The required length of the third side of the right-angle triangle is 15 units.
What are Pythagorean triplets?In a right-angled triangle, its sides, such as the hypotenuse, are perpendicular and the base is Pythagorean triplets.
Here,
For the right triangle lenght of the two legs to be 9 and 12 units, a measure of the third side is to be determined.
So, Applying the Pythagoras theorem,
h² = p² + b²
h = √9² + 12²
h = √225
h = 15
Thus, the required length of the third side of the right-angle triangle is 15 units.
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Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05. y' = (x − y)2, y(0) = 0.6; y(0.5) y(0.5) ≈ (h = 0.1) y(0.5) ≈ (h = 0.05)
Using Euler's method, y(0.5) ≈ 0.7052 when h = 0.1 and y(0.5) ≈ 0.6803 when h = 0.05.
What is Euler's method?
Ordinary differential equations (ODE) can be solved using the Euler's method, a first-order numerical technique, with a specified beginning value.
We are given an equation y' = f(x,y), a point ([tex]x_{o} ,y_{o}[/tex]) and a step value h.
We have
f(x,y) = (x − y)^2
Point is (0,0.6)
We need to find y(0.5).
So, using Euler's method with step h = 0.1, we get
⇒[tex]x_{1}[/tex] = [tex]x_{o}[/tex] + h
⇒[tex]x_{1}[/tex] = 0 + 0.1 = 0.1
⇒[tex]y_{1}[/tex] = [tex]y_{o}[/tex] + h * f([tex]x_{o} ,y_{o}[/tex])
⇒[tex]y_{1}[/tex] = 0.6 + 0.1 * (0 - 0.6)^2
⇒[tex]y_{1}[/tex] = 0.636
⇒[tex]x_{2}[/tex] = [tex]x_{1}[/tex] + h
⇒[tex]x_{2}[/tex] = 0.1 + 0.1 = 0.2
⇒[tex]y_{2}[/tex] = [tex]y_{1}[/tex] + h * f([tex]x_{1} ,y_{1}[/tex])
⇒[tex]y_{2}[/tex] = 0.636 + 0.1 * (0.1 - 0.636)^2
⇒[tex]y_{2}[/tex] = 0.6647296
⇒[tex]x_{3}[/tex] = [tex]x_{2}[/tex] + h
⇒[tex]x_{3}[/tex] = 0.2 + 0.1 = 0.3
⇒[tex]y_{3}[/tex] = [tex]y_{2}[/tex] + h * f([tex]x_{2} ,y_{2}[/tex])
⇒[tex]y_{3}[/tex] = 0.6647296 + 0.1 * (0.2 - 0.6647296)^2
⇒[tex]y_{3}[/tex] ≈ 0.6863
⇒[tex]x_{4}[/tex] = [tex]x_{3}[/tex] + h
⇒[tex]x_{4}[/tex] = 0.3 + 0.1 = 0.4
⇒[tex]y_{4}[/tex] = [tex]y_{3}[/tex] + h * f([tex]x_{3} ,y_{3}[/tex])
⇒[tex]y_{4}[/tex] ≈ 0.6863 + 0.1 * (0.3 - 0.6863)^2
⇒[tex]y_{4}[/tex] ≈ 0.7012
⇒[tex]x_{5}[/tex] = [tex]x_{4}[/tex] + h
⇒[tex]x_{5}[/tex] = 0.4 + 0.1 = 0.5
⇒[tex]y_{5}[/tex] = [tex]y_{4}[/tex] + h * f([tex]x_{4} ,y_{4}[/tex])
⇒[tex]y_{5}[/tex] ≈ 0.7012 + 0.1 * (0.5 - 0.7012)^2
⇒[tex]y_{5}[/tex] ≈ 0.7052
So, y(0.5) ≈ 0.7052
Similarly, with step h = 0.5, we will calculate the values till [tex]x_{10}[/tex] and [tex]y_{10}[/tex]
On doing this, we get y(0.5) ≈ 0.6803
Hence, using Euler's method, y(0.5) ≈ 0.7052 when h = 0.1 and y(0.5) ≈ 0.6803 when h = 0.05.
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What is the last one?
The value of the last one is 18.
How to solve equation?The solution of the diagram can be modelled with an equation. Therefore, let's find the last value.
we will use variable to represent the fruit names.
