Answer:
31
Step-by-step explanation:
the next number is 31 in this series
Please help with these to questions its urgent please!
Answer:
First problem: x = 5.7√2 = (57√2)/10
Second problem: x = 10√3
Use the quadratic formula to solve -2x^2+2x+1
The solutions for -2x²+2x+1 are x = 1 + √(3)/2 and x = 1 - √(3)/2.
To use the quadratic formula to solve -2x²+2x+1, we first identify the coefficients a, b, and c of the quadratic equation in standard form: ax²+bx+c=0. In this case, a=-2, b=2, and c=1. We then plug these values into the quadratic formula
x = (-b ± √(b²-4ac)) / 2a
Substituting the values, we get
x = (-2 ± √(2²-4(-2)(1))) / 2(-2)
Simplifying the expression inside the square root
x = (-2 ± √(4+8)) / (-4)
x = (-2 ± √12) / (-4)
Simplifying the radical
x = (-2 ± 2√3) / (-4)
Reducing the fraction
x = (1 ± √3/2)
Therefore, the two solutions are x = (1 + √3/2) and x = (1 - √3/2).
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A motorist travelled from Town A to Town B. After travelling for 1 1/2 hours , he passed
a cyclist travelling at an average speed of 45 Km/h in the opposite direction.
when the motorist reached Town B 2 h later, the cyclist was 30 km away from Town A.
a) Find the average speed of the motorist.
b) Find the distance between Town A and Town B.
Answer:
the formula: Speed = Distance / Time.
Let's break down the problem into two parts:
a) Find the average speed of the motorist.
Let's assume the average speed of the motorist is "v" km/h.
The distance the motorist traveled in 1 1/2 hours is given by: Distance = Speed × Time = v × (3/2) km.
The distance the cyclist traveled in 1 1/2 hours (opposite direction) is given by: Distance = Speed × Time = 45 × (3/2) km.
Since they passed each other, the sum of their distances should be equal to the distance between Town A and Town B.
So, we can set up the equation: v × (3/2) + 45 × (3/2) = Distance between Town A and Town B.
b) Find the distance between Town A and Town B.
When the motorist reached Town B 2 hours later, the cyclist was 30 km away from Town A.
We can set up the equation: Distance between Town A and Town B = v × 2 + 30.
Now, let's solve the equations:
v × (3/2) + 45 × (3/2) = v × 2 + 30.
Simplifying the equation, we have: (3v + 135)/2 = 2v + 30.
Multiplying both sides of the equation by 2 to eliminate the fraction, we get: 3v + 135 = 4v + 60.
Subtracting 3v from both sides of the equation, we have: v = 75.
Therefore, the average speed of the motorist is 75 km/h.
To find the distance between Town A and Town B, we substitute the value of v into the equation:
Distance between Town A and Town B = v × 2 + 30 = 75 × 2 + 30 = 150 + 30 = 180 km.
Therefore, the distance between Town A and Town B is 180 km.
