appropriate functions above). The first step is to calculate the length of a side of the regular pentagon by dividing the perimeter by 5, since there are 5 sides in a pentagon.
In this case, we are given that the perimeter is 140 cm, so we get 5s = 140, which simplifies to s = 28 cm. We can now use the formula for the area of a regular pentagon, which is\(A = (1/4)(5 + sqrt(5))a^2\), where a is the length of a side of the pentagon.
However, we need to substitute a with s since that is the value we have calculated. Therefore, we get\(A(s) = (1/4)(5 + sqrt(5))s^2.\) This is the function notation that represents the area of a regular pentagon whose perimeter is 140 cm.
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8
Which does NOT describe the quadrilateral below?
O A. trapezoid
B. square
C. parallelogram
О
D. rectangle
Answers
1., express the following properties in propositional logic:
(a) For every location that is a cliff, there is an
adjacent location to it that contains some
non null quantity of resource r3.
(b) For every location that contains some
non null quantity of resource r2,
there is exactly one adjacent location that is a hill
.
(c) Resource r1 can only appear in the corners of the
grid (the corners of the grid are the locations
(1, 1), (K, 1), (1, K), (K, K)).
Answers
(a) The proposition can be expressed as ∀x(Cliff(x) → ∃y(Adjacent(x, y) ∧ NonNull(y, r3))).
(b) The proposition can be expressed as ∀x(NonNull(x, r2) → (∃y(Adjacent(x, y) ∧ Hill(y)) ∧ ¬∃z(Adjacent(x, z) ∧ Hill(z) ∧ ¬(z = y)))).
(c) The proposition can be expressed as ∀x(Resource(x, r1) → (Corner(x) ∧ ¬∃y(Resource(y, r1) ∧ ¬(x = y) ∧ Adjacent(x, y)))).
(a) In propositional logic, we use quantifiers (∀ for "for every" and ∃ for "there exists") to express the properties. The proposition (a) states that for every location that is a cliff (Cliff(x)), there exists an adjacent location (Adjacent(x, y)) to it that contains some non-null quantity of resource r3 (NonNull(y, r3)).
(b) The proposition (b) states that for every location that contains some non-null quantity of resource r2 (NonNull(x, r2)), there is exactly one adjacent location (y) that is a hill (Hill(y)), and there are no other adjacent locations (z) that are hills (¬(z = y)).
(c) The proposition (c) states that resource r1 (Resource(x, r1)) can only appear in the corners of the grid (Corner(x)), and there are no other adjacent locations (y) that contain resource r1 (Resource(y, r1)).
By using logical connectives (∧ for "and," ∨ for "or," ¬ for "not"), quantifiers (∀ for "for every," ∃ for "there exists"), and predicate symbols (Cliff, NonNull, Resource, Hill, Corner), we can express these properties in propositional logic to represent the given statements accurately.
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If a loan is taken out for $278 at 10% and costs the borrower $174.12 in simple interest, how many years was the loan for? ROUND YOUR ANSWER TO THE NEAREST WHOLE YEAR
Answers
Data:
Loan=$278 at 10%
A simple interest is calculated for payments on the initial capital.
If the total interest was $174.12.
If the interest is on a year. You calculated the interes of one year. The 10% of $278:
\(278\cdot\frac{10}{100}=27.8\)In a year the interest is $27.8.
Then, $174.12 divided into $27.8 is the number of years of the loan:
\(\frac{174.12}{27.8}=6.26\approx6\)Then, the loan was for 6 yearsMultiply: (6a³3b³) (5a² + 46²)
O 11a³ + 10a³b² + 2a²b³ + b³
O 30a5 +24a³b² - 15a²b³ - 1265
O 30a5 +9a5b5 - 1265
O30a6 +24a³ b² - 15a² b3 - 1266
Answers
30a^5 + 24a^3b^2 + 15a^2b^3 + 1386a^3b^3
Step by step explanation:
To get this answer, we need to distribute the terms of the first expression (6a^3b^3) to the terms of the second expression (5a^2 + 46^2) using the distributive property:
(6a^3b^3) (5a^2 + 46^2) = 30a^5b^3 + 276a^3b^3 + 15a^2b^4 + 1386a^3b^3
We can simplify this expression by combining like terms, which gives us:
30a^5 + 24a^3b^2 + 15a^2b^3 + 1386a^3b^3
Therefore, the correct answer is 30a^5 + 24a^3b^2 + 15a^2b^3 + 1386a^3b^3.
