a. Φ(t)= P.D.[tex]P^{-1}[/tex] is the fundamental matrix of the homogeneous system.
b. [tex]P^{-1}[/tex]= 1/√5 || 1 2 || || -2 1 || is the inverse
c. x(t)= Φ(t)∫ Φ(t)-1.b(t) dt + Φ(t).[tex]x_0[/tex]
d. The solution to the system is x(t)= || e-t/√5 (1-e2t)/2√5 - (2e2t -3)/2√5 || || 1/√5 e-t/√5 +2/√5 (e2t-1)/2√5 ||
In this problem, we will solve the nonhomogeneous system.
Firstly, we will write a fundamental matrix for the associated homogeneous system 1 1 2 0.
a) Fundamental matrix for the homogeneous system
The homogeneous system of equations can be written as:
X'=AX
Here, X is the vector of unknowns, and A is a matrix of constants.
For the given homogeneous system of equations, we have:
1 1 2 0X1' X2'= 1 2 X1X2
X'= AX
Thus, the matrix A can be written as:
A= 1 2 1 0
Now, let us solve the eigenvalue problem for matrix A as follows:
| A - λI |= 0
⇒| 1 - λ 2 || 1 || λ - 0 || 1 || 2 || 0 - λ || 0 |
⇒ (1-λ)(0-λ)-2⋅1=0
⇒ λ2 - λ - 2 = 0
On solving, we get:
λ1 = -1 and λ2 = 2
Thus, the eigenvectors corresponding to these eigenvalues are:
For λ1 = -1, A - (-1)I= 2 2 || x || = 0 1 || y ||
⇒2x+2y = 0y = -2x
Thus, the eigenvector corresponding to λ1 is:
x1= || 1 || and x2 = || -2 ||
The normalized eigenvector corresponding to λ1 is:
x1= || 1/√5 || and x2 = || -2/√5 ||
For λ2 = 2, A - 2I= -1 2 || x || = 0 -2 || y ||
⇒-x+2y = 0y = 1/2x
Thus, the eigenvector corresponding to λ2 is:
x1= || 2 || and x2 = || 1 ||
The normalized eigenvector corresponding to λ2 is:
x1= || 2/√5 || and x2 = || 1/√5 ||
Thus, the matrix P of eigenvectors is:
P= || 1/√5 -2/√5 || || 2/√5 1/√5 ||
Hence, the fundamental matrix of the homogeneous system is:
Φ(t)= P.D.[tex]P^{-1}[/tex]
where D= diagonal matrix of eigenvalues= || -1 0 || || 0 2 ||and [tex]P^{-1}[/tex] is the inverse of matrix P.
We will now find the inverse of matrix P.
b) Computation of inverse of matrix P
The inverse of matrix P is:
[tex]P^{-1}[/tex]= 1/√5 || 1 2 || || -2 1 ||
c) Multiplication by and integration
Multiplying Φ(t) by e
At, we have:e AtΦ(t)= PeDt.
[tex]P^{-1}[/tex].eAt= P.|| e-1t 0 || || 0 e2t ||.[tex]P^{-1}[/tex]
The solution to the system is:
x(t)= Φ(t)∫ Φ(t)-1.b(t) dt + Φ(t).[tex]x_0[/tex]
where b(t) is the vector of inhomogeneous terms, and [tex]x_0[/tex] is the initial condition.
d) Solution to the system
The nonhomogeneous system of equations can be written as:
X'=AX + b
where b= 1 [tex]e^{-t}[/tex]
The solution to the system is:
x(t)= Φ(t)∫ Φ(t)-1.b(t) dt + Φ(t).x0= P.|| e-t/√5 (e2t -1)/2√5 || || -2(e2t-1)/2√5 e-t/√5 ||.P-1 .|| 0 || || 1 ||+ P.|| 1/√5 -2/√5 || || e-t/√5 (e2t -1)/2√5 || || -2(e2t-1)/2√5 e-t/√5 ||.P-1 .|| -1/√5 || || 2/√5 ||= || 1/√5 -2/√5 || || e-t/√5 (1-e2t)/2√5 || || (e2t-3)/2√5 e-t/√5 ||.|| 0 || || 1 ||+ || 1/√5 -2/√5 || || e-t/√5 (e2t -1)/2√5 || || -2(e2t-1)/2√5 e-t/√5 ||.|| -1/√5 || || 2/√5 ||
The final answer is: x(t)= || e-t/√5 (1-e2t)/2√5 - (2e2t -3)/2√5 || || 1/√5 e-t/√5 +2/√5 (e2t-1)/2√5 ||
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A merry-go-round rotates from rest with cons†an t angular acceleration 'alpha'. ratio of time to rotate first 2 revolutions and next 2 revolutions i.
