question 6: orthogonal matrices let r be an n×northogonal matrix. 1. what values can the determinant of r take? 2. what values can the real eigenvalues take?

Answer:

a function is said to be positive if all the values of y are above the x-axis and it is said to be negative if all the values of y are below the x-axis.

Answer:

  C. Exponential growth

Step-by-step explanation:

You want to know how to describe the relation that represents a 10% increase per year.

The problem statement tells you the number of guests "increased by approximately 10% each year". The nature of the function depends on what "10%" refers to.

If the amount of increase is constant year after year (10% of the original number of guests), then the relationship is linear.

If the amount of increase is 10% of the previous year's number, so the amount increases each year, then the relationship is exponential.

When the number increases year on year, the number is said to be growing. If the relationship is exponential, it is described as exponential growth.

If the number were decreasing by 10% per year, it would be an exponential decay relationship.

__

Additional comment

Whenever you're dealing with percentages, you always need to understand what is being used as the base amount. (What does 100% represent?)

Usually, a percentage increase or decrease is referring to the current value. Not always.

Sometimes a change from 10% to 6% is referred to as a 4% decrease, and sometimes it is referred to as a 40% decrease. It depends on who is selling what to whom.

The flight-path angle is equal to 45' when v =90' on all parabolic trajectories is proved using the trajectories.

Using trajectories equation :

r = p/(1+cosv)     at v= 90 degrees

r{90} = p/ (1+cos90°)

        = p       since, cos 90 = 0

also, v= √2mue/r

        v{90} = √2mue/p

where r {90}  and v{90} are radius and velocity at v = 90 degrees.

the angular momentum is given as , h = r.v cosФ

and at periapsis on a parabolic trajectory , rp = p/2

  v{90} = √2mue/p

applying the concentration of angular momentum,

rp.vp = r90.v90.cosФ

p/2.√2mue/p = p.√2mue/p.cosФ

cos Ф = 1/√2

therefore,Ф = 45

hence the flight-path angle is equal to 45' when v =90' on all parabolic trajectories is proved.

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The equation of direct variation given that y varies directly as x is 125= 25k.

The relationship between two variables in which one is a constant multiple of the other is referred to as direct variation. For instance, two variables are said to be in proportion when one affects the other. If b and an are directly proportional, then b = ka is the equation.

The equation of direct variation is given as:

y = kx

where, k is the proportionality constant.

Substituting the value of y = 125 and x = 25:

125 = k (25)

Hence, the equation of direct variation given that y varies directly as x is 125= 25k.

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The change in speed equals 2m/s and Magnitude of acceleration = 2.

What is meant by zero acceleration is the rate of change in speed. Thus, the rate of change in speed is zero while the car is moving at a constant pace. As a result, there is no acceleration. Magnitude is the length of any vector that points in the direction of the vector and is denoted by its unit. As a result, the acceleration's magnitude is equal to the acceleration vector's length and is directed in the same direction. For any vector quantity, this is true.

the sense, or the direction the vector is moving (left/right, up/down); the orientation, or the angle(s) governing the vector's alignment (transversal, longitudinal); and

The "strength" of the function is represented by the magnitude, or the value derived from the scalar quantities.

Change in speed = 5 m/s - 3m/s = 2m/s

As acceleration = rate of change in speed

Magnitude of acceleration = 2

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The area, in square units, between the two functions over the interval [−3,3]. is 36 square units

integral is defined to be exactly the limit and summation that we looked at in the last section to find the net area between a function and the

x-axis.

Also note that the notation for the definite integral is very similar to the notation for an indefinite integral.

the fact that a and b were given as an interval the lower limit does not necessarily need to be smaller than the upper limit. Collectively we’ll often call a and b the interval of integration.

[tex]\int\limits^3_3 {f(x)} \, dx -\int\limits^3_3 {g(x)} \, dx\\=\int\limits^3_3 {f(x) - g(x) } \, dx \\= \int\limits^3_3 {(x^{2} -9x-18) - (-9x-9)} \, dx \\=\int\limits^3_3 {x^{2} -9x-18 + 9x+9} \, dx \\\\=\int\limits^3_3 {x^{2} -9} \, dx[/tex]

=[tex]\left \{ {{y=3} \atop {x=-3}} \right. (\frac{x^{3} }{3} -9x)\\= \frac{3^{3} }{3} -9*3 -( \frac{3^{-3} }{3} -9*-3)\\=9-27 +9-27\\= -36[/tex]

The area, in square units, between the two functions over the interval [−3,3]. is 36 square units

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Answer:

Look at the y axis

Step-by-step explanation:

Each division is 2 units, so look and mark 8,2 carefully
If it confuses you, take 16, 4 as well because I dunno how flexible are your drawing tools
Hope this helps :)

Edit: Sorry but I have just realized what had happened, I took the time variable for the y axis. (Sorry, I am used to using the time variable on the y axis

Answer: It would cost $3.35 ?more than the leave cloth.