Hence,
yellow fruit = x
red fruit = y
green fruit = z
Therefore,
x + y + z = 29
2y + z = 26
2z + y = 25
Using substitution,
y = 25 - 2z
2(25 - 2z) + z = 26
50 - 4z + z = 26
50 - 3z = 26
-3z = 26 - 50
-3z = -24
z = 8
y = 25 - 2(8)
y = 25 - 16
y = 9
Therefore,
x + 9 + 8 = 29
x = 29 - 17
x = 12
Finally,
x + 1 / 2x = 12 + 1 / 2(12) = 12 + 6 = 18
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A) Find the time tH it takes the projectile to reach its maximum height H.
B) Find tR, the time at which the projectile hits the ground after having traveled through a horizontal distance R.
C) Find H, the maximum height attained by the projectile.
D) Find the total distance R (often called the range) traveled in the x direction; see the figure in the problem introduction.
The Max height H = [tex]\frac{u^{2} sin^{2} }{2g}[/tex] and the range is R = [tex]\frac{u* sin2thetha }{2g}[/tex].
What is projectile?Any object sent into space with only gravity acting on it is considered a projectile. Gravity is the main force working on a projectile. This doesn't necessary imply that the other troops don't affect it; it merely means that their impact is far less than that of gravity. A projectile's route is known as its trajectory. When a particle is hurled obliquely near the surface of the earth, it flows along a curved route with constant acceleration directed towards the planet's center (we assume that the particle stays close to the surface). Such a particle's motion is referred to as projectile motion, and its route is a projectile.
(A) Time taken by the projectile to reach its maximum height which will be given as -
using the equation of motion (1), we get
Vy= V0 + a*t
So the Max height H = [tex]\frac{u^{2} sin^{2} }{2g}[/tex]
And the range is R = [tex]\frac{u* sin2thetha }{2g}[/tex]
Hence the Max height H = [tex]\frac{u^{2} sin^{2} }{2g}[/tex] and the range is R = [tex]\frac{u* sin2thetha }{2g}[/tex].
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Question 1 of 10
If AABC= ADBC, then BC bisects the angle ZACD.
OA. True
OB. False
D
SUBMIT
solve it please it's a math questions
Using the principle of alternate angles we can say that the value of x will be 120°.
What are Alternate angles?The intersection of two parallel lines by a transversal produces alternate angles.
The alternate angles are equivalent when a transversal cuts two parallel lines.
The total of all the angles created on a transverse that are inside either of the two parallel lines is always 180 degrees.
Alternate interior angles for non-parallel lines don't have any special characteristics.
Angles of the same size that are found on opposing sides of the transversal line are known as alternate angles.
So, do not observe the given image:
We know that it is the pair of alternate angles that is equal.
Then, x = 120°.
Therefore, using the principle of alternate angles we can say that the value of x will be 120°.
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Camille took a friend for a birthday dinner. The total bill for dinner was $31.09 (including tax and a tip). If Camille paid a 20.7% tip, what was her bill before adding the tip?
Answer: Let's call the bill before the tip "x".
The tip was 20.7% of the bill, so the amount of the tip can be expressed as:
tip = 0.207 * x
The total bill, including the tip, was $31.09, so we can write an equation to represent this:
x + tip = 31.09
Substituting the expression for the tip into this equation, we get:
x + 0.207x = 31.09
Expanding and solving for x, we get:
1.207x = 31.09
x = 31.09 / 1.207
x = 25.76
So the bill before adding the tip was $25.76.
Step-by-step explanation:
6, Jason and David are having a competition to see who can earn the most money in one month. Jason works as a server at Buffalo Wild Wings and averages 18% tips of each table he serves. Each week his total average sales are $2100. David works at Kohl's and earns a weekly salary of $185, as well as 6% commission on whatever he sells. David's average weekly sales total $3500. Who makes more money in one month?
Answer: First, let's calculate Jason's weekly earnings:
Jason's weekly tips = $2100 * 18% = $378
Jason's total weekly earnings = $2100 + $378 = $2478
Next, let's calculate David's weekly earnings:
David's weekly commission = $3500 * 6% = $210
David's total weekly earnings = $185 + $210 = $395
To find the monthly earnings, we need to multiply their weekly earnings by 4 (since there are 4 weeks in a month):
Jason's monthly earnings = $2478 * 4 = $9912
David's monthly earnings = $395 * 4 = $1580
So, Jason makes more money in one month, earning approximately $9912, compared to David's $1580.
Step-by-step explanation:
The individual who makes more money in a month between Jason and David is David.