The required ratio of the time to rotate first and next 2 revolution is given by option 2. (√2+1) : 1.
Merry go round starts from rest.
Let us assume that the initial angular velocity and displacement are zero. Final angular velocity after 2 revolutions is ω.
This implies,
ω₀ = 0 rad/s.
For first two revolutions of the merry go round we have,
θ = 2×2π rad
⇒ θ = ω.t₁ + 1/2 × α × t₁²
⇒ 2×2π = 0×t₁ + 1/2 × α × t₁²
⇒t₁² = 8π/α
⇒ t₁ = √(8π/α)
For first four revolutions of the merry go round we have,
θ = 4×2π rad
⇒ θ = ω.t₂ + 1/2 α.t₂²
⇒ 4×2π = 0×t₂ + 1/2 × α × t₂²
⇒ t₂² = 16π/α
⇒t₂ = √(16π/α)
Time taken for 3rd and 4th revolution of the merry go round is equals to,
t₃ = t₂ - t₁
Substitute the values we get,
⇒ t₃ = √(16π/α) - √(8π/α)
⇒t₃ = (√2-1) √(8π/α)
Ratio of time to rotate first 2 revolutions and next 2 revolutions is written as ,
t₁/t₃ = √(8π/α) / [(√2-1)√(8π/α)]
⇒ t₁/t₃ = 1/(√2-1)
Multiply and divide by the conjugate of √2-1 is equals to,
⇒ t₁/t₃ = 1/(√2-1) × (√2+1)/(√2+1)
⇒ t₁/t₃ = √2+1
Therefore, ratio of time to rotate first 2 revolutions & next 2 revolutions is equals to option 2. (√2+1) : 1.
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The above question is incomplete, the complete question is:
A merry go round rotates from rest with constant angular acceleration `α` . Ratio of time to rotate first 2 revolutions & next 2 revolutions is
1) 1:1
2) (√2+1):1
3) √2:1
4) 1:√2
Kevin's GPS tracker tells him that he just ran 7 miles. He actually ran 10 miles. What was the percent error? HELP PLEASE!
Answer:42.8571429%
Step-by-step explanation:
His GPS tracker got 7 miles while he actually ran 10. 10 - 7 = 3. Then you divide 3 by 7 which is [tex]\frac{3}{7}[/tex] and that is around 42.8571429%.
a lattice point is a point with integer coordinates such as 2, 3. in how many ways can we pick 3 lattice points such that both coordinates of each point are positive integers less than 5, and the three points form a triangle?
The possible triangle whose vertices are given by the lattice points with positive integer coordinates less than 5 is 20.
To solve this, we apply principles of permutation and combination to count the number of possible triangles that can be formed from the given set of lattice points.
Here are the different ways of arranging the lattice points with lattice point coordinates less than 5:
If we want to choose any three lattice points, we can pick the first point from the 20 given in the table above.
Then, we can pick the second point from those lattice points that lie strictly above and strictly to the right of the first point.
For the third point, we can only pick lattice points lying above and to the right of the line segment connecting the first two points.
So, there are 20 possible triangles whose vertices are positive integer coordinates less than five.
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Hong knit scarves for his friends. Altogether, the scarves had a total length of
436.8 in. If he knit 7 scarves, and each scarf was the same length, how long
was each scarf? Write your answer in feet.
Use the table of conversion facts as necessary, and do not round your answer
Answer:
5.2 feet
Step-by-step explanation:
436.8/7/12 feet
consider a 20-foot chain that weighs 2 pounds per foot hanging from a winch 20 feet above ground level. find the work done by the winch in running the winch until the bottom of the chain is at the 10-foot level.
The work done by the winch in running the winch until the bottom of the chain is at the 10-foot level is 200 foot-pounds.