4 less than a number g equals 9
Answer fast I will name you the most Brainly please help
Answers
9514 1404 393
Answer:
g - 4 = 9
Step-by-step explanation:
If the number is g, then four less than the number is g-4.
If that is 9, then you have the equation ...
g - 4 = 9
_____
Additional comment
If you add 4 to both sides of the equation, you have its solution.
g -4 +4 = 9 +4
g = 13
So, 4 less than 13 is 9.
Dustin has a spinner that is divided into 5 equal-size sections colored pink, blue, orange, white, and green. What is the probability that Dustin spins pink on the 1st spin and then spins a color other than pink on the 2nd spin?
Answers
Answer:
1/6
Step-by-step explanation:
1st spin:
1 out of the 5 sections are pink, so the probability of spinning a pink on the 1st spin is 1/5.
2nd spin:
5 out of the 6 sections are not pink, so the probability of spinning a pink on the 2nd is 5/6.
Multiply the probabilities of the two spins to get the answer.
\(\frac{1}{5} *\frac{5}{6} =\frac{5}{30} =\frac{1}{6}\)
Pablo is studying the function f(x) shown in the graph. He claims that he can transform the function to include the
ordered pair (2, 2).
Select the function that supports his claim.
A. q(x) = f(x - 2)
B. r(x) = f(x) - 2
C. s(x) = f(x) + 2
D. t(x) = f(x + 2)
Answers
Answer:
The first option will be your answer
Step-by-step explanation:
A. q(x) = f(x - 2)
have a nice day!!
Find Cos(A)
Thanks!!!
Answers
Answer:
Cos(A) = 12 / 13
Step-by-step explanation:
Cos(A) = Adjacent / Hypotenuse
Cos(A) = 12 / 13
Hope this helps!
11. Engineering The maximum load for a certain elevator is 2000 pounds. The total
weight of the passengers on the elevator is 1400 pounds. A delivery man who weighs
243 pounds enters the elevator with a crate of weight w. Write, solve, and graph an
inequality to show the values of w that will not exceed the weight limit of the elevator.
Answers
The inequality to show the values of [w] that will not exceed the weight limit of the elevator is w + 1643 ≤ 2000. On solving the inequality, we get w ≤ 357. The graph of the inequality is attached.
What is inequality?In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size.An inequality is a mathematical relationship between two expressions and is represented using one of the following -≤ : less than or equal to
≥ : greater than or equal to
< : less than
> : greater than
≠ : not equal to
Given is the maximum load for a certain elevator is 2000 pounds. The total weight of the passengers on the elevator is 1400 pounds. A delivery man who weighs 243 pounds enters the elevator with a crate of weight [w].
We can write the inequality as follows -1400 + 243 + w ≤ 2000
w + 1643 ≤ 2000
Solving the inequality, we get -w + 1643 ≤ 2000
w ≤ 2000 - 1643
w ≤ 357
Refer to the graph attached.Therefore, the inequality to show the values of [w] that will not exceed the weight limit of the elevator is w + 1643 ≤ 2000. On solving the inequality, we get w ≤ 357. The graph of the inequality is attached.
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ms. forsythe gave the same algebra test to her three classes. the first class averaged $80\%$, the second class averaged $85\%$, and the third $89\%$. together, the first two classes averaged $83\%$, and the second and third classes together averaged $87\%$. what was the average for all three classes combined? express your answer to the nearest hundredth.
Answers
Let x= the average for all three classes combined. So x=85.25.
What is average?When you add two or more numbers and divide the result by the number of numbers you added together, you obtain an average.
How to calculate average?The arithmetic mean is determined by adding a collection of numbers, dividing by their count, and obtaining the result.
average = total points / number of students
total points = average*number of students
Let the total number of students in each class = a , b and x
The total number of points the first class amassed was 80a
The total number of points amassed by the second class was 85b
The total number of points amassed by the third class = 89c
For the first two classes we have
[ 80a + 85b ] / [ a + b] = 83
80a + 85b = 83a + 83b
subtract 80a, 83b from both sides
3a = 2b
a = 2/3b
For the second two classes we have
[ 85b + 89c ] / [ b + c ] = 87
[ 85b + 89c ] = 87 [b + c]
85b + 89c = 87b + 87c
subtract 85b, 87c from both sides
2c = 2b
b = c
For the three classes.....