Step-by-step explanation:so for 2.5 yards the leaves of fabric would cost $16.6 because 6.64×2=$13.28+6.64/2=16.6

Then for the pumpkin which costs $19.95 because you would multiply 7.98 by 2 then add 7.98 divided by 2. So it would cost 19.95-16.6=$3.35

Answer:  15 (cupcakes)

Step-by-step explanation:

So, as you understand, 3 cupcakes represent 20% of the order.

20% = 20/100 = 1/5

3=1/5

3x5=15

1/5x5=1

Therefore, your one whole order is 15 cupcakes

a) The limiting value of the temperature of the coffee is 25°C.

b) The limiting value of the rate of cooling is equals to 0.

c) The value of constant k in the

differential equation is -0.05.

d) The temperature of the coffee at time t = 10 is 66.33°C.

Newton's Law of Cooling: The rate at which an object's temperature changes is proportional to the difference in temperature between the object and its surroundings. For an object that has a temperature very close to that of its surroundings, the rate of heating or cooling becomes very small. We have the temperature of the coffee, T(t), satisfies the differential equation,

dT/dt= k(T − T (room)),

where T(room) = 25 is the room temperature, and k is some constant.

A) Limiting value of the temperature

= 25°C

B)limiting value of dT/dt = k[(limit temp) - (room temp)] = k(25 - 25)= 0

C) dT/dt = k(T - 25)

=> -2 = k (65 - 25)

=> -2 = 40k

=> k = -0.05

D) T' = -0.05 x (95-25) = - 3.5

h = 2, ∆T = -3.5x 2 = -7

T = 95- 7 = 88

Then T' = -0.05 x (88-25) = - 3.15

=> T = 88 - 2 x 3.15 = 81.7

=> 81.7 after 4 minutes

Then T' = -0.05x (81.7-25) = -2.835

T = 81.7 - 2x 2.835 = 76.03 after 6 mins

Then T' = -0.05 x (76.03-25) = -2.5515

T = 76.03 - 2 x 2.5515 = 70.92700 after 8 mins

Then T' = -0.05 x (70.927-25) = -2.29635

T = 70.927 - 2 x 2.29635

= 66.3343 after 10 minutes

so temperature = 66.33 after 10 mins by Euler's method.

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Complete question:

Suppose you have just poured a cup of freshly brewed coffee with a temperature of 90∘C in a room where the temperature is 25∘C. Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. Therefore, the temperature of the coffee, T(t), satisfies the differential equation

dT/dt= k(T−Troom), where T(room) = 25 is the room temperature, and k is some constant.

Suppose it is known that the coffee cools at a rate of 2∘C per minute when its temperature is 65∘C.

A. What is the limiting value of the temperature of the coffee?

B.What is the limiting value of the rate of cooling?

C. Find the constant k in the differential equation.

D. Use Euler's method with step size h = 2 minutes to estimate the temperature of the coffee after 10 minutes_ Answer (in Celsius): T(10)

Answer:

See proof below

Step-by-step explanation:

We will denote the sum of n elements of the sequence as S(n)

What is proof by induction?

We first assume that for n=1 , S(1) is true
This step is called the basis

Then for n= k ≥ 1, prove S(n) is true and then prove S(k+1) also true. This is called the induction step

For this specific problem, we assume S(1) is true

[tex]\mbox{\large S(1) = 2}[/tex]

[tex]\mbox{\large S(k) = 2 + 4 + 6 + ..... + 2k = k(k+1) }[/tex]

We are using 2k because the values are twice the index values 1, 2, 3...
2.1, 2.1, 2.3 ...2k

[tex]\mbox{\large We now show }[/tex] [tex]\mbox{\large S(k + 1) }[/tex]  [tex]\mbox{\large is true }[/tex]

[tex]\mbox {\large S(k + 1) = S(k) + 2(k + 1)}[/tex]

[tex]\mbox{\large 2+4+6+...+2k+2(k+1)}\\\\\mbox{\large=k(k+1)+2(k+1)}\\\\\mbox{\large=(k+1)(k+2)}\\\\= \mbox{\large (k+1)(k+1+1)}}\\\\[/tex]

[tex]\mbox{\large Substituting n for k + 1 we get }[/tex]
[tex]\mbox{\large S(n) = n(n+1)}[/tex]

I hope the proof is comprehensible. If not please ask for clarifications

Answer:

  the proof is below

Step-by-step explanation:

You want a proof by induction of the formula for the sum of consecutive even numbers.