Given, The weight of the chain per foot = 2 pounds
Total length of the chain = 20 feet
The chain is hanging from a winch 20 feet above ground level.
The work done by the winch in running the winch until the bottom of the chain is at the 10-foot level is to be determined.
The force required to lift the chain is equal to the weight of the chain. Therefore, the force needed is F = 2 × 20 = 40 pounds. The work done is the product of force and distance.
Therefore, Work done = force × distance= 40 × (20 − 10)= 400 foot-pounds
Thus, the work done by the winch in running the winch until the bottom of the chain is at the 10-foot level is 200 foot-pounds.
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pls help asap The following is a right angle. Find X:
x =
Answer:
x = 11
Step-by-step explanation:
3x + 5x + 2 =90
(3x + 5x) + (2) = 90
8x + 2 = 90
8x + 2 -2 = 90 -2
8x = 88
8x/8 = 88/8
x = 11
Which object weighs 15 kilograms?
A. A bag of potatoes
B. Story book
C. Laptop D. Washing machine
It is highly possible for a bag of potatoes to weigh 15 kilograms.
Out of the given options, the object that weighs 15 kilograms is most likely to be a bag of potatoes. Generally, bags of potatoes weigh anywhere between 5 kilograms to 25 kilograms, depending on the size and quantity of the potatoes. Therefore, it is highly possible for a bag of potatoes to weigh 15 kilograms.
A story book, on the other hand, usually weighs between 200 grams to 1 kilogram, which is significantly lighter than 15 kilograms. Similarly, a laptop typically weighs between 1 kilogram to 3 kilograms, while a washing machine usually weighs around 50 kilograms to 100 kilograms, making them unlikely to weigh 15 kilograms.
It is important to note that the weight of an object can vary greatly depending on its size, material, and other factors. Therefore, it is always best to weigh an object using a scale to accurately determine its weight.
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100 POINTS!
A soccer player kicks a ball from the ground with a speed of 16 m/s at an angle of 63°.
a. What is the horizontal component of the velocity?
b. What is the vertical component of the velocity?
c. How long does it take the ball to reach its highest point?
d. What is the maximum height of the ball?
e. What is the total amount of time that the ball is in the air?
f. How far is the ball from the soccer player when it lands? (2 points)
Please show work for all questions
Answer:
a) v.h=v*cos(63)=7.26 m/s
b) v.v=v*sin(63)=14.26 m/s
c) t=v.v/g=14.26/9.8=1.46 s since time the body takes to go upward is defined by the acceleration due to gravity and the vertical component of the speed
d) h=1/2*gt^2=7.15 m
e) T=2t=2.92 s
f) R=v.h*t=7.26*1.46=10.6 m since horizontal displacement is defined by the horizontal component of the speed only.
Step-by-step explanation:
Hope this helps! =D
Mark me brianliest! =D
Tell whether the angles are adjacent or vertical. Then find the value of x . (4x -25) and 75
The angles are vertical and the value of x from the angles (4x -25) and 75 can be expressed as 25.
How can the value of x be calculated?A pair of angles that are vertically opposed to one another are always equal. Moreover, a vertical angle and the angle to which it is adjacent are supplementary angles, meaning their sum is 180 degrees. When two lines join to form an angle, say X=45°, the opposite angle is also 45°, for instance.
From the fiqure we can see that the angles the vertically opposite angles, base on this condition we can form the equation as;
(4x-25)=75
4x=(75+25)
4x=100
x=100/4
x=25
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Write a polynomial function in standard form with zero's -5,-4,and 3
The equation of the polynomial function in standard form is P(x) = x³ + 6x² - 7x - 60
Calculating the polynomial function in standard formGiven that we have the following
zeros = -5,-4,and 3
The polynomial function in standard form is calculated as
P(x) = (x - zeros)
So, we have
P(x) = (x + 5)(x + 4)(x - 3)
Opening the brackets, we have the following equation
P(x) = x³ + 6x² - 7x - 60
Hence, the polynomial is P(x) = x³ + 6x² - 7x - 60
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Can't Seem to be able to solve this help
An angle measurement, m∠BOC = 60° degrees where O is the center of the corcle.
What is an angle?
Since the ΔABC has three similar sides, we know that it is an equilateral triangle, meaning that all three angles are equal and each angle measures 60°.
