Let U and V be nxn orthogonal matrices. Explain why UV is an orthogonal matrix. [That is, explain why UV is invertible and its inverse is (UV)"]
Why is UV invertible?
A. Since U and V are nxn matrices, each is invertible by the definition of invertible matrices. The product of two invertible matrices is also invertible.
B. UV Is Invertible because it is an orthogonal matrix, and all orthogonal matrices are invertible.
C. Since U and V are orthogonal, each is invertible by the definition of orthogonal matrices. The product of two invertible matrices is also invertible.
D. UV is invertible because the product of any two matrices is always invertible

The rabbit travels a distance of 138 m, and the average speed of the hiker for the entire trip is 7.2 km/h.

Given that,

Speed = 12.0 m

Time = 11.5 seconds

Now we need to calculate the distance by using the formula of distance:

d=v*t

d=12.0*11.5

d=138m

The rabbit travels a distance of 138 m.

Distance = 5.10 km

Time = 1 hour

Distance 16.5 km

Time = 2 hours

To calculate the average speed of the hiker for the entire trip using the formula of average speed:

v=D/T

Where, D = Total distance

T = Total time

Put the value into the formula:

v=5.10+16.5/1+2

v=7.2km/h

Therefore, the average speed of the hiker for the entire trip is 7.2 km/h.

Hence, This is the required solution.

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Answer: Yes

Step-by-step explanation:

So, first of all, convert all the given quantities in the same unit.

6m = 600cm

4m = 400cm

2.5m = 250cm

Now, volume of the garage = 600 x 400 x 250

                                              = 60,000,000cm³

And, volume of a brick = 7 x 22 x 10

                                      = 1,540cm³

Number of bricks needed = 60,000,000 ÷ 1,540

                                           ≈ 39,000

So yes, Natalia's estimate is correct.

If sin(4x-8)=cos36°, the value of X is 15.5 since sin x =cos(90-x).

An equation is a formula in mathematics that expresses the equivalence of two expressions by linking them with the equals symbol =. In its most basic form, an equation is a mathematical statement that indicates that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign. A mathematical equation that depicts the connection between two expressions on opposite sides of the sign. It generally consists of one variable and one equal to symbol. A mathematical statement made up of two expressions joined by an equal sign is known as an equation.

Here,

90-(4x-8)=36

4x-8=54

4x=62

x=15.5

The value of X if sin(4x-8)= cos36° is 15.5 as sin x =cos(90-x).

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The solution to the system of equations in this problem is given as follows:

(4.5, -1.5).

The intercept of the system of equations is equivalent to the solution of the system of equations.

The system in this problem is defined as follows:

On the second equation, we isolate y, as follows:

y = 7.5 - 2x.

Replacing on the first equation, the value of x is obtained as follows:

-0.4x + 0.25(7.5 - 2x) = -2.175

0.9x = 4.05

x = 4.05/0.9

x = 4.5.

Hence the solution for y is given as follows:

y = 7.5 - 2(4.5)

y = 7.5 - 9

y = -1.5.

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According to the problem the mass of the prism is 207.875 g.

The measure of an object's matter content is its mass. It is typically measured in kilograms, grams, pounds, or ounces. Mass is related to weight, which is the measure of the force of gravity on an object, but the two terms are not interchangeable.

The mass of the prism can be calculated using the formula mass = density x volume. To calculate the volume of the prism, we can first calculate the area of the faces using the formula area = length x width. The area of the faces of the prism is 7.8 cm x 2.5 cm = 19.5 cm2. The volume of the prism is the area of the faces multiplied by the height, which is 2.5 cm. The volume of the prism is then 19.5 cm2 x 2.5 cm = 48.75 cm3. The mass of the prism is then density x volume, which is 4.3 g/cm3 x 48.75 cm3 = 207.875 g. Therefore, the mass of the prism is 207.875 g.

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Answer:

40d > 600

Step-by-step explanation:

In order to find the number of days the store needs to sell, you need to determine the number of shirts the store sells in a day, which is 40. If you multiply the number of shirts the store sells in a day (40) with the number of days its open (d), you get 40d. 40d must be greater than 600, since the store needs to sell more than 600 shirts.

Answer:

6xy is the answer

Step-by-step explanation:

As per the given triangle RT = AS, RS = AT, the statement < TSA = < STR is proved.

Given two triangles, if the length of one side is proportional to the length of a corresponding side in another triangle, then the two triangles are similar.

This relationship is expressed as "RT = AS," where R and S are corresponding vertices in the two triangles and T is the point where the two sides intersect.

To prove that < TSA = < STR, we need to show that corresponding angles in the two triangles are equal.

