To find the equation of the line that passes through the point (1, 2) and is perpendicular to the line y=3x-1, we can first find the slope of the line y=3x-1. The slope of this line is 3, so a line perpendicular to it will have a slope of -1/3. We can then use the point-slope form of the line to write the equation of the line that passes through (1,2) and has a slope of -1/3. The point-slap form of a line is given by y-y₁ = m(x-x₁), where (x₁,y₁) is a point on the line and m is the slope of the line. Plugging in the coordinates of the point (1,2) and the slope -1/3, we get the equation y-2 = -1/3(x-1), which simplifies to y-2 = -x/3 + 1/3. Multiplying both sides of the equation by 3 to get rid of the fractions, we get 3y-6 = -x+1. Finally, we can move all the terms to one side of the equation to get the standard form of the line: 3y-x-5 = 0. Therefore, the equation of the line that passes through the point (1, 2) and is perpendicular to the line y = 3x - 1 is x-3y-5=0, so the correct answer is A.
34) An airplane pilot is planning a flight to Hawaii. In his flight plan, he will include fuel estimates for the trip.
Explain the way he can use estimates in his plan without causing any danger to the flight
The airplane pilot should include and follow the relation y > [D][x] for his fuel estimates.
What is a mathematical function, equation and expression?
Function : In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function
Expression : A mathematical expression is made up of terms (constants and variables) separated by mathematical operators
Equation : A mathematical equation is used to equate two expressions.
Given is that an airplane pilot is planning a flight to Hawaii. In his flight plan, he will include fuel estimates for the trip.
Assume that the distance of Hawaii from the initial location of pilot is [D] miles. If the amount of fuel burnt per miles is [x] liters/miles. Let the total amount of fuel needed to reach Hawaii is [y] liters. Than, his estimates should include and follow the given relation as -
y > [D][x]
Therefore, the airplane pilot should include and follow the relation y > [D][x] for his fuel estimates.
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How to turn x = 3y -1 into slope intercept form?
Answer:
y=1/3(x)+1/3
Step-by-step explanation:
x=3y-1
x+1=3y
(x+1)/3=(3y)/3
y=1/3x+1/3
Written as a mixed number, the improper fraction 113⁄8 would appear as
Answer:
[tex]14 \frac{1}{8}[/tex]
Step-by-step explanation:
112 goes into 8 14 times, there is then 1/8 leftover
Find the nth term of this number sequence 7, 10, 13, 16, ...
Answer:
31
Step-by-step explanation:
7, 10, 12, 16, 19, 22, 25, 28, 31
Adding by 3
Hope this works
Solve: 6г - 17r = -66
-6
-11
11
Answer:
r = 6
Step-by-step explanation:
6г - 17r = -66
-(17г - 6r) = -66
-11r = -66
-11r/(-1) = -66/(-1)
11r = 66
11r/11 = 66/11
r = 66/11
r = 6
One angle of a triangle meaure 10 degree greater than the mallet angle and the third angle meaure 10° le than twice the mallet angle find the meaure of the three angle
One angle in a triangle is 10 degrees larger than the smallest angle, while the third angle is 10 degrees less than twice the smallest angle. As a result, the three angles are 55°, 45°, and 80°.
Three vertices and three sides make up a triangle. The angles of the triangle are formed by the connection of the three sides end to end at a single point. The sum of the triangle's three angles is 180 degrees.
Let's consider the three angles of the triangle as ∠P, ∠Q, and ∠R. Let's consider Q as the smallest angle. The sum of angles P, Q, and R is given as ∠P+∠Q+∠R=180°
Then, ∠P = ∠Q + 10° and ∠R = 2∠Q - 10°. Substituting these two angle values in the above equation,
[tex]\begin{aligned}\angle Q + 10^{\circ}+\angle Q+2\angle Q - 10^{\circ}&=180^{\circ}\\4\angle Q&=180^{\circ}\\\angle Q&=45^{\circ}\end{aligned}[/tex]
Then,
∠P = ∠Q + 10° = 45°+ 10° = 55°
∠R = 2∠Q - 10° = 2(45°) - 10° = 90° - 10° = 80°
The answers are 55°, 45°, and 80°.