Total points by all three classes / number of class members = the average for all three classes
[ 80a + 85b + 89c ] / [ a + b + c ] =[sub for a and c ]
[80 (2/3)b + 85b + 89b] / [ (2/3)b + b + b ]
b [ 80 (2/3) + 85 + 89 ] / [ b [( 2/3) + 1 + 1] ] [cancel the b's ]
[ 80 (2/3) + 85 + 89] [ 8/3 ]
[ 160/3 + 85 + 89 ] / [8/3] =
[ 160/3 + 255/3 + 267/3] / [8/3] multiply top/bottom by 3
[160 + 255 + 267 ] / 8 =
85.25 = average for all three classes
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6-4 properties of rhombuses rectangles and squares
Answers
Quadrilaterals, rectangles, and squares are all quadrilaterals, which means they have four sides. However, there are some differences in their properties. Here are the 6 properties of rhombuses, rectangles, and squares:
Properties of Rhombuses:
A rhombus has four congruent sides.Opposite angles of a rhombus are congruent.Diagonals of a rhombus bisect each other at right angles.The diagonals of a rhombus are perpendicular.The area of a rhombus can be calculated as A = ½ (d1 x d2), where d1 and d2 are the lengths of the diagonals.The perimeter of a rhombus is equal to four times the length of one side.Properties of Rectangles:
A rectangle has four right angles.Opposite sides of a rectangle are congruent.The diagonals of a rectangle are congruent.The area of a rectangle can be calculated as A = l x w, where l is the length and w is the width.The perimeter of a rectangle can be calculated as P = 2(l + w), where l is the length and w is the width.The diagonals of a rectangle bisect each other.Properties of Squares:
A square has all four sides congruent.All four angles of a square are right angles.The diagonals of a square are congruent and bisect each other at right angles.The area of a square can be calculated as A = s^2, where s is the length of one side.The perimeter of a square can be calculated as P = 4s, where s is the length of one side.The diagonals of a square divide it into four congruent right triangles.For more details about quadrilaterals click here:
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The cost to fill a car's tank with gas and get a car wash is a linear function of the capacity of the tank. The costs
of a fill-up and a car wash for three different customers are shown in the table. Write an equation for the
function in slope-intercept form. Then, find the cost of a fill-up and a car wash for a customer with a truck
whose tank size is 25 gallons.
Answers
Answer:
Option (2)
Step-by-step explanation:
Table given in the question shows the relation between the size of tank (x) and cost of gas to fill the tank f(x).
Since, relation is a linear function in the form of f(x) = mx + b
Graph of the function will be straight line.
Here 'm' = slope of the line
b = y-intercept
Since, two points (9, 23.55) and (15, 36.75) lie on this line.
Slope of the line = \(\frac{y_2-y_1}{x_2-x_1}\)
= \(\frac{36.75-23.55}{15-9}\)
= \(\frac{13.2}{6}\)
= 2.2
By substituting the value of 'm' in the equation,
f(x) = 2.2x + b
Since, the given line passes through (9, 23.55),
23.55 = 2.2(9) + b
b = 23.55 - 19.8
b = 3.75
Equation of the function will be,
f(x) = 2.2x + 3.75
Cost to fill the tank and a car wash with size of the tank = 25 gallons,
f(25) = 2.2(25) + 3.75
= 55 + 3.75
= $58.75
Therefore, Option (2) will be the answer.
Solve |x + 4| - 5 =6.
Answers
Answer:
The answer is x =7 and x= - 15
Explanation
\( |x + 4| - 5 = 6 \\ |x + 4| = 6 + 5 \\ |x + 4| = 11 \\ x + 4 = 11 \: \:and \: \: x + 4 = - 11 \\ x = 11 - 4 \: \: and \: \: x = - 11 - 4 \\ x = 7 \: \: and \: \: x = - 15\)
122. which of the following statements about influential scores are true? i. influential scores have large residuals. ii. removal of an influential score sharply affects the regression line. i. an x-value that is an outlier in the -variable is more indicative that a point is influential than a y-value that is an outlier the y-variable. (a) i and ii (b) 1 and iii (c) ii and iii (d) i, il, and ii (e) none of these are true
Answers
Influential scores, indicated by large residuals, have a significant impact on the regression line when removed from the data. These points, characterized by extreme x or y-values, can alter the slope and intercept of the regression line, emphasizing their importance in regression analysis.