Who makes more money in one month?Jason's weekly tips = 18% of $2100
= $378
David weekly salary= $185
David's weekly tips = 6% of $3400
= $210
Total weekly salary of David = $185 + $210
= $395
Total monthly salary of Jason= $378 × 4
= $1,512
Total monthly salary of David= $395 × 4
= $1,580
Therefore, David makes more money than Jason in a week.
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A CFL player has an annual salary of $305,000. The player donates 1/8 of his salary to charity. What amount is this?
Answer:
The player has an annual salary of $305,000, and they have decided to donate 1/8 of their salary to charity. To calculate this, we simply divide their salary by 8.
305,000 / 8 = $38,125
So, the player is donating $38,125 to charity every year. This is calculated by taking 1/8 of their annual salary.
The fraction 1/8 represents one out of eight equal parts of the player's salary, and when this fraction is multiplied by the player's salary, it gives the exact amount that they are donating to charity.
Find the length of side x.
Give your answer to 1 decimal place.
13 cm
98°
X
17 cm
The value of the x is 19.9.
What is the Law of Cosines?A planar triangle's side's square is equal to the sum of the squares of the other sides minus the product of those sides and the cosine of the angle between them, divided by two.
Given:
The angle C is 98°.
Applying law of cosines:
c² = a² + b² -2ab cosC.
Substituting the values,
c² = (17)² + (13)² -2(17)(13) cos98°.
c² = 458 - (-0.13917)442
c² = 396.48686
c = 19.9
Hence, x = 19.9.
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In Exercises 1 and 2, determine which matrices are in reduced echelon form and which others are only in echelon form.a. 1 1 0 10 0 1 10 0 0 0b. 1 1 0 00 1 1 00 0 1 1c. 1 0 0 01 1 0 00 1 1 00 0 1 1d. 0 1 1 1 10 0 2 2 20 0 0 0 30 0 0 0 0
In the given exercise option a, b and c are in reduced echelon form and option d is in echelon form.
What is echelon form and reduced echelon form?Row echelon form suggests that: Each row's leading (first) entry must be
Each following row's first entry must be on a new column to the right.
All rows with zero entries are listed below rows with non-zero items.
If the leading entry in each column is the only nonzero element in that column, reduced echelon form also results from echelon form conditions.
According to the definitions we see that, in the given exercise option a, b and c are in reduced echelon form and option d is in echelon form.
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Find a vector equation and parametric equations for the line. (Use the parameter t.)
a) The line through the point (1,0,7) and perpendicular to the plane x + 3y + z = 2.
b) The line through the point (3, 2.3, 3.6) and parallel to the vector 3i + 3j - k.
a. The vector equation for the given line is (x,y,z) = (1,0,7) + t(1,3,1) and the parametric equations are x = 1+t, y = 3t, z = 6+t.
b. The vector equation for the given line is
(3 + 3t)i + (2.3 + 3t)j + (3.6 - t)k and the parametric equation is
x = 3 + 3t, y = 2.3 + 3t, z = 3.6 - t.
What is a parametric equation?
A series of equations known as a parametric equation specifies a point's location in the xy plane in terms of a third variable, which is typically referred to as a parameter.
a. We are given that a line passes through the point (1,0,7) and is perpendicular to the plane x + 3y + z = 2.
Now, since the line is perpendicular to the plane, the normal vector (1,3,1) for the plane is parallel to the line.
So the vector equation for the line is
(x,y,z) = (1,0,7) + t(1,3,1)
The parametric equations for the line are
x = 1+t, y = 3t, z = 6+t
b. We are given that a line passes through the point (3, 2.3, 3.6) and parallel to the vector 3i + 3j - k.
So the vector equation "r" for the line is
r = (3i + 2.3j + 3.6k) + t(3i + 3j - k)
r = (3 + 3t)i + (2.3 + 3t)j + (3.6 - t)k
The parametric equations for the line are
x = 3 + 3t, y = 2.3 + 3t, z = 3.6 - t
Hence, the required vector and parametric equations have been obtained.
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25. Mr. Polar Bear asks if dilations will result in similar figures that differ by a scale factor.
Do you tell him "yes" or "no"?.
Answer:
Yes
Step-by-step explanation:
Dilations shrink or expand a figure. If the scale factor is one, the two figures will be identical.
graph this equation on the graph then, use the check step to prove the solution works in both equations. show all work
y = -1/2x + 3
y = 5
check:
The solution that works for both the equations is (-4, 5) and the two lines intersect at this point.
What is system of equations?A system of equations in algebra consists of two or more equations and looks for common answers to the equations. "A set of equations satisfied by the same set of variables is called a system of linear equations."