The complete question is -
One angle of a triangle measure 10 degrees greater than the smallest angle and the third angle measure 10 degrees less than twice the smallest angle find the measure of the three angles.
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Move the yellow dots in order to make the segments intersect. Let the intersection be called point V. Make the lines intersect in such a way that angle ∠SVT is an obtuse angle less than 135°. Afterwards, use the protractor to determine the precise measure of all four angles. You should redraw the lines if angle ∠SVT is not the proper size.
All four measures of the angles are mentioned in the figure.
What is an angle?The inclination is the separation seen between planes or vectors that meet. Degrees are another way to indicate the slope. For a full rotation, the angle is 360 °.
If the measure of the angle is more than the right angle that is 90°. Then the angle is known as the obtuse angle.
Move the yellow dabs to make the portions cross. Allow the convergence to be called point V. Cause the lines to cross so that point ∠SVT is a harsh point under 135°.
Let the measure of the angle ∠SVU be 'x' which is greater than 45°. Then the measure of the angle ∠SVT will be '180 - x'.
All four measures of the angles are mentioned in the figure.
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The cash price of a refrigerator is Rs 34 500. The hire purchase price is a deposit of Rs 6500 and 10 % simple interest on the outstanding balance, to be paid in 24 monthly installments. Calculate the amount of each monthly instalment
Answer:
Rs 1400
Step-by-step explanation:
You want the monthly installment payment if the balance of a Rs 34500 purchase after a 6500 deposit earns 10% simple interest and is paid in 24 months.
OutstandingThe outstanding balance is ...
Rs 34500 -6500 = Rs 28000
InterestThe interest on this balance for 24 months is ...
I = Prt
I = Rs 28000(0.10)(2) = Rs 5600
Monthly paymentThe monthly payment will be 1/24 of the total due:
(28000 +5600)/24 = 1400
Each monthly installment will be Rs 1400.
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fido's leash is tied to a stake at the center of his yard, which is in the shape of a regular hexagon. his leash is exactly long enough to reach the midpoint of each side of his yard. if the fraction of the area of fido's yard that he is able to reach while on his leash is expressed in simplest radical form as $\frac{\sqrt{a}}{b}\pi$, what is the value of the product $ab$?fido's leash is tied to a stake at the center of his yard, which is in the shape of a regular hexagon. his leash is exactly long enough to reach the midpoint of each side of his yard. if the fraction of the area of fido's yard that he is able to reach while on his leash is expressed in simplest radical form as $\frac{\sqrt{a}}{b}\pi$, what is the value of the product $ab$?
Product ab has a value of 18 and is shaped like a regular hexagon.
What is area?The quantity area indicates the extent of a region on a planar or curved surface. Surface area refers to the area of an open surface or the border of a three-dimensional object, whereas area of a plane region or plane area refers to the area of a form or planar lamina. The area is the region defined by an object's form. The area of a form is the space covered by a figure or any two-dimensional geometric shape in a plane.
Here,
Let the side of the hexagon be 2.
so the radius of the circle is sqrt(3),
area of hexagon=sqrt(3) * 2 * 3
= 6 sqrt(3)
area of circle is pi (sqrt(3))^2 = 3 pi
3 pi / 6 sqrt(3) = sqrt(3)/ 6
3 * 6 = 18
The value of product ab is 18 which is in the shape of a regular hexagon.
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A total of 699 tickets were sold for the school play. They were either adult tickets or student tickets. There were 51 fewer student tickets sold than adult tickets. How many adult tickets were sold?