In tetrahedron $ABCO,$ $\angle AOB = \angle AOC = \angle BOC = 90^\circ.$ A cube is inscribed in the tetrahedron so that one of its vertices is at $O,$ and the opposite vertex lies on face $ABC.$ Let $a = OA,$ $b = OB,$ and $c = OC.$ Show that the side length of the cube is
\[\frac{abc}{ab + ac + bc}.\]
[asy]
import three;
size(180);
currentprojection = orthographic(6,3,2);
real a, b, c, s;
triple A, B, C, O;
a = 6;
b = 3;
c = 2;
s = a*b*c/(a*b + a*c + b*c);
A = (a,0,0);
B = (0,b,0);
C = (0,0,c);
O = (0,0,0);
draw(O--A,dashed);
draw(O--B,dashed);
draw(O--C,dashed);
draw(A--B--C--cycle);
draw((0,0,s)--(s,0,s)--(s,0,0)--(s,s,0)--(0,s,0)--(0,s,s)--cycle,dashed);
draw((s,s,0)--(s,s,s),dashed);
draw((s,0,s)--(s,s,s),dashed);
draw((0,s,s)--(s,s,s),dashed);
label("$A$", A, SW);
label("$B$", B, E);
label("$C$", C, N);
dot("$O$", O, NW);
dot((s,s,s));
[/asy]
pls help me
Answer:
Step-by-step explanation:To solve this problem, we can use similar triangles. Let the side length of the cube be s, and let the midpoints of AB, AC, and BC be P, Q, and R, respectively. Then OP, OQ, and OR are the diagonals of the faces of the cube, and so they are equal to .Consider triangle AOB. By the Pythagorean theorem, we have , so . Since angle AOB is 90 degrees, we can use similarity to see that triangle OAP is similar to triangle OAB, so . Solving, we find that .Using similar triangles, we can find that . Since the sum of the squares of the lengths of the diagonals of a cube is equal to three times the square of the side length, we have
$$(0P2 + 0Q? + OR?) = 35388 Substituting the values we found for $OP$, $OQ$, and $OR$, we haveSimplifying and solving for , we find that . Thus, the side length of the cube inscribed in the tetrahedron is .
PLEASE HELP SOLVE 30 PTS
Answer:
The polynomial has a unique zero in the interval [1,2] by using intermediate value theorem
Enter your answer by filling in the boxes to correctly complete the statements. If necessary, round to the nearest hundredth.
The practical domain of the situation is [0, ∞].
The practical range of the situation is [0, 100].
What is a domain?In Mathematics and Geometry, a domain refers to the set of all real numbers (x-values) for which a particular function (equation) is defined.
How to identify the domain any graph?In Mathematics and Geometry, the horizontal portion of any graph is used to represent all domain values and they are both read and written from smaller to larger numerical values, which simply means from the left of any graph to the right.
By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:
Domain = [0, ∞] or 0 ≤ x ≤ ∞.
Range = [0, 100] or 0 ≤ y ≤ 100.
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USE THE FACTOR THEOREM
YES OR NO
Answer:
yes
Step-by-step explanation:
the factor theorem says if (x + a) is a factor of a polynomial f(x) then f(- a) = 0
given (x+ 5) then if f(- 5) = 0 , (x + 5) is a factor
4[tex](-5)^{4}[/tex] + 16(- 5)³ - 6(- 5)² + 73(- 5) + 15
= 4(625)+ 16(- 125) - 6(25) - 365 + 15
= 2500 - 2000 - 150 - 350
= 500 - 500
= 0
then (x + 5) is a factor of the given polynomial
what is the difference between an angle bisector and a segment bisector ?
The clear difference between an angle bisector and a segment bisector is that angle bisector passes through the apex of an angle while segment bisector passes through the midpoint of the line segment.
What is an angle bisector?An angle bisector is defined as the line the divides an angle into two equal parts by passing through its apex.
A segment bisector is defined as the line that divides a segment at the midpoint into two parts.
The difference between angel bisector and segment bisector is that angle bisector passes through the apex of an angle while segment bisector passes through the midpoint of the line segment.
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Step 1: Understand the Problem
Keep the Old Car or Buy a Used Car
Manny is an online student who currently owns an older car that is fully paid for. He drives, on average, 190 miles per week to commute to work. With gas prices currently at $
2.9 per gallon, he is considering buying a more fuel-efficient car, and wants to know if it would be a good financial decision.
The old car Manny owns currently gets 18 miles per gallon for average fuel efficiency. It has been a great vehicle, but with its age, it needs repairs and maintenance that average $
770 per year (as long as nothing serious goes wrong).
The newer, more fuel-efficient car that he is looking at to purchase will cost a total of $
6,500 over a three-year loan process. This car gets 32 miles per gallon and would only require an average of $
10 per month for general maintenance. To help make a decision, Manny wants to calculate the total cost for each scenario over three years. He decides to use the quantitative reasoning process to do this.