The given equation are:
y = -1/2x + 3
y = 5
Substitute the value of y in first equation:
5 = -1/2x + 3
5 = -x + 6 / 2
10 = -x + 6
10 - 6 = -x
x = -4
The solution that works for both the equations is (-4, 5) and the two lines intersect at this point.
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Six oranges cost R18,00. how much will 5 oranges and 15 oranges cost?
The costs of 5 oranges and 15 oranges are R15 and R45
How to determine the costsFrom the question, we have the following parameters that can be used in our computation:
Six oranges cost R18,00
This means that the unit rate is
Unit rate = R18,00/6 = R3.00
For 5 oranges, we have
Cost = 5 * R3.00 = R15.00
For 15 oranges, we have
Cost = 15 * R3.00 = R45.00
So, the costs are R15.00 and R45.00
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bonnie wrote 2 fractions that are equivalent to 1/3.Are bonnie's equivalent fractions which are 2/6 and 3/12 correct?
One common system for computing a grade point average (GPA) assigns 4 points to an A, 3 points to a B, 2 points to a C, 1 point to a D, and 0 points to an F. What is the GPA of a student who gets an A in a 3-credit course, a B in each of two 2-credit courses, a C in a 4-credit course, and a D in a 3-credit course?
The mean grade point score is
Round to the nearest tenth as needed.)
As stated in the preceding statement When 4 points are given in one method for calculating grade point averages (GPA), the resulting mean grade rating is 3.41.
How can we determine the average?Average While calculating the arithmetic mean, a group of integers is added, their count is divided, and the result is then taken. For instance, the average of 2, 3, 5, 7, and 10 is 5, this is the result of 30 reduced by 6.
The quality points by each grade every course must first be calculated before being added together to determine the GPA.
3 credit hours times 4 points for A equals 12 quality points.
B - ( 3 points x 2 credit hours ) x 2 (because there are 2 courses) = 18 quality points
C - 2 points x 4 credits = 8 quality points
D - 1 point x 3 credits = 3 quality points
Now add up all the quality points = 12 + 18 + 8 + 3 = 41
Add the credit hours as well = 3 + 2 + 4 + 3 = 12
Divide the total kit by total semester credit to calculate the GPA = 41/12 = 3.41
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i need this by today please
Translate the following verbal statement into an algebraic equation and then solve:
Use x for your variable.
The quotient of four more than a number and seven is ten.
the quotient of four (4) more than a number (x) and seven (7) is ten (10) equals (x +4)/7 = 10.
What is algebraic Expression?Any mathematical statement that includes numbers, variables, and an arithmetic operation between them is known as an expression or algebraic expression. In the phrase 4m + 5, for instance, the terms 4m and 5 are separated from the variable m by the arithmetic sign +.
Given The quotient of four is more than a number and seven is ten.
also given Use x for your variable.
In Mathematics, quotient literally means dividing a particular number by another number i.e the division of two (2) numbers.
Next, we would translate the word problem (sentence) into an algebraic expression as follows;
=> (x +4)/7 = 10
In conclusion, the algebraic expression for the quotient of four (4) more than a number (x) and seven (7) is ten (10) equals (x +4)/7 = 10.
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What did Nixon's visit to China prove to the US?
A. If you can deal with one communist nation, others will likely join in
B. The Soviet Union was close to falling apart and communism would leave Eastern Europe
C. Most communist nations actually practice capitalism and have free elections
D. The communist world was not a single entity, and had many different positions
According to the information, it can be inferred that US President Nixon's visit to China proved that the communist world was not a single entity and had many different positions.
What was Nixon's visit to China?The visit of the President of the United States Richard Nixon to the People's Republic of China is an important event in the political and foreign relations sphere of the United States because it was a way to establish a formal relationship between the United States and the People's Republic of China.
This was significant because it was the first time a US President had visited the PRC, who at the time considered the US one of his main enemies. Additionally, this allowed him to verify to the other states that there were different ways of proposing communism. In this case, Chinese communist thought was moderated, so the relationship with the United States was allowed.
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Two vertical poles, one 16 ft high and the other 24 ft high, stand 30 feet apart on a flat field. A worker wants to support both poles by running rope from the ground to the top of each post. If the worker wants to stake both ropes in the ground at the same point, where should the stake be laced to use the least amount of rope?
The ropes should be staked 12 ft away from 16 ft pole and 18 ft away from 24 ft pole.