The correct statements about influential scores are i and ii.
i. Influential scores have large residuals because they have a strong effect on the regression line. This means that if we remove an influential score from the data, it will significantly change the slope and intercept of the regression line.
ii. Removal of an influential score sharply affects the regression line because influential scores have a large impact on the regression line. If we remove an influential score, it will significantly change the slope and intercept of the regression line.
iii. This statement is not true. An x-value that is an outlier in the x-variable may be indicative of a point being influential, but a y-value that is an outlier in the y-variable can also be indicative of a point being influential.
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what type of graph or chart is used to compare relative parts of a whole?
Answers
A pie chart is commonly used to compare relative parts of a whole. It is a circular chart that is divided into sectors, where each sector represents a proportionate part of the whole.
The size of each sector is proportional to the value it represents, and the whole circle represents the total value. Pie charts are frequently used in business, finance, and statistics to show the relative sizes of different categories or data points. They are easy to understand and can quickly convey complex information in a simple visual format. However, it's important to note that pie charts are not always the most effective way to display data and should be used appropriately depending on the type of data being presented.
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Which of the following ordered pairs represents the unit rate?
Answers
Answer:
The answer might be C
Step-by-step explanation:
The graph hits the unit rate of (4,5) every time
Correct Me If I am Wrong
Answer my question im being timed Please!!
Answers
[ (10)(x^3)(y^2) / (5)(x^-3)(y^4) ]^-3
[ (2)(x^3)(y^2) / (x^-3)(y^4) ]^-3
[ (2)(x^6)(y^2) / (y^4) }^-3
[ (2)(x^6)(y^-2) ]^-3
(2^-3)(x^-18)(y^6)
---Not simplified (contains negative exponents)
(1/8)(x^-18)(y^6)
---Fully simplified
(y^6) / (8)(x^18)
Hope this helps!
Ali must mix 6 tablespoons of plant food for every 2 gallons of water if she has 6 gallons of water how much plant food should she use
Answers
What kind of sequence is this? –149, 447, –894, 894, ...
Answers
Answer:
multiply by -3, multiply by -2, multiply by -1, ...
Step-by-step explanation:
the scaling assumptions underlying a question determine which measure is appropriate. a. descriptive b. inference c. difference d. association
Answers
The scaling assumptions underlying a question determine which descriptive measure is appropriate.
The descriptive degree can be defined as the sort of measure handling the quantitative information in a mass that famous certain preferred characteristics. The descriptive measure has different types, all depending on the one-of-a-kind characteristics of the statistics. One very critical degree is the correlation coefficient, now and again referred to as Pearson's r. The correlation coefficient measures the diploma of linear association between two variables.
A statistic is a numerical descriptive degree computed from sample information. A parameter is a numerical descriptive measure of a populace. Descriptive facts encompass measures of the count, together with; frequencies and chances, measures of vital tendency including; imply, median, and mode, measures of variability including; trendy deviation, variance, and kurtosis, and measures of role, along with; percentiles and quartiles.
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could you help me with 11% and 9% thank you Assuming that the current interest rate is 10 percent, compute the present value of a five-year, 10 percent coupon bond with a face value of $1,000. What happens when the interest rate goes to 11 percent? What happens when the interest rate goes to 9 percent?
Answers
As the interest rate increases from 10 percent to 11 percent, the present value of the bond decreases from $1,074.47 to $1,058.31. Conversely, when the interest rate decreases to 9 percent, the present value increases to $1,091.19. This is because the discount rate used to calculate the present value is inversely related to the interest rate, meaning that as the interest rate increases, the present value decreases, and vice versa.
To compute the present value of a five-year, 10 percent coupon bond with a face value of $1,000, we need to discount the future cash flows (coupon payments and face value) by the appropriate interest rate.