Step 3: Apply Quantitative Tools
Use computational and algebraic tools to quantify the total costs (gas, maintenance/repairs, purchase price) for each scenario over the three years.
Round your answers to the nearest dollar.
Scenario
Total Cost for Three Years
Keep the old car
$
Number
Buy the fuel-efficient used car
$
Number
The sound wave of two music notes can be represented by the function y = 2sin2x and y = sin x, where x represents the time since the note is played. At what times will the frequencies of the sound waves be the same in the interval [0°, 360°)?
Group of answer choices
x = 0°, 30°, 150°, 180°
x = 0°, 180°, 210°, 330°
x = 30°, 90°, 150°, 270°
x = 90°, 210°, 270°, 330°
The correct answer is:
x = 0°, 30°, 150°, 180°
This is because the sine function equals 0 at 0° and 180° and equals 1/2 at 30° and 150°.
How to solveFor the frequencies to be the same, we need to find values of n and m such that:
n = m
Solving for n and m, we get:
n = m = 2
So both functions complete two cycles in the interval [0, 2π).
We can find the times when this occurs by setting the angles in radians equal to multiples of 2π/2, or π:
2sin2x = sin x
2sin2x - sin x = 0
sin x (2sin x - 1) = 0
sin x = 0 or 2sin x - 1 = 0
sin x = 0 when x is an integer multiple of π.
2sin x - 1 = 0 when sin x = 1/2, which occurs when x = π/6 or x = 5π/6.
Therefore, the frequencies of the sound waves are the same at x = π/6, x = π/2, x = 5π/6, and x = 3π/2.
The correct answer is:
x = 0°, 30°, 150°, 180°
This is because the sine function equals 0 at 0° and 180° and equals 1/2 at 30° and 150°.
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Complete the following using compound future value. (Use the Table provided.)
Note: Do not round intermediate calculations. Round your final answers to the nearest cent.
Time
6 years
Principal
$ 15,300
Rate
8 %
Compounded
Quarterly
What is amount &
Interest?
The total is $24,947.06 plus $9,647.06 in compound interest.
Define compound interest?One way to earn or pay interest on interest is through compound interest. It denotes that interest will be charged on both the main amount and the accrued interest in the subsequent period. Savings or investments can expand over time with the aid of compound interest.
We can use the following formula to determine compound future value:
FV = P * (r/n + 1)(n*t)
where P is the principal, r is the yearly interest rate, n is the number of times interest is compounded annually, and t is the amount of time in years.
Your values have led us to:
FV = 15300 * (1 + 0.08/4)²⁴
FV = $24,947.06
The total is $24,947.06 plus $9,647.06 in interest.
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The space available for a relay race at a carnival is 92.75 yards. A rules sign is placed 5 yards before the starting line. If there are 5 sections of th
start to finish and each section covers the same distance, how long is each section of the race?
Each section of the race is (blank) yards long
If the total distance available is 92.75 yards, and there is a sign 5 yards before the starting line, the actual distance available for the race is:
92.75 - 5 = 87.75 yards
Since there are 5 sections of the race, each section covers:
87.75 / 5 = 17.55 yards
Therefore, each section of the race is 17.55 yards long.
When doubling the dimensions of a rectangle, why does the area quadruple and the perimeter only doubles?
If you double the length and the width, you are quadrupling the area.
This is because 2 × 2 = 4.
Perimeter of new rectangle= 2 × perimeter of initial rectangle.
The perimeter will be double of if we double the length and breadth of rectangle
→ When doubling the dimensions of a rectangle, why the area is quadruple:
We know that :
Area of rectangle = l × b
where, l is length and b is breadth
A = L × W
This means area equals length times width.
if you double the length and you double the width, the formula becomes:
A = 2 × L × 2 × W
A = 2 × 2 × L × W
A = 4 × L × W
if you double the length and the width, you are quadrupling the area.