The given formula is ...

  Sn = n(n+1)

For n=1, the one-term sequence is the first term: 2. Its sum is 2.

  2 = (n)(n+1) = (1)(1+1) = 2 . . . . . true

For this step, we assume the formula is correct. Then ...

  Sk = k(k+1)

The sequence sum for k+1 terms is ...

  2 + 4 + 6 + ... + 2k = k(k+1) . . . . . assumption from step 2

  (2 + 4 + 6 + ... + 2k) +2(k+1) . . . . . one more term added; sum of k+1 terms

  k(k+1) +2(k+1) . . . . . . . . substitute assumed value for (2 + 4 + ...)

  (k +2)(k +1) . . . . . . . . . . factor out k+1

And the formula for k+1 terms is ...

  = (k+1)((k+1) +1) . . . . . substitute (k+1) for n

  = (k+1)(k+2) . . . . . . . perform the addition

Note that the sum of terms is (k+2)(k+1), and is identical to the formula for the same sum: (k+1)(k+2). (Remember, multiplication is commutative.)

  (k+2)(k+1) = (k+1)(k+2) . . . . . QED

Answer: I think its X intercept is (-0.75,0) and then y is (0,0.429)

Step-by-step explanation:

Tell me if I'm wrong

Answer:

Senior ticket = $10

Child's ticket = $8

Step-by-step explanation:

s = senior ticket

c = child's ticket

We need to write 2 equations, and we have 2 unknowns, s and c.

12s + 12c = 216

12() + 12c = 216

(70 - 5c) + 12c = 216

4 (70 - 5c) + 12c = 216

280 - 20c + 12c = 216

280 - 8c = 216  Add 8c to each side

280 - 8c + 8c = 216 + 8c

280 = 216 + 8c   Subtract 216 from each side.

280 - 216 = 216 - 216 + 8c

280 - 216 = 8c

64 = 8c   Divide each side by 8

64/8 = 8c/8

64/8 = c

8 = c

Now plug c into the first equation and solve for s.

3s + 5c = 70

3s + 5(8) = 70

3s + 40 = 70    Subtract 40 from each side.

3s + 40 - 40 = 70 - 40

3s = 70 - 40

3s = 30  Divide each side by 3

3s/3 = 30/3

s = 30/3

s = 10

So a senior ticket costs $10 and a child's ticket costs $8.

The equation to model the depth (d) of the snow n days after the storm

d = 15-3n

What is the percentage?

It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol %.

An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used. In the equation 3x + 5 = 14, for instance, the two expressions 3x + 5 and 14 are separated.

20% of 15 = (20/100)*15 = 3

The equation to model the depth (d) of the snow n days after the storm

d = 15-3n

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This statement is called an Income Statement.

December 3 I Income Statement

Revenue

Sales:          $1,027,800

Rental of Equipment to Customers:   $47,000

Total Revenue:        $1,074,800

Cost of Goods Sold:       $537,400

Gross Profit:        $537,400

Expenses

Salaries and Wages:       $15,000

Rent of Facilities:       $70,600

Depreciation of Equipment:     $3,000

Electricity:        $2,900

Supplies:         $1,850

Other Expenses:        $700

Total Expenses:        $93,150

Net Profit:         $444,250

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It would take 17 years for the value of the account to reach $2,550. The solution has been obtained by using the concept of compound interest.

What is compound interest?

The principal as well as the interest that has accrued over the previous period are both used to compute interest. Compound interest factors the principal into the calculation for determining the interest for the subsequent month, in contrast to simple interest, which does not. In mathematics, the symbol for compound interest is typically the letter C.I.

We are given the principal amount as $790 paying an interest rate of 6.9% compounded continuously.

The final balance is $2,550.

We know that formula for continuous compounding interest is

[tex]P(t) = P_{0} e^{rt}[/tex]

So, using this, we get

⇒[tex]2550 = 790 e^{0.069*t}[/tex]

Solving this, we get

t = 17

Hence, it would take 17 years for the value of the account to reach $2,550.

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The probability of obtaining 5 impulses in one minute is 0.067.

The calculation of the probability of obtaining 5 impulses in one minute is based on the Poisson distribution. The Poisson distribution is used to calculate the probability of a given number of events occurring in a fixed interval of time or space, when the average rate of occurrences is known. In this case, the given rate of occurrences is 10 impulses per minute.