Answer:
Sold:
adults: 375 tickets
student: 324 tickets
Step-by-step explanation:
a + s = 699 Eq. 1
a = s + 51 Eq. 2
a = adult tickets sold
s = student tickets sold
Replacing Eq. 2 im Eq. 1
(s+51) + s = 699
2s + 51 = 699
2s = 699 - 51
2s = 648
s = 648/2
s = 324
From Eq. 2
a = 324 + 51
a = 375
Check:
From Eq. 1:
a + s = 699
375 + 324 = 699
Use the given information to find the new price. Round to the nearest cent Original Price: $28 Discount Rate: 25%
Answer:
To find the new price after a discount, you can use the formula: new price = original price * (1 - discount rate). In this case, the new price would be $28 * (1 - 0.25) = $28 * 0.75 = $<<28*0.75=21>>21. So the new price, rounded to the nearest cent, would be $21.
Step-by-step explanation:
b)
Given that n is an integer greater than 1, explain why the largest prime factor of
2(7^n) - 2(7^n-1)+7^n+1 is 61.
Answer:
See below.
Step-by-step explanation:
Given expression:
[tex]2(7^n) - 2(7^{n-1})+7^{n+1}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^{b+c}[/tex]
[tex]\implies 2(7^n) - 2(7^{n-1})+7^{n} \cdot 7^1[/tex]
[tex]\textsf{Apply exponent rule} \quad \dfrac{a^b}{a^c}=a^{b-c}[/tex]
[tex]\implies 2(7^n) - 2\left(\dfrac{7^{n}}{7^{1}}\right)+7^{n}\cdot 7^1[/tex]
Simplify:
[tex]\implies 2(7^n) - \dfrac{2}{7} (7^n)+7(7^{n})[/tex]
Factor out the common term 7ⁿ:
[tex]\implies \left(2- \dfrac{2}{7} +7\right)7^{n}[/tex]
Therefore:
[tex]\dfrac{61}{7}(7^n)[/tex]
A prime number is a whole number greater than 1 that cannot be made by multiplying other whole numbers. Therefore, the factors of a prime number are 1 and the number itself.
If n is an integer greater than 1, the number will always have at least 4 factors (1, 7, 61 and itself) and therefore cannot be a prime number by definition.
For example:
[tex]n = 2 \implies \dfrac{61}{7}(7^2)=61 \times 7=427[/tex]
[tex]n = 3\implies \dfrac{61}{7}(7^3)=61 \times 7^2=2989[/tex]
Therefore, the largest prime number is when n = 1:
[tex]n = 1 \implies \dfrac{61}{7}(7^1)=61[/tex]
The length of a rectangle is 2m longer than its width. If the perimeter of the rectangle is , 60 find its length and width.
Answer:
Let the width of the rectangle be w.
Then the length of the rectangle is w+2.
The perimeter of the rectangle is the sum of all its sides, so it is 2(w) + 2(w+2) = 60.
Expanding the parentheses and combining like terms, we get:
2w + 2w + 4 = 60
4w + 4 = 60
4w = 56
w = 14
The width of the rectangle is 14.
The length of the rectangle is 14+2 = 16.
Therefore, the length of the rectangle is 16 and the width is 14.
CORRECT ME IF I'M WRONG
An agricultural land of 2 hectares is planted with cereals. 20% of it is planted with corn, 10% with oats, and the rest with wheat. Find how many dynyms is each surface
First, we need to find the total area of the land that is planted with cereals. To do this, we multiply the total area of the land, which is 2 hectares, by the percentage of the land that is planted with cereals, which is 100% - 20% - 10% = 70%. This gives us a total area of 2 hectares * 70% = 1.4 hectares.
Next, we need to divide the total area of 1.4 hectares into three parts, one for each type of cereal. Since 20% of the land is planted with corn, the area of land planted with corn is 1.4 hectares * 20% = 0.28 hectares. Similarly, since 10% of the land is planted with oats, the area of land planted with oats is 1.4 hectares * 10% = 0.14 hectares. Finally, the area of land planted with wheat is 1.4 hectares - 0.28 hectares - 0.14 hectares = 0.98 hectares.