This is because 2 × 2 = 4.
→When doubling the dimensions of a rectangle, why the perimeter is doubles?
Let length be equal to ′l′
And breadth be equal to ′b′
Perimeter of rectangle = 2 (l + b) ____(equation i )
The perimeter if the length and breadth of the rectangle will increase by two, or will it double? So new parameters of rectangle when its length and breadth will get double on initial one.
Double length of rectangle of initial one= 2 × l=2l
Double breadth of rectangle of initial one = 2×b=2b
So the perimeter of new rectangle whose length and breadth are double of initial one are:
As we know that:
Perimeter of rectangle = 2 (l + b)
So,
Perimeter of new rectangle= 2(2l+2b), (by taking 2 common)
Perimeter of new rectangle = 2×2(l + b) ____ equation (ii)
As from equation (i) perimeter of initial rectangle is = 2(l + b)
So from equation (i) and (ii)
Perimeter of new rectangle= 2 × 2(l + b)
Perimeter of new rectangle= 2 × perimeter of initial rectangle.
Hence, the perimeter will be double of if we double the length and breadth of rectangle.
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A house is purchased for $313,000 with a 15% down payment. A mortgage is secured at 7% for 25 years. Find the monthly payment.
If house is purchased for $313,000 with a 15% down payment, the monthly payment for the mortgage is $1,906.33.
To find the monthly payment for a mortgage secured at 7% for 25 years on a house purchased for $313,000 with a 15% down payment, we can use the formula for calculating mortgage payments:
P = L[c(1 + c)ⁿ]/[(1 + c)ⁿ - 1]
Where P is the monthly payment,
L is the loan amount (which is the purchase price minus the down payment),
c is the monthly interest rate (which is the annual interest rate divided by 12),
n is the total number of monthly payments (which is the number of years multiplied by 12).
Substituting the given values, we get:
L = $313,000 - 0.15($313,000) = $266,050
c = 0.07/12 = 0.00583
n = 25 x 12 = 300
Plugging these values into the formula, we get:
P = $266,050[0.00583(1 + 0.00583)³⁰⁰]/[(1 + 0.00583)³⁰⁰ - 1]
Simplifying the expression, we get:
P = $1,906.33
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A system of equations is shown.
{ x= = 10
3x + 5y = 20
What is the solution x of the system of equations?
Answer:
(r, y)=(10,-2)
Step-by-step explanation: Hope this helps :)
Find the midpoint of the segment formed by the points D(8, 6) and E(4, 10). A. (2, –2) B. (6, 8) C. (9, 5) D. (12, 16)
Answer:
B
Step-by-step explanation:
let the mid-point of DE be ( x , y )
x = (8+4)/2 = 6
y = (6+10)/2 = 8
the mid- point of DE is ( 6 , 8 )
Working with Formulas
HELP PLS
.
the rate of interest is 11.25%.we can use simple interest formula get answer as above
what is simple interest ?
Simple interest is a type of interest that is calculated as a percentage of the principal amount of a loan or investment, and is based only on the original amount of money that was borrowed or invested.
In the given question,
We can use the formula for simple interest to solve for the rate:
Simple Interest = (Principal x Rate x Time) / 100
Substituting the given values, we get:
$900 = (400 x Rate x 20) / 100
Multiplying both sides by 100 and dividing by 400 x 20, we get:
Rate = $900 / (400 x 20) = 0.1125 or 11.25%
Therefore, the rate of interest is 11.25%.
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Please I need some help as soon as possible
Answer:
[tex] {m}^{2} + 7m - 30 = [/tex]
[tex](m - 3)(m + 10)[/tex]
Mo is running the tracks where one lap is 400m. After completing some laps, he found that he has completed 40% of the total distance. After another 2, 50% is done. What is the total distance that he needs to cover in meters?
I REALLY NEED THE ANSWER SOON,
IF YOU CAN DO IT WELL THANK YOU.