Since A, B, and C are points on a circle with O as the center, we know that each of the distances OA, OB, and OC is equal to the radius of the circle.
Let's call this radius r.
Since ΔABC is equilateral, we know that each side has length r. We can also draw radii from the center O to points A, B, and C to form three congruent triangles AOB, BOC, and COA, each with two sides of length r and one angle of 60°.
Since the sum of the angles in a triangle is 180° degrees, each of the other two angles in each of these congruent triangles must measure 60° as well.
Thus, we see that ∠BOC is one of the angles in the isosceles ΔBOC, which has two angles of 60° and a third angle x, which we want to find.
We can use the fact that the sum of the angles in a triangle is 180° to find x:
x + 60 + 60 = 180
Simplifying, we get:
x = 60
Therefore, angle BOC measures 60°
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Complete question is: m∠BOC = 60° degrees where O is the center of the corcle.
Alexander owes \$3,200$3,200 on his credit card. The bank charges an annual interest rate of 16. 8%, compounded monthly. If Alexander wants to pay off his credit card using equal monthly payments over the next 8 months, what would the monthly payment be, to the nearest dollar?
Alexander would need to make monthly payments of $430 for the next 8 months to pay off his credit card debt.
To calculate the monthly payment that Alexander needs to make, we can use the formula for the present value of an annuity, which is:
[tex]PMT=PV*(r*(1+r)^n)/(1+r)^n-1)[/tex]
Where:
PMT = Monthly payment
PV = Present value of the debt
r = Monthly interest rate
n = Total number of payments
First, we need to calculate the monthly interest rate. The annual interest rate is 16.8%, which means the monthly interest rate is 16.8%/12 = 1.4%.
Next, we can calculate the total number of payments that Alexander needs to make over 8 months.
n = 8
Now, we can plug in the values:
[tex]PMT = 3200*\frac {(0.014*(1+0.014)^8)}{((1+0.014)^8-1)}[/tex]
This works out to approximately $430 per month (rounded to the nearest dollar).
Therefore, Alexander would need to make monthly payments of $430 for the next 8 months to pay off his credit card debt.
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If y varies directly as x and y = 32 when x = 4, find the value of y when x = 5.
Answer:
first find constant term
y=kx
32=k(4)
32/4=4k/4
k=8
find value of y when x=5
y=kx
y=8×5
y=40
I WILL GIVE SOMEONE BRAINLIEST IF YALL ANSWER THIS!!!!
Answer:
v<-4/35. If you want interval notation it is (-infinity, -4/35)
Step-by-step explanation:
A plane is located at C on the diagram. There are two towers located at A and B. The distance between the towers is 7,600 feet, and the angles of elevation are given. a. Find BC, the distance from Tower 2 to the plane, to the nearest foot. b. Find CD, the height of the plane from the ground, to the nearest foot.
In linear equation, 6499 ft CD, the height of the plane from the ground, to the nearest foot.
What in mathematics is a linear equation?
An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.
Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.
Part A) Find BC, the distance from Tower 2 to the plane, to the nearest foot.
in the triangle ACD
sin16=CD/(7600+BD)--------> CD=sin16*(7600+BD)---------> equation 1
in the triangle BCD
sin24=CD/BD-----------> CD=sin24*BD---------------> equation 2
equation 1=equation 2
sin16*(7600+BD)=sin24*BD-----> sin16*7600+sin16*BD=sin24*BD
sin24*BD-sin16*BD=sin16*7600----> BD=[sin16*7600]/[sin24-sin16]
BD=15979 ft
in the triangle BCD
cos24=BD/BC---------> BC=BD/cos24-------> 15979/cos24-------> 17491
BC=17491 ft
= BC is 17491 ft
Part 2) Find CD, the height of the plane from the ground, to the nearest foot.
CD=sin24*BD ( remember equation 2)
BD=15979 ft
CD=sin24*15979 -----------> CD=6499 ft
= CD is 6499 ft
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Dan and Paul share some money in the ratio 11:5.
Dan decides gives Paul £39 of his share to make the ratio 1:1.
How much did Paul originally have?