Since 1 hectare is equal to 10,000 square meters, the area of land planted with corn is 0.28 hectares * 10,000 square meters/hectare = 2,800 square meters. The area of land planted with oats is 0.14 hectares * 10,000 square meters/hectare = 1,400 square meters, and the area of land planted with wheat is 0.98 hectares * 10,000 square meters/hectare = 9,800 square meters. So, each surface of land planted with corn is 2,800 square meters, each surface of land planted with oats is 1,400 square meters, and each surface of land planted with wheat is 9,800 square meters.
match the parametric equations with the correct graph. x = t cos t, y = t, z = t sin t, t ≥ 0
[tex]x^{2} + y^{2} = t^{2}[/tex] represent the correct graph which is the first figure graph.
Equations with Parameters If x and y are continuous functions of t on an interval I, the equations x = x(t) and y = y(t) are referred to as parametric equations, and t is also known as the parameter. The graph of the parametric equations is the set of points (x, y) obtained as t varies over the interval I.
Since x = tsint
And z= tcost
We have a circle in the x-z plane a
[tex]x^{2} + z^{2} = t^{2}[/tex]
Now, as y = t
The graph will be a helix with movement along the y-axis. Furthermore, the circle made on the x-z plane will have a variable radius since the radius is equal to t which by itself is changing.
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Help on logic and proof homework geometry
find the value of y
Answer:
y = 10 cm
Step-by-step explanation:
A = 12 · 5 / 2
A = 6 · 5 = 30 cm²
y = 30 · 2 / 6
y = 5 · 2
y = 10 cm
Complete each congruence statement by naming
the corresponding angle or side.
ABCA = AFGA
B
AB = ?
C
? = side
A
side type your answer...
F
G
If the congruence statement ABCA = AFGA is true, then the following must also be true:
B = G, since corresponding angles are congruent
AB = FG, since corresponding sides are congruent
C = F, since corresponding angles are congruent
Thus, the complete congruence statement is ABCA = AFGA, where B = G, AB = FG, and C = F.
(2+√-4) (3 +5i)(-4-i)
Answer:
[tex]32-60i[/tex]----------------------------------------
Simplify the expression in below steps:
[tex](2+\sqrt{-4} ) (3 +5i)(-4-i)=[/tex][tex](2+\sqrt{-1*4} ) (3 +5i)(-4-i)=[/tex][tex](2+2i ) (3 +5i)(-4-i)=[/tex][tex]2(1+i ) (3 +5i)(-4-i)=[/tex][tex]2(1+i ) (3(-4) -3i+5i(-4)-5i*i)=[/tex][tex]2(1+i ) (-12 -3i-20i-5(-1))=[/tex][tex]2(1+i ) (-12 -23i+5)=[/tex][tex]2(1+i ) (-7 -23i)=[/tex][tex]2(-7 -23i -7i-23i*i)=[/tex][tex]2(-7 -30i +23)=[/tex][tex]2(16 -30i)=[/tex][tex]32-60i[/tex]Answer:
[tex]32-60i[/tex]
Step-by-step explanation:
Given expression:
[tex](2+\sqrt{-4})(3 +5i)(-4-i)[/tex]
[tex]\textsf{Rewrite $\sqrt{-4}$ as $\sqrt{4 \cdot -1}$}:[/tex]
[tex]\implies (2+\sqrt{4 \cdot -1})(3 +5i)(-4-i)[/tex]
[tex]\textsf{Apply radical rule} \quad \sqrt{ab}=\sqrt{a}\sqrt{b}:[/tex]
[tex]\implies (2+\sqrt{4}\sqrt{-1})(3 +5i)(-4-i)[/tex]
[tex]\implies (2+2\sqrt{-1})(3 +5i)(-4-i)[/tex]
[tex]\textsf{Apply imaginary number rule} \quad \sqrt{-1}=i:[/tex]
[tex]\implies (2+2i)(3 +5i)(-4-i)[/tex]
Multiply:
[tex]\implies (6+16i+10i^2)(-4-i)[/tex]
[tex]\implies -24-6i-64i-16i^2-40i^2-10i^3[/tex]
[tex]\implies -24-70i-56i^2-10i^3[/tex]
[tex]\textsf{Apply imaginary number rule} \quad i^2=-1:[/tex]
[tex]\implies -24-70i-56(-1)-10(-1)i[/tex]
Simplify:
[tex]\implies -24-70i+56+10i[/tex]
[tex]\implies 32-60i[/tex]
mr. dawson is making a grocery budget for the month of april. he plans to split the budget equally among 4 shopping trips. to stay under budget, mr. dawson figures he should spend less than $180 each trip.let x represent how much mr. dawson wants to spend on groceries in april. which inequality describes the problem?