Answer:
Total distance is 8000 m------------------
According to the problem, 2 laps make a difference of 10% as:
50% - 40% = 10%Hence the total distance in terms of laps is:
100% / 10% = 10 times the 2 laps:Find the total distance:
Total distance = 10*2 = 20 lapsTotal distance = 20*400 m = 8000 mknow how I can find the measurements in Bloxburg?
Answer:
I'm sorry, but there is no feature for this in this game. But, there is a grid and a sizing tool. You'll just have to make an educated guess based on the size of your character and your character's stage of life (baby, toddler, kid, teen, adult). Hope I helped! I once had the same question, but just did this. I think each big grid square maybe is 3-5 ft?
Find the are of the shaded shape below it has two lines of symmetry (vertical and horizontal) that pass through its centre
Give your answer in cm^2
The area of the shaded shape is 62.5cm^2.
To find the area of the shaded shape, we need to break it down into simpler shapes that we can calculate the area of. From the image, we can see that the shape is composed of a rectangle and two right-angled triangles.
First, we need to find the dimensions of the rectangle. Since the shape has vertical and horizontal lines of symmetry passing through its center, we know that the rectangle is divided into four equal parts.
Therefore, the length of the rectangle is twice the length of one of the triangles, which is 5cm. So, the length of the rectangle is 2 x 5cm = 10cm. The width of the rectangle is also 5cm.
Next, we need to find the area of each of the triangles. We know that the base of each triangle is 5cm, and the height is half of the width of the rectangle, which is 2.5cm. Therefore, the area of each triangle is:
0.5 x base x height = 0.5 x 5cm x 2.5cm = 6.25cm^2
So, the total area of the two triangles is 2 x 6.25cm^2 = 12.5cm^2.
Finally, we can find the area of the shaded shape by adding the area of the rectangle and the area of the triangles:
Area = rectangle area + triangle area = 10cm x 5cm + 12.5cm^2 = 62.5cm^2.
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Brennan knew his parents had set up their will some years ago. Neither he nor his sister, Maureen, had ever discussed the will with their parents. When both parents passed away, they met with the family lawyer about the will. What they found was that the money their parents had been giving them over the last few years was a gift and part of their estate planning. The gifts reduced the size of the estate to ensure the value of the estate upon their passing would be under the tax-free limit.
Their parents had not really talked with Brennan or Maureen about how the “gift” money they had been receiving would affect their final inheritance. As they listened to
the estate lawyer give them details of their inheritance, they were shocked to learn there was less than what they had thought.
1. Discuss your thoughts on how Brennan’s and Maureen’s parents handled the informationin the will prior to their passing. Should the parents have explained more clearly about the “gift” money and how it affected their inheritance?
2. What responsibility did Brennan and Maureen have to discuss the will with their parents?
3. What experience do you have with a family member’s will? How would you approach
the topic of writing a will with a family member?
Open and clear communication is crucial when discussing estate planning and wills with family members to prevent misunderstandings and hurt feelings.
1. It is understandable that Brennan and Maureen feel shocked and disappointed with the reduced inheritance.
While it is not required for parents to disclose every detail of their estate planning to their children, it would have been helpful if they had discussed the impact of the gifts on their estate and their final inheritance.
Clear communication can prevent misunderstandings and hurt feelings in the future.
2. Brennan and Maureen have a responsibility to discuss the will with their parents. It is essential to have an open and honest conversation about their parents' wishes and intentions to avoid any confusion or disputes in the future.
While it can be uncomfortable to talk about death and inheritance, it is a crucial conversation to have.
3. It can be a difficult and emotional topic to discuss, but it is important to have open communication to ensure that everyone's wishes are respected and honored. It may be helpful to consult with a lawyer or financial advisor to ensure that the will is properly drafted and that everyone's needs are addressed.
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Can someone help pleaseeee
The result of the row 1 multiplication by 1/4 is determined as [¹/₂ 0 ].