D = Dan's share
P = Paul's share
[tex]\stackrel{D}{11}~~ : ~~\stackrel{P}{5}\qquad \implies \qquad \cfrac{D}{P}=\cfrac{11}{5}\implies D=\cfrac{11P}{5}[/tex]
now Dan decides to give Paul £39, so that means that Dan has £39 less whilst Paul has £39 more, so the new shares will be D - 39 and P + 39, and that puts them in a 1 : 1 ratio.
[tex]\stackrel{D-39}{1}~~ : ~~\stackrel{P+39}{1}\qquad \implies \qquad \cfrac{D-39}{P+39}=\cfrac{1}{1}\implies D-39=P+39 \\\\\\ \stackrel{\textit{substituting from the 1st eqution}}{\left(\cfrac{11P}{5} \right)-39~~ = ~~P+39}\implies \cfrac{11P}{5}=P+78\implies 11P=5P+390 \\\\\\ 6P=390\implies P=\cfrac{390}{6}\implies \boxed{P=65}[/tex]
pleasee help me fast
Therefore, the length of side AC in triangle ABC is approximately 14.6 units.
What is triangle?A triangle is a closed two-dimensional geometric shape that is formed by connecting three non-collinear points with straight line segments. Each of these line segments is called a side, and the point where two sides meet is called a vertex. A triangle has three vertices, three sides, and three angles. Triangles can be classified based on the length of their sides and the size of their angles.
Here,
We can use the sine rule to find the length of side AC in triangle ABC, where angle A = 32° and AB = 8. The sine rule states that for any triangle ABC:
a/sin(A) = b/sin(B) = c/sin(C)
where a, b, and c are the lengths of the sides opposite to the angles A, B, and C, respectively.
In this case, we know the length of side AB (a = 8) and the measure of angle A (A = 32°), so we can set up the following proportion:
a/sin(A) = c/sin(C)
Substituting the values, we get:
8/sin(32°) = AC/sin(C)
Solving for AC, we get:
AC = (8/sin(32°)) * sin(C)
To find sin(C), we can use the fact that the sum of the angles in a triangle is 180°. So, we have:
C = 180° - 90° - A
C = 180° - 90° - 32°
C = 58°
Substituting this value, we get:
AC = (8/sin(32°)) * sin(58°)
AC ≈ 14.6
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b. John took out a loan from the bank for $221 and was told he would have to pay back
some interest on top of the $221 dollars. John had to pay back 321 dollars, how much.
money did the bank make after John Paid them back?
The bank made $100 in profit from the loan.
What is simple interest?
Simple Interest is an easy method of calculating the interest for a loan/principal amount. Simple interest is a concept that is used in many sectors such as banking, finance, automobile, and so on.
If John had to pay back $321 in total and the original loan was $221, then the bank made $100 in interest from John.
Interest = Amount John had to pay - Original Loan
Interest = 321 - 221
Interest = $100
So, the bank made $100 in profit from the loan.
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Function f is defined by the equation f(x)=y?.
a. What is f(2)?
Answer:
Function f is defined by the equation f(x)=y?. this isn't a thing
i believe you mean... Function f is defined by the equation f(x) = x².
What is f(2)?
What is f(3)?
in which case the answer is The value of f(2) and f(3) for the given functions are 4 and 9 respectively.
The given function is f(x)=x².
What is the function?
Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.
We need to find the value of f(2) and f(3)
Put x=2 in the given function, we get
f(2)=2²
= 4
Put x=3 in the given function, we get
f(3)=3²
= 9
Hence, the value of f(2) and f(3) for the given functions are 4 and 9 respectively.
Identify the radius and the center of a circle whose equation is (x â€" 5)² y² = 81. the radius of the circle is units. the center of the circle is at ( , ).
The radius of the circle is 9 units. The center of the circle is at (5, 0).
The given equation is (x – 5)²y² = 81.
To identify the radius and center of this circle, we can rewrite the equation in standard form.
First, let's divide both sides by 81 to get (x – 5)²y²/81 = 1.
Next, we can take the square root of both sides to eliminate the squared terms.
This gives us the equation (x – 5)/9 · y/3 = 1.
The standard form of a circle is (x – h)² + (y – k)² = r², where (h, k) is the center of the circle and r is the radius.
Comparing this form to our equation, we can see that (x – 5)/9 and y/3 correspond to (x – h) and (y – k), respectively.
Therefore, we can solve for h and k by setting (x – 5)/9 = 0 and y/3 = 0, respectively.