What are similar triangle?Two triangles are similar if they have the same ratio of corresponding sides and equal pair of corresponding angles. If two or more figures have the same shape, but their sizes are different, then such objects are called similar figures.
Given that, two vertical poles, one 16 ft high and the other 24 ft high, stand 30 feet apart on a flat field. A worker wants to support both poles by running rope from the ground to the top of each post, we need to find where should the stake be laced to use the least amount of rope,
Let rope of the 16 ft high pole be at x ft and that of 24 ft high pole at (30-x)
The structure of rope and poles are making a set of similar triangles, (refer to figure attached)
We know that, the corresponding sides of the similar triangle are in same ratio,
Therefore,
16 / x = 24 / (30-x)
2 / x = 3 / (30-x)
60 - 2x = 3x
5x = 60
x = 12
30-12 = 18
Hence, the ropes should be staked 12 ft away from 16 ft pole and 18 ft away from 24 ft pole.
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Why does a reflection make my phone look like theres a scratch when there is none on my phone? answer asap for vrainlist
Do u have a screen protector cuz if u do thats probably why
Step-by-step explanation:
Answer:
light diffraction and a dirty phone screen.
Guys does someone know why are checks debit cards electronic payments all considered cash forms of payment even though the y do not insolvente bills and coins ???
Checks, debit cards, and electronic payments are considered forms of cashless payments because they allow for the transfer of funds between individuals or businesses without the need for physical currency, such as bills and coins.
Checks are considered cash because they are legal tender and can be deposited into a bank account, allowing the funds to be transferred between parties. Debit cards and electronic payments work in a similar way by deducting funds directly from a bank account or transferring funds electronically. While these forms of payment may not involve physical currency, they represent a transfer of value and can be used to purchase goods and services in the same way as cash.
In summary, checks, debit cards, and electronic payments are considered forms of cash because they facilitate the transfer of value between parties, even though they do not involve physical currency.
Answer:
Debit cards account for 28% of all payments, and credit cards account for 23% of all payments. For these payments, payment details are entered manually during a checkout process, or the physical card is processed at a point of sale (POS) system. There’s usually a debit or credit card processing fee, which the business owner needs to pay.
Step-by-step explanation:
pls tell me if wrong thank you have a great night or day
the makers of skittles claim that 20% of skittles candies are orange. suppose this claim is true. you select a random sample of 30 skittles from a large bag. let the proportion of orange skittles in the sample.
The sample size that required to reduce the standard deviation of the sampling distribution to one-half that standard deviation (0.073) is 120.
As per the data given:
The makers of skittles claim that 20% of skittles candies are orange.
20 % = 0.20
Select a random sample of 30 skittles from a large bag.
[tex]\hat p[/tex] = the proportion of orange skittles in the sample.
The standard deviation of the sampling distribution of [tex]\hat p[/tex] = 0.073
The Standard deviation of the Sampling distribution was reduced to [tex]$= \frac{1}{2} \sigma_{\hat{p}}$[/tex].
[tex]$ \sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}=0.073$[/tex]
Divide both sides by 2
We get:
[tex]$\sqrt{\frac{p(1-p)}{n}}=\frac{0.073}{2}[/tex]............... (1)
Now it should become be one - half.
Let [tex]$n_1[/tex] be the new sample size.
[tex]$\sqrt{\frac{p(1-p)}{n_1}}=\frac{0.073}{2}[/tex] ..................2
From (1) and (2) RHS of both equations is equal.
Then equate both equations
[tex]$\frac{1}{2} \sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{p(1-p)}{n_1}}$[/tex]
Squaring on both sides
We get:
[tex]$\Rightarrow\left(\frac{1}{2}\right)^2 \frac{p(1-p)}{n}=\frac{p(1-p)}{n_1}\end{aligned}$[/tex]
[tex]$\Rightarrow \frac{1}{4 n}=\frac{1}{n_1}$[/tex]
[tex]$\Rightarrow n_1=4 n$[/tex]
[tex]$n_1-[/tex] The new sample size required = 4n
n-The old sample size.
The sample size that required = 4n
= 4 (30)
= 120
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The makers of skittles claim that 20% of skittles candies are orange. Suppose this claim is true. You select a random sample of 30 skittles from a large bag. Let p-hat = the proportion of orange skittles in the sample. The standard deviation of the sampling distribution of p-hat = .073
What sample size would be required to reduce the standard deviation of the sampling distribution to one-half that standard deviation (.073)
EASY POINTS!
Two runners are training for their next competitions. Each runner is able to run a certain
distance in proportion to the time. The table on the left displays the distance Runner A
can run in relation to the time. Meanwhile, the graph on the right displays the distance
Runner B can run in relation to time.