What is the result of the row multiplication?The resultant of the row multiplication in the Matrice is calculated by applying the following method;
row 1 in the given matrices = [2 0]
To multiply row by 1/4, we will multiply each entity by 1/4 as shown below;
[2 0] x 1/4 = 1/4(2 0)
= [¹/₂ 0 ]
Thus, the result of the row multiplication is determined by multiplying each entry in row 1, that is 2, and 0 by 1/4.
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Triangle QRS is rotated 180° about the origin.
On a coordinate plane, triangle Q R S has points (negative 4, 1), (negative 4, 4), (2, 1).
What are the coordinates of point S’?
Therefore, the coordinates of [tex]S'[/tex] are [tex](-2, -1)[/tex] .
What are coordinates?A series of numbers called coordinates is used to describe a point's location in a two- or three-dimensional space.
Typically, coordinates in a two-dimensional space are expressed as a pair of integers (x, y), where x denotes the point's horizontal location (also known as the abscissa) and y denotes its vertical position (also known as the ordinate).
Coordinates are commonly expressed as a triplet of numbers (x, y, z) in a three-dimensional space, where x, y, and z stand for the locations along the x, y, and z axes, respectively.
In order to describe the location of objects or points in space, coordinates are often employed in a variety of disciplines, including mathematics, physics, engineering, and computer graphics.
To rotate a point 180° about the origin, we need to negate the x-coordinate and the y-coordinate of the point.
So to find the coordinates of S', we can apply this rule to the coordinates of S:
x-coordinate of [tex]S' = -x-[/tex] coordinate of [tex]S = -(2) = -2[/tex]
y-coordinate of [tex]S' = -y-[/tex] coordinate of [tex]S = -(1) = -1[/tex]
Therefore, the coordinates of S' are [tex](-2, -1).[/tex]
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What expression represents a x 5
A: Product
B: Sum
C: Difference
D:Quotient
8 The events committee wants to survey students about a school dance. The committee is
meeting in the gym, where the girls' basketball team is practicing. They survey the
players on the girls' basketball team.
The events committee is surveying only 1.5% of the student population by asking the girls' basketball team about the school dance.
How to solveTo calculate the percentage of the student population being surveyed, you can use the formula:
Percentage = (Number of students surveyed / Total number of students) * 100
In this case, the number of students surveyed is 15 (girls' basketball team), and the total number of students is 1,000.
Percentage = (15 / 1,000) * 100 = 1.5%
So, the events committee is surveying only 1.5% of the student population by asking the girls' basketball team about the school dance.
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The events committee wants to survey students about a school dance. The committee is
meeting in the gym, where the girls' basketball team is practicing. They survey the
players on the girls' basketball team.
If there are 1,000 students in the school, and the girls' basketball team has 15 players, what percentage of the student population is being surveyed by the events committee about the school dance?
This table shows all of the values for the function y = f(x).
y = f(x)
Checkpoint Function transformations (L
-5-28
-3
-2
0
0
7
1 26
Based on those values, complete a table for the function y=f(x)-5.
=F(x)-5
The table for the function y = f(x) - 5 is
x f(x) - 5
5 -33
-3 -5
-2 2
0 21
Creating a table for the function y = f(x) - 5From the question, we have the following parameters that can be used in our computation:
x f(x)
5 -28
-3 0
-2 7
0 26
To create a table for the function y = f(x) - 5, we subtract 5 from the values of f(x)
using the above as a guide, we have the following:
x f(x) - 5
5 -28 - 5
-3 0 - 5
-2 7 - 5
0 26 - 5
Evaluate
x f(x) - 5
5 -33
-3 -5
-2 2
0 21
This represents the table for the function y = f(x) - 5
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Find the area of each of the figures
In the given diagram, the areas of the figures are as follows:
Area of the rectangle is 1/6 cm²
Area of the parallelogram is 12 m²
Calculating the area of a plane shapeFrom the question, we are to calculate the areas of the given plane shapes
From the diagram, we have a rectangle and a parallelogram
The area of a rectangle is given by the formula
Area = l × w
Where l is the length
and w is the width
From the given information,
l = 1/2 cm
w = 1/3 cm
Thus,
Area = 1/2 cm × 1/3 cm
Area = 1/6 cm²
Hence, the area of the rectangle is 1/6 cm²
For the parallelogram
The area of a parallelogram is given by the formula
Area = b × h
Where b is the base length
and h is the perpendicular height
From the given information,
b = 6 m
h = 2 m
Thus,
Area = 6 m × 2 m
Area = 12 m²
Hence, the area of the parallelogram is 12 m²
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Ilana is training for a half marathon. For the first three days of her training plan, she will need to run 4.125 miles each day. For the next two days she will increase her mileage to 5.675 miles per day. How many miles total did Ilana run during her first training week?