This gives us h = 5 and k = 0, so the center of the circle is (5, 0). Finally, we can solve for the radius r by noting that the
distance from the center (5, 0) to the edge of the circle is 3 units in the x-direction and 9 units in the y-direction.
Therefore, the radius of the circle is 9 units, and we can express our final answer as:
The radius of the circle is 9 units. The center of the circle is at (5, 0).
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The sum of two numbers is -10. Negative three times the first number minus the second number equals 2.
Find the numbers.
Answer:
x = 4, y = -14
Step-by-step explanation:
1. x + y = -10 and -3x - y = 2
2. y= -10 - x > plug in for y > -3x - (-10 - x) = 2
3. solve for x > -3x + 10 + x = 2
-2x + 10 = 2 > -2x = -8 > x= 4
4. plug x into original equation > 4 + y = -10 > y = -14
The GCF of 15
and 30
Answer:
15
Step-by-step explanation:
The factors of 15 are 1,3,5,15
The factors of 30 are 1,2,3,5,6,10, 15, 30
The largest factor that both of these numbers share is 15
what is the volume of the right circular cylinder with a diameter of 10 meters and a height of 16 meters. leave the answer in terms of pi.
if it has a diameter of 10, that means its radius is half that or 5.
[tex]\textit{volume of a cylinder}\\\\ V=\pi r^2 h~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=5\\ h=16 \end{cases}\implies V=\pi (5)^2(16)\implies V=400\pi[/tex]
Answer: The answer is A, 400π m3
Step-by-step explanation:
Joe has $2500 in a savings account that earns 9% interest per year. How much amount in dollars will he have in 3 years?
To calculate the amount of money Joe will have in 3 years, we can use the formula:
A = P(1 + r/n)^(nt)
where:
A = the amount of money Joe will have in 3 years
P = the principal amount = $2500
r = the annual interest rate = 9% = 0.09 (expressed as a decimal)
n = the number of times the interest is compounded per year (assuming annual compounding, n = 1)
t = the number of years = 3
Plugging in these values, we get:
A = $2500(1 + 0.09/1)^(1*3)
= $2500(1.09)^3
= $2500(1.29503)
= $3237.58 (rounded to two decimal places)
Therefore, Joe will have $3237.58 in 3 years.
A house is valued at 210000 correct to 2 significant figures
What is the lowest possible value of the house?
A house with value at 210000 correct to 2 significant figures. The lowest possible value of the house is equals to the £205000.
The second significant figure of a number is the digit after the first significant figure. This is true even in case of the digit is zero. We have a house is valued at £210000 correct to 2 significant figures. So, it is bounded question. Since, two significant figures is correct then there is error in 10000 range in £210000. The rounded up value is £2.10 × 10⁵. Steps to determine these bounds
Let assumes that a certain number x must be rounded by a number y. Divide y by 2, i.e. y/2.Upper bound is addition of specific number and y/2, i.e. upper bound = x + y/2.To calculate the lowest value subtract y/2 from the given number, i.e. lower limit = x - y/2.here x = £210000, y = 10000
Since 2 significant digits are correct,
is about 10,000 of £210,000. The lowest possible number rounded to 2 significant digits in the £210,000 range is £205,000, because anything below that looks like £204....that doesn't round up to £21. Hence required value is £205000.
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[100 POINTS!!!]
According to the Fundamental Theorem of Algebra, how many zeros does the function f(x) = 5x^6+2x^3−4x + 1
There are____________ Zeros.
Step-by-step explanation:
According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n complex zeros (counting multiplicities).
The degree of the polynomial f(x) = 5x^6+2x^3−4x + 1 is 6, which means that it is a sixth-degree polynomial. Therefore, it has exactly 6 complex zeros (counting multiplicities).
Note that the zeros may be real or complex, and they may not all be distinct. To find the exact number of zeros and their values, we would need to use additional methods such as factoring, the rational zeros theorem, or numerical methods.
Answer:
The Fundamental Theorem of Algebra states that every non-constant polynomial function has at least one complex root. In the case of the polynomial function f(x) = 5x^6+2x^3−4x + 1, it is a sixth-degree polynomial function, which means it has at most six complex roots. However, the Fundamental Theorem of Algebra does not tell us the exact number of roots, nor does it tell us whether the roots are real or complex.