What are the unit rates for the two runners? Which runner can run faster?
Answer:
12 / 1 and runner B
Step-by-step explanation:
at 13 seconds runner a had only run 156 ft, however runner b had run 200 ft.
A national diet company markets a special diet plan which requires dieters to use frozen meals prepared by the company. The company advertises that those who follow their plan for 10 weeks will lose, on the average, at least 10 lbs. A consumer group is suspicious of this claim, believing that the weight lose is, on average, much less. If m represents the mean weight loss the population of all American adults would experience after 10 weeks on the plan, what is the appropriate alternative hypothesis that the consumer group wishes to test?a. Ha: mu < 10 b. Ha: mu > 10 c. Ha: mu (notequal)10 d. Ha: mu (notequal) 0
Ha:mu<10 is the appropriate alternative hypothesis that the consumer group wishes to test .
How would you define hypothesis?
An assumption or notion is called a hypothesis when it is put forth with the purpose of debating whether it might be true.
In the scientific process, the hypothesis is developed prior to the completion of any relevant study, other than a brief background review.
Ha:mu<10
Because A consumer group is suspicious of this claim, believing that the weight lose is, on average, much less.
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What quadrant is the point (2,-3) in
A man holds a note of $4,000 that has an interest rate of 12% annually. The note was made on March 16 and is due November 14. He sells the note to a bank on June 11 at a discount rate of 11% annually. Find the proceeds on the third-party discount note. (Use the banker's rule.)
The proceeds (discounted value) on the third-party discount note are $4,116.90.
What is the discounted value of a note?The discounted value of a note is a value that is less than the maturity value.
The maturity value is the face value of the note plus the accumulated interest.
Face value of the note receivable = $4,000
Annual interest rate = 12%
Note issuance date = March 16
Note due date = November 14
The total number of days = 243 days
Discount date = June 11
Discount rate = 11%
March 16 to June 11 = 87 days
Interest on the note = $320 ($4,000 x 243/365 x 0.12)
Maturity value = $4,320 ($4,000 + $320)
Discount period = 156 days (243 - 87)
Discount on the note = $203.10 ($4,320 x 156/365 x 11%)
Discounted value = $4,116.90 ($4,320 - $203.10)
Thus, the man, who holds a note of $4,000 at 12% interest but sells it at a discount rate of 11%, will receive $4,116.90.
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Solve the system of equations below. You may want to use the method by substitution starting with the second equation.
3x−2y
=34
5x+10y
=−10
Answer:
(8,-5)
x=8
y=-5
Step-by-step explanation:
In Problems 31 and 32 find values of m so that the function y = xm is a solution of the given differential equation. xy'' + 2y' = 0
m = -2, 0. The function y = xm is a solution when m = -2 or 0.
For the differential equation xy'' + 2y' = 0, the function y = xm is a solution if m = -2 or 0. This can be seen by substitution and solving the resulting equation. When m = -2, the equation simplifies to 0 = 0, which is true. When m = 0, the equation simplifies to 0 = 2y', which is true when y' = 0. Therefore, the function y = xm is a solution when m = -2 or 0.
To solve for m, first substitute the function y = xm into the differential equation, giving xm'' + 2mx' = 0. Then move all terms containing m to one side of the equation, leaving 0 on the other side. This gives m'' + 2m' = 0, which is a second-order linear differential equation. This equation can be solved using the characteristic equation, which is found by setting the discriminant of the equation equal to 0. The discriminant for this equation is 4 - 4(1)(0) = 4, which is greater than 0. Thus, the equation has two distinct real solutions, which are m = -2 and m = 0. Therefore, the function y = xm is a solution when m = -2 or 0.
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A woman 6 ft tall is standing near a street lamp that is 13 ft tall, as shown in the figure. Find a function that models the length L of her shadow in terms of her distance d from the base of the lamp.
Answer:
Step-by-step explanation:
We can use similar triangles to find the function that models the length of the woman's shadow. If the woman is standing at a distance d from the base of the lamp, the length of her shadow will be proportional to her height, so the ratio of the shadow length to the height will be equal to the ratio of the distance from the base of the lamp to the height of the lamp.
So, if h is the woman's height and L is the length of her shadow, we have:
L/h = d/13
And since h = 6 feet, we can write:
L = (d/13) * 6
So the function that models the length of the woman's shadow in terms of her distance from the base of the lamp is:
L = (d/13) * 6