Ilana ran a total distance of 23.725 miles during her first week of training for a half marathon.
What does distance mean?Distance refers to the amount of space or physical separation between two points or objects. It is a measure of how far apart two objects or points are from each other.
According to the given informationIlana will run 4.125 miles each day for the first three days, which gives a total distance of:
4.125 miles/day × 3 days = 12.375 miles
For the next two days, she will run 5.675 miles per day, which gives a total distance of:
5.675 miles/day × 2 days = 11.35 miles
To find the total distance that Ilana ran during her first training week, we simply add the distances from the first three days and the next two days:
12.375 miles + 11.35 miles = 23.725 miles
Therefore, Ilana ran a total of 23.725 miles during her first week of training.
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Given v = - 5i + 7j and w = - 1 - j a. Find pro*l_{w}*v b. Decompose v into two vectors v_{1} and v_{2} where v_{1} is parallel to w and v_{2} is orthogonal to w
The value of proᵥᵥv is (-30/37)i + (42/37)j and the decomposed vector v into v₁ and v₂ are 6 + 6j and -5i + j - 6 respectively.
a. To find the projection of w onto v (proᵥᵥw), we can use the formula,
proᵥᵥw = (w · ȳ)ȳ where ȳ is the unit vector in the direction of v.
First, let's find the unit vector in the direction of v,
|v| = √((-5)² + 7²) = √(74)
ȳ = v/|v| = (-5/√(74))i + (7/√(74))j
Next, let's find the dot product of w and ȳ,
w · ȳ = (-1)(-5/√(74)) + (-1)(7/√(74))
w · ȳ = 12/√(74)
Finally, we can find the projection of w onto v,
proᵥᵥw = (w · ȳ)ȳ = (12/√(74))((-5/√(74))i + (7/√(74))j)
(w · ȳ)ȳ = (-60/74)i + (84/74)j
(w · ȳ)ȳ = (-30/37)i + (42/37)j
Therefore, the projection of w onto v is (-30/37)i + (42/37)j.
b. v₁ = ((v · w)/|w|²)w
v₂ = v - v₁ where |w| is the magnitude of w. First, let's find |w|,
|w| = √((-1)² + (-1)²)
|w| = √(2)
Next, let's find v · w,
v · w = (-5)(-1) + (7)(-1)
v · w = -12
Using the formula, we can find v₁,
v₁ = ((v · w)/|w|²)w = (-12/2)(-1 - j)
v₁ = 6 + 6j
Now, finding v₂,
v₂ = v - v₁
v₂ = (-5i + 7j) - (6 + 6j)
v₂ = -5i + (7 - 6)j - 6
v₂ = -5i + j - 6
As a result, the vector v can be divided into two parts: v₁ = 6 + 6j and v₂ = -5i + j - 6, where v₁ is parallel to w and v₂ is orthogonal to w.
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Complete question - Given v = - 5i + 7j and w = - 1 - j.
a. Find proᵥᵥv
b. Decompose v into two vectors v₁ and v₂ where v₁ is parallel to w and v₂ is orthogonal to w.