The inequality that describes the problem is x < 180 × 4, which means that the total amount of money Mr. Dawson wants to spend on groceries in April should be less than $180 multiplied by 4, or $720.
Mr. Dawson is making a grocery budget for the month of April. He plans to split the budget equally among 4 shopping trips and wants to stay under budget. To figure out the maximum amount of money he should spend on groceries in April, he needs to calculate the total amount of money he should spend for all 4 shopping trips. To do this, he multiplies the maximum amount of money he should spend per trip, $180, by the number of trips, 4. This gives us $180 × 4 = $720. Therefore, Mr. Dawson should spend less than $720 on groceries in April. The inequality that describes the problem is x < 180 × 4, which means that the total amount of money Mr. Dawson wants to spend on groceries in April should be less than $720. This inequality ensures that Mr. Dawson will stay under budget and be able to complete his 4 shopping trips for the month of April.
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a five-foot prep casts a shadow that is 40 feet long while standing 200 feet from a streetlight. how high above the ground is the lamp?
30ft high above the ground is the lamp.
What is an angle?
An angle is a shape created by two rays that share a terminus and are referred to as the angle's sides and vertices, respectively. Angles created by two rays are in the plane where the rays are located. The meeting of two planes also creates angles. We refer to these as dihedral angles.
Here, we have
BC⊥AD and DE⊥AD
∠ABC = ∠ADE(=90°)
Also,∠A=∠A
ΔABC ΔADE (AA similarity)
AB/AD = BC/DE
40/(200 + 40) = 5ft/DE
DE = 5ft×6 = 30ft
Hence, 30ft high above the ground is the lamp.
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1. A model car travels horizontally after being released. The car travels a distance d metres in a time of t seconds. d is directly proportional to t. The car travels 20 metres in a time of 2 seconds. a) Calculate the distance the car travels in 3 seconds.
If d is directly proportional to t, then we can say that d = k * t, where k is a constant of proportionality. We can find the value of k by substituting the given values for d and t into this equation:
d = k * t
20 = k * 2
Solving for k, we get k = 10.
Now that we know the value of k, we can calculate the distance the car travels in 3 seconds by substituting 3 for t in the equation d = k * t:
d = 10 * 3
d = 30
Therefore, the car travels 30 meters in a time of 3 seconds.
Help I need to turn this in urgent please help me
The lengths of the segments SH is 16, HM is 8, TH is 8, HR is 12,
TD = 12 and ER = 18.
What is a triangle?A triangle is a three-sided closed-plane figure formed by joining three noncolinear points. Based on the side property triangles are of three types they are Equilateral triangle, Scalene triangle, and Isosceles triangle.
We know a centroid divides a line segment in the ratio of 2 : 1 from vertices
to the other side.
Given, ΔSTR, H is the centroid EH = 6, DH = 4, and SM = 24.
SH = (2/3)×24 = 16.
HM = 8.
As DH = 4 hence TH = 8.
As EH = 6, HR = 12.
TD = (DH + TH) = 12.
ER = 6 + 12 = 18.