The probability of obtaining 5 impulses in one minute can be calculated by using the following formula:

P(x) = (e-λ) (λ^x) / x!

Where,

P(x) is the probability of x events occurring in a given interval

e is the base of the natural logarithm, approximately 2.718

λ is the average rate of occurrences, which is 10 in this case

x is the number of events, which is 5 in this case

Therefore, the probability of obtaining 5 impulses in one minute can be calculated as:

P(5) = (2.718-10) (10^5) / 5!

P(5) = 0.067

Therefore, the probability of obtaining 5 impulses in one minute is 0.067.

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The equation can be used to determine the ellipse's area, circumference, and other parameters because it is symmetrical.

For this to be a valid probability model, p must be a positive integer between 0 and 1, and p + p2 + p must equal 1. This random experiment must have a probability rea, circumference, and other parameters because it is symmetrical. of each of the three outcomes adding up to 1 in order to be a reliable probability model. This implies that the probabilities p, p2, and p associated with outcomes a, b, and c must all add up to 1. In other words, p + p2 + p = 1 would be the equal. Since p must be a real value between 0 and 1 and satisfy the equation p + p2 + p = 1, this random experiment must be a legitimate probability model.

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To find the distance traveled by a car given the average speed and time traveled, we can use the formula:

distance = speed x time

In this case, the average speed of the car is 48 miles per hour (mph) and the time traveled is 5 hours and 45 minutes.

Before we use this formula, we need to convert the time of 5 hours and 45 minutes into hours. Since there are 60 minutes in 1 hour, we can convert 45 minutes into hours by dividing it by 60:

45 minutes / 60 minutes/hour = 0.75 hours

Now, we can add the time in hours and minutes to get the total time traveled in hours:

5 hours + 0.75 hours = 5.75 hours

Now we can use the formula to find the distance traveled:

distance = speed x time

distance = 48 mph x 5.75 hours

distance = 277 miles

So, the car travels 277 miles in 5 hours and 45 minutes.

Answer: 275 Miles

Step-by-step explanation:

1. Convert the hours into minutes:
We'll take the 5 hours and 45 minutes and divide the minutes by 60, giving you 5.75 hours.

2. Multiply the hours by the rate
The 48 mph is given to us, so we should multiply that by the converted hours we got in the first step. 5.75*48 = 275

The 272 demonstrates the number of miles that are driven in that total time span.

The missing side is 29, because in a triangle, the sum of the angles is 180°.  

A triangle is a three-sided polygon, a two-dimensional shape with three straight sides and three angles. It is one of the most basic shapes in geometry, and can be used to construct more complex shapes and figures such as rectangles and circles. Triangles come in many different varieties, including equilateral, isosceles, and scalene. The angles in a triangle add up to 180°, and the sides of a triangle are always connected.

In this case, 39° + X° = 180°, and X° = 180° - 39° = 141°. Since the angle is given in degrees, the side length can be found using the law of cosines, which states c^2 = a^2 + b^2 - 2ab cos C. In this case, c = 29, a = b = 11, and C = 141°. Plugging this into the equation, we get 29^2 = 11^2 + 11^2 - 2(11)(11)cos(141°). Simplifying and solving for c, we get c = 29. Therefore, the missing side is 29.

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The equation of a line y=2x+8.

The equation of a straight line can be represented in the slope-intercept form, y = mx + c

Where c = intercept

Slope, m =change in the value of y on the vertical axis/change in the value of x on the horizontal axis

Looking at the given line,

y = 2x + 7

Compared with the slope-intercept equation,

Slope, m = 2

If a line is parallel to another line, it means that both lines have equal or the same slope. This means that the slope of the line passing through the point (-2, 4) is 2

Substituting m= 2, y = 4 and x = -2 into the equation, y = mx + c , it becomes

4 = 2 × - 2 + c

4 = - 4 + c

c = 4 + 4 = 8

The equation becomes y=2x+8.

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The length of a rectangle= (10/3) cm or 3.334 cm

The area can be defined as the amount of space covered by a flat surface of a particular shape

The formula for the area of rectangle= length × breadth

It is given in the question that the area of a rectangle= 20 sq. cm

Also, Breadth is given as 6 cm.

We have to find the length.

Putting the given values in the above formula:

20 sq. cm = 6cm × length

length = 20 sq. cm/ 6cm

length =10/3 cm = 3.334 cm

Final length = (10/3) cm or 3.334 cm

Therefore, the length of a rectangle is 3.334cm.