Step 1: Calculate the present value of each coupon payment.
Since the bond has a 10 percent coupon rate, it pays $100 (10% of $1,000) annually. To calculate the present value of each coupon payment, we need to discount it by the interest rate.
Using the formula: PV = C / (1+r)^n
Where PV is the present value,
C is the cash flow,
r is the interest rate, and
n is the number of periods.
At an interest rate of 10 percent, the present value of each coupon payment is:
PV1 = $100 / (1+0.10)^1 = $90.91
Step 2: Calculate the present value of the face value.
The face value of the bond is $1,000, which will be received at the end of the fifth year. We need to discount it to its present value using the interest rate.
At an interest rate of 10 percent, the present value of the face value is:
PV2 = $1,000 / (1+0.10)^5 = $620.92
Step 3: Calculate the total present value.
To find the present value of the bond, we need to sum up the present values of each coupon payment and the present value of the face value.
Total present value at an interest rate of 10 percent:
PV = PV1 + PV1 + PV1 + PV1 + PV1 + PV2
PV = $90.91 + $90.91 + $90.91 + $90.91 + $90.91 + $620.92
PV = $1,074.47
When the interest rate goes to 11 percent, we would repeat the above steps using the new interest rate.
Total present value at an interest rate of 11 percent:
PV = PV1 + PV1 + PV1 + PV1 + PV1 + PV2
PV = $90.91 + $90.91 + $90.91 + $90.91 + $90.91 + $620.92
PV = $1,058.31
When the interest rate goes to 9 percent, we would repeat the above steps using the new interest rate.
Total present value at an interest rate of 9 percent:
PV = PV1 + PV1 + PV1 + PV1 + PV1 + PV2
PV = $90.91 + $90.91 + $90.91 + $90.91 + $90.91 + $620.92
PV = $1,091.19
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Find the diameter of a circle with a radius of 13 ft.
A
6.5 ft
B
26 ft
C
81.64 ft
D
530.66 ft
Answers
Answer:
B
Step-by-step explanation:
A radius is half of a diameter so multiply by 2
5-3(5x+2)+15x
Show and explain work
Answers
Answer:
-1
Step-by-step explanation:
5-3(5x+2)+15x
Distribute
5 - 3*5x - 3*2 +15x
5 - 15x -6+15x
Combine like terms
5-6 -15x+15x
-1
5−3(5x+2)+15x
Distribute:
=5+(−3)(5x)+(−3)(2)+15x
=5+−15x+−6+15x
Combine Like Terms:
=5+−15x+−6+15x
=(−15x+15x)+(5+−6)
=−1
what is the Area of the finished window
Answers
Use Green's Theorem to evaluate the line integral along the given positively oriented curve. integral_c ye^x dx + 2e^x dy, C is the rectangle with vertices (0, 0), (3, 0), (3, 4), and (0, 4)
Answers
Applying Green's Theorem, the line integral can be evaluated as the double integral over the region enclosed by the curve. By calculating the double integral, we find the value of the line integral to be 6e^3 - 6.
The line integral along the given curve can be expressed as:
∮C ye^x dx + 2e^x dy
We apply Green's Theorem, which states that for a vector field F = (M, N):
∮C M dx + N dy = ∬R (dN/dx - dM/dy) dA
First, let's calculate the partial derivatives of the given functions:
∂/∂x (2e^x) = 2e^x
∂/∂y (ye^x) = e^x
Now, we substitute these derivatives into Green's Theorem:
∮C ye^x dx + 2e^x dy = ∬R (e^x - 2e^x) dA
The region R is defined by the vertices of the rectangle (0, 0), (3, 0), (3, 4), and (0, 4).
To evaluate the double integral, we integrate with respect to x and y over the region R:
∬R (e^x - 2e^x) dA = ∫[0,3] ∫[0,4] (e^x - 2e^x) dy dx
Integrating with respect to y first, we get:
∫[0,3] [(e^x - 2e^x)y] [0,4] dx
Simplifying, we have:
∫[0,3] (4e^x - 8e^x) dx
Combining like terms, we obtain:
∫[0,3] (-4e^x) dx
Integrating, we have:
[-4e^x] [0,3] = -4e^3 + 4
Thus, the value of the line integral along the given curve is 6e^3 - 6.
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