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The price of a venti vanilla bean Frappuccino from Starbucks is $4.45, but you have only $3 with you. Your friend is working and gets you a discount of 20%. How much does the drink cost with the discount? Do you have enough to buy the drink? How much money are you short?
The cost of the drink after discount is 3.56 dollars.
You don't have enough money to buy the drink.
You are short of 0.89 dollars.
How to find the cost of the drink?The price of a venti vanilla bean Frappuccino from Starbucks is $4.45, but you have only $3 with you. Your friend is working and gets you a discount of 20%.
The cost of the drink with discount can be calculated as follows:
amount discounted = 20% of 4.45
amount discounted = 20 / 100 × 4.45
amount discounted = 89 / 100
amount discounted = 0.89 dollars
Therefore,
cost of the drink after discount = 4.45 - 0.89
cost of the drink after discount = 3.56 dollars
Therefore, you don not have enough money to buy the drink.
You are short of 4.45 - 3.56 = 0.89 dollars.
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An online music store is offering a promotion. When you spend $40 or more, you receive 3 songs for free. You buy an album for $8. Write and solve an inequality that represents how much more money you must spend to receive the three free songs.
Answer:[tex]x+8\geq 40[/tex]
Step-by-step explanation:
[tex]x+8\geq 40[/tex]
subtract 8 from both sides to isolate x
[tex]x\geq 32[/tex]
You must spend 32 more dollars to receive three free songs.
In many countrie, cigarette moking ha decreaed in the 21" century. The percentage of moker in Ruia can be
repreented by the equation 36. * 100 y = 346, and the percentage of moker in Poland i repreented by the
year ince 20000
equation 18 r 50 y = 176. The equation ue y for the percentage of adult who moke and * for the number of year ince 2000,
Determine the olution to the ytem and how your work
Based on the provide system of linear equations, the solution is that there is no solution.
A system of linear equations refers to a collection of one or more linear equations involving the same variables. The solution to a system of linear equations refer to the point at which the lines representing the linear equations intersect. If the lines are parallel, they will never intersect hence, the system has no solution.
According to the provided information, the percentage of smokers in Russia is given by the equation 36x+100y=346 and the percentage of smokers in Poland is given by the equation 18x+50y=176. In order to determine the solution of the system of linear equations, multiply the Poland equation by 2 and subtract it from the Russian equation. Hence,
2*(18x+50y=176)
36x + 100y = 352
-(36x+100y=346)
0 = 6
It means the original equations are parallel. As both the variables are eliminated, the system has no solution.
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determine the horizontal and vertical components of force at pins a and c of the two-member frame.
a and C of the two-member frame=Ax=-300N,Ay=300N, cx=300n, cy=300N.
momentum of all forces about A is
=-600x1.5+Cxx3=0
cx=900/3=300n
cx=300n.
momentum of all forces about c is
-600x1.5-AXx3=0
Ax=-900/3=-300n
Ax=-300n
forces in Ab components is
600x1.5-Ayx3=0
ay=300n
momentum of all forces about b
600x1.5-ayx3-cyx3+cxx3=0
900-900-900+cxx3=0
cxx3=900
cx=300n
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Find all angles 0 < theta < 360 that satisfy the equation to the nearest 10th
4cos2x + cosx -3= -5cosx - 8
The angles that satisfy the trigonometric equation 4 · cos 2x + cos x - 3 = - 5 · cos x - 8 are described below:
x₁₁ ≈ 104.5°, x₁₂ ≈ 255.5° x₂₁ = 120°, x₂₂ ≈ 240°How to find the solution of a trigonometric equation
In this problem we find a trigonometric equation involving both cos 2x and cos x, we need to simplify the expression such that only cos x is present. This can be done by means of trigonometric formulas. First, write the entire expression:
4 · cos 2x + cos x - 3 = - 5 · cos x - 8
Second, use trigonometric formulas to eliminate cos 2x:
4 · (2 · cos² x - 1) + cos x - 3 = - 5 · cos x - 8
8 · cos² x - 4 + cos x - 3 = - 5 · cos x - 8
8 · cos² x + 6 · cos x + 1 = 0
Third, use the quadratic formula to find all possible roots:
cos x₁ ≈ - 1 / 4, cos x₂ ≈ - 1 / 2
Fourth, use inverse trigonometric functions to determine all possible angles between 0° and 360°:
Case 1: - 1 / 4
x₁₁ ≈ 104.5°, x₁₂ ≈ 255.5°
Case 2: - 1 / 2
x₂₁ = 120°, x₂₂ ≈ 240°
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Mary bought an antique clock.