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The translation of 5 units to the left and 3 units down,The new coordinates are therefore (5, 1).

The coordinates of the image of point B of quadrilateral ABCD after the translation of 5 units to the left and 3 units down.

By multiplying the value of the horizontal translation by the original x-coordinate, get the new x-coordinate of the translated point (adding a negative number if translating to the left).

Let (A,B)=(3,5) serve as the new origin and (x,y)=(3,4) serve as the provided point.

new coordinates are therefore (X,Y).

x = X+A and y = Y+B

such as 3=X2 and 4=Y+5.

This results in X=5 and Y=1

The new coordinates are therefore (5, 1).

The translation of 5 units to the left and 3 units down,The new coordinates are therefore (5, 1).

Determine the coordinates of the image of point B of quadrilateral ABCD after the translation of 5 units to the left and 3 units down. (3,4) will become

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The value of a is 3

The equation of the quadratic function is

f(x) =  3(x+2) (x+1) in factored form and

f(x) = 3x² + 9x+2 in standard form.

What is a parabola?

A parabola is an approximately U-shaped, mirror-symmetrical plane curve in mathematics. It corresponds to a number of seemingly unrelated mathematical descriptions, all of which can be shown to define the same curves. A parabola can be described using a point and a line.

The x-intercepts of the parabola are (-2,0) and (-1,0).

So the factors are (x- (-2)) and (x- (-1))

That is, (x+2) and (x+1)

Now the equation of the parabola,

y = a(x+2) (x+1)

To find a substitute the values -3 and 6 for x and y as these points are on the parabola.

6 = a(-3+2)(-3+1)

6 = a ( -1)(-2)

a = 3

So the equation of the parabola is y = 3(x+2) (x+1)

Therefore the complete statements regarding the parabola is:

The value of a is 3

The equation of the quadratic function is

f(x) =  3(x+2) (x+1) in factored form and

f(x) = 3x² + 9x+2 in standard form.

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Answer: The total volume of concrete that Jimmy will need to create the 4 spheres is 38.25 cubic feet, rounded to the nearest whole number.

Step-by-step explanation:

To find the total volume of concrete needed for the 4 spheres, we need to find the volume of each sphere individually and then add them up. The formula for the volume of a sphere is (4/3)πr^3, where r is the radius of the sphere.

First, we need to find the radius of the largest sphere, which is 4 feet in diameter. The radius is half the diameter, so it is 2 feet.

The radius of the next sphere is half of the radius of the first sphere, which is 1 foot.

The radius of the next sphere is half of the radius of the second sphere, which is 0.5 feet.

The radius of the last sphere is half of the radius of the third sphere, which is 0.25 feet.

We can now use the formula to find the volume of each sphere and then add them up.

The volume of the first sphere = (4/3)π(2^3) = (4/3)π(8) = 33.49 cubic feet

The volume of the second sphere = (4/3)π(1^3) = (4/3)π(1) = 4.19 cubic feet

The volume of the third sphere = (4/3)π(0.5^3) = (4/3)π(0.125) = 0.52 cubic feet

The volume of the last sphere = (4/3)π(0.25^3) = (4/3)π(0.015625) = 0.05 cubic feet

Total volume of all spheres = 33.49 + 4.19 + 0.52 + 0.05 = 38.25 cubic feet

The total volume of concrete that Jimmy will need to create the 4 spheres is 38.25 cubic feet, rounded to the nearest whole number.

By dividing the output power (work completed) by the input power ratio, the motor's efficiency may be determined. In this instance, the input power is 300 W, and the output power is 2400 joules (400 N * 6.0 m). As a result, the motor has an efficiency of 2400 J/300 W, or 8.0.

The input power is 300 W.

Time (t) = 20 s

Height is 6.0 metres.

(F) = 400 N Load

Output power (Pout) is calculated as follows: F x h = 400 N x 6.0 m = 2400 J

Efficiency (e) is calculated as Pout / P = 2400 J / 300 W = 8.0.

A motor's efficiency can be determined by dividing its output power by its input power. Assuming that the input power is 300 W and the output power equals the amount of work completed, the load (F) and height can be multiplied to determine the output power (h). The output power in this example is 2400 joules (400 N * 6.0 m) since the load is 400 N and the height is 6.0 m. The motor's efficiency is 2400 J/300 W, or 8.0, as a result. Knowing the input power, output power, and length of time it took the motor to lift the load will help you determine the efficiency. In this illustration, the time is 20 s, and the input power is 300 W.

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