To determine the number of real roots of the function f(x), we can use the Intermediate Value Theorem or graph the function to find the number of times it crosses the x-axis. However, determining the number of complex roots requires more advanced techniques, such as the use of the Fundamental Theorem of Algebra in combination with the Factor Theorem, Rational Root Theorem, or other methods.
Therefore, based on the Fundamental Theorem of Algebra alone, we can say that the function f(x) = 5x^6+2x^3−4x + 1 has at least one complex root, but we cannot determine the exact number or nature of the roots without additional analysis.
An equation is modeled. What value of x makes the equation true?
An equation is modeled. The value of x makes the equation true is -1.
Equation:
Conditional comparisons apply only to specific values of variables. An equation consists of two expressions joined by an equal sign ("="). Expressions for both sides of the equals sign are called the "left side" and the "right side" of the equation. Usually the right side of the equation is assumed to be zero. If this is accepted, it does not reduce the generality, since it can be done by subtracting the right side from the two sides.
According to the Question:
5x + 6 = 1
5x = -5
x = -1
Complete Question:
An equation is modeled. What value of x makes the equation true?
(1) 1
(2) 7
(3) -5
(4) -1
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pentagon abcde is the image of pentagon abcde under
The length of segment A'E' of the new formed pentagon is 5 units.
How to find the length of the line segment?We are given that:
Pentagon A’B′C′D′E’, is the image of pentagon ABCDE under a dilation with a scale factor of 5/2.
From the attached graph, we see that the coordinates of the points A and E, in the pentagon ABCDE with reference to point A are;
A- (0, 0)
E- (0, 2)
The length of the segment AE is:
AE = √((2-0)² + (0 - 0)²)
AE = √2²
AE = 2
When a pentagon ABCDE is dilated to pentagon A'B'C'D'E' with a scale factor of 5/2, we have;
Length of segment A'E' = 5/2 × Length of segment AE
Thus;
Length of segment A'E' = 5/2 × 2
Length of segment A'E' = 5 units.
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Complete question is:
Pentagon A’B′C′D′E’, is the image of pentagon ABCDE under a dilation with a scale factor of 5/2.
Find the length of the line segment A'E'.
I need helppppppppp please help me
The measures of angles in a rhombus are m∠2=130, m∠3=50 and m∠4=50.
What is angle measure?Angle measure is a term used in geometry to describe the size of an angle. It is usually expressed in degrees, which is a unit of measurement for angles. A degree is defined as 1/360th of a complete revolution around a circle.
According to question:In the rhombus, m∠1 =130
In a rhombus, opposite angles are congruent.
The angle opposite to angle 1 is angle 2.
so m∠1 =m∠2
m∠2= 130 degrees
Other pair of opposite angles are also congruent.
so m∠3=m ∠4
We know that the sum of 4 angles = 360
m∠1+m∠2+m∠3+m∠4=360
130+130+m∠3+m∠4=360
260+m∠3+m∠4=360
m∠3+m∠4=360-260
m∠3+m∠3=100
2m∠3=100
m∠3=50
Now, m∠3=50
m∠4=50
Therefore, m∠2=130, m∠3=50 and m∠4=50
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Bentley wants to ride his bicycle 39. 6 miles this week. He has already ridden 8 miles. If he rides for 4 more days, write and solve an equation which can be used to determine
x
x, the average number of miles he would have to ride each day to meet his goal
This gives us the answer of 1.9 miles, meaning that he must ride an average of 1.9 miles each day for the remaining 4 days of the week in order to reach his goal of 39.6 miles.
(39.6 - 8) ÷ 4 = x
7.6 ÷ 4 = x
x = 1.9 miles
Bentley wants to ride his bicycle 39.6 miles this week. He has already ridden 8 miles, leaving him with 31.6 miles left to ride. He has 4 days left to ride, so we need to figure out how many miles he must ride each day in order to meet his goal. We can do this by subtracting the 8 miles he has already ridden from the total goal of 39.6 miles. This gives us 31.6 miles remaining to be ridden. We then divide this number by 4, since he has 4 days left in the week to ride. This gives us the answer of 1.9 miles, meaning that he must ride an average of 1.9 miles each day for the remaining 4 days of the week in order to reach his goal of 39.6 miles.
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