In the first year, its value increased by 30%
In the second year, its value decreased by 30%
Work out whether the clock increased in value, decreased in value or remained the
same value over the two years.
You must show your working clearly.
The overall price considering the percentages is decreased.
What are percentages?Percentage is defined as "out of 100." Similar to fractions and decimals, percentages are used in mathematics to represent subsets of a whole. When expressing a sum as a percentage, 100 equal components are regarded to make up the whole. A percentage can be shown by the symbol % or, less frequently, by the abbreviation 'pct'.
In stores, online, in commercials, and in the media, you can see percentages practically anywhere. Understanding what percentages signify is a crucial skill that could help you save time and money and raise your employability.
Calculation:Let us suppose mary bought that antique clock for 100$.
So now in the first year its value is increased by 30% which is 100+30 = 130$ after the 1 st year.
Now the clock price is 130$
So now in the second year its value is decrease by 30% which is 130- 39 = 91 $
So the overall clock price has been decreased by $9 since mary bought it.
The overall price considering the percentages is decreased.
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solve the given differential equation by undetermined coefficients. 1 4 y'' + y' + y = x2 − 3x
The general solution will be:
[tex]$$y(x)=e^{-x / 28}\left(c_1 \cos \left(\frac{\sqrt{55}}{28} x\right)+c_2 \sin \left(\frac{\sqrt{55}}{28} x\right)\right)+x^2-5 x-23$$[/tex]
[tex]$$14 y^{\prime \prime}+y^{\prime}+y=x^2-3 x$$[/tex]
Consider the homogenous equation [tex]$14 y^{\prime \prime}+y^{\prime}+y=0$[/tex]
The auxiliary equation is [tex]$14 m^2+m+1=0$[/tex]
[tex]m=-\frac{1}{28}+i \frac{1}{28} \sqrt{55}[/tex]
The homogenous solution is
[tex]$$y_c(x)=e^{-x / 28}\left(c_1 \cos \left(\frac{\sqrt{55}}{28} x\right)+c_2 \sin \left(\frac{\sqrt{55}}{28} x\right)\right) \text {. }$$[/tex]
Consider the non-homogenous de [tex]$14 y^{\prime \prime}+y^{\prime}+y=x^2-3 x$[/tex] by using the method of undetermined coefficients
consider [tex]$y_p=A x^2+B x+C \Rightarrow y_p^{\prime}=2 A x+B \Rightarrow y_p^{\prime \prime}=2 A$[/tex]
by plugging the values in [tex]$14 y^{\prime \prime}+y^{\prime}+y=x^2-3 x$[/tex]
[tex]$$\begin{aligned}& 28 A+(2 A x+B)+A x^2+B x+C=x^2-3 x \\& \Rightarrow x^2(A)+x(2 A+B)+28 A+B+C=x^2-3 x\end{aligned}$$[/tex]
On comparing both sides we get
A=1.
2A+B =-3
=>B=-3-2A
=>B=-5
28A+B+C=0
=>C=-28A-B
=>C=-23
by plugging the values [tex]y_p=x^2-5 x-23[/tex].
The general solution will be:
[tex]$$y(x)=e^{-x / 28}\left(c_1 \cos \left(\frac{\sqrt{55}}{28} x\right)+c_2 \sin \left(\frac{\sqrt{55}}{28} x\right)\right)+x^2-5 x-23$$[/tex]
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