This is because corresponding angles in similar triangles are equal, and this is one of the ways to prove that two triangles are similar.

To prove that RT = AS and RS = AT, we can use the function of proportionality. This function states that if two sides are proportional, then the ratio of corresponding lengths will be equal.

Therefore, if RT = AS and RS = AT, then we can conclude that < TSA = < STR and the two triangles are similar.

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The radius of the oil slick as a function of time t is R(t) = √((10/8) × 10000t/π), and after approximately 31.83 hours, the oil slick will have a radius of 1 kilometre.

(a) Let's assume that the oil spread rate is constant. Then, the volume of oil that has leaked into the ocean after t hours is 10000t litres. If this oil forms a circular slick of radius R, then its surface area is given by A = πR². If 8 litres of oil spread out over 10 square meters of the ocean surface, then 1 litre of oil spreads out over 10/8 square meters. Therefore, the surface area of the circular slick is given by

A = (10/8) × 10000t

Equating this to πR², we have:

πR² = (10/8) × 10000t

R² = (10/8) × 10000t / π

R = √((10/8) × 10000t / π)

So, the radius of the oil slick as a function of time t is:

R(t) = √((10/8) × 10000t / π)

(b) To find the time at which the oil slick has a radius of 1 kilometre, we set R(t) = 1 km and solve for t:

1 km = √((10/8) × 10000t / π)

1² km² = (10/8) × 10000t / π

π / (10/8) = 10000t / (1 km)²

π / (10/8) = 10000t / (1000 m)²

π / (10/8) = 10000t / (10⁶ m²)

Solving for t, we find:

t = π / (10/8) × (10⁶ m²) / 10000

t = π / (10/8) × 1000

t = (π × 1000) / (80)

So, after approximately 31.83 hours, the oil slick will have a radius of 1 kilometre.

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Step-by-step explanation:

just look at the graph ! what answer options can be possible, when you look at the first and the last point ?

the position km are 40 and 15.

the first answer option is the right one.

No such vector exists because the directional cosines must be unit vector, with a magnitude of 1.

The directional cosines of a vector are ratios of the components of the vector in the x, y, and z directions. Each directional cosine is a unit vector, meaning it has a magnitude of 1. Therefore, if the directional cosines of a vector are 5 and 6, as in the question, the magnitude of the vector would be the square root of (5^2 + 6^2), which is 7.81. Similarly, if the directional cosines are 2 and 3, the magnitude of the vector will be the square root of (2^2 + 3^2), which is 3.61. Therefore, the magnitude of the vector would be neither 5 nor 6, so no such vector exists.

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The answers to each part of the question is given below.

Given is that Kumiko has $50 gift card for a website.

{ 1 }

We can write the inequality as -

2.5g + 1.5a ≤ 50

{ 2 }

The graph of the inequality is plotted. The shaded region represents the possible set of solutions.

{ 3 }

We can write the combinations as -

Therefore, the answers to each part of the question is given above.

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In order for a graph to be considered a time series graph, it must have a series of data points that are displayed in chronological order over a period of time.

Graph is a data structure which is used to represent a set of connected objects. It is composed of nodes and edges which represent the relationships between the objects. Graphs are used to represent different types of data such as networks, trees, and more. They can be used to model different types of relationships such as relationships between people, objects, and even ideas. Graphs are also used in a variety of applications including route-finding, graph theory, and data mining.

This means that the data must be plotted on the graph according to when it was collected, with the earliest data points on the left and the most recent data points on the right. The data points must be consistently spaced apart; for example, if the data points represent daily measurements, then the data points should be one day apart. Additionally, the graph should show the overall trend of the data over the period of time. Ideally, the graph should also show the variability of the data points over the period of time.

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The solution to the inequality is -5 <= x <= -3.

From the question, we have the following parameters that can be used in our computation:

x^2 + 8x + 15 <= 0

Expand

x^2 + 3x + 5x + 15 <= 0

Factorize the expression

(x + 3)(x + 5) <= 0

Expand

x + 3 <= 0 and x + 5 <= 0

So, we have

x <= -3 and x <= -5

When combined, we have

-5 <= x <= -3.

Hence, the solution is  -5 <= x <= -3.

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The integer solutions of the graph are x = -1, 0, 1, 2, 3 and 4

The complete question is added as an attachment

To determine the solution of f(x) at tthe points where f(x) is more than or equal to 0.

We make use of the positive y-axis

On the positive y axis, the values of x are

x = -1 to 4

This means that the solution is x = -1, 0, 1, 2, 3 and 4

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The sum of vectors a and b where a = <3, 2> and b = <-1, 4> is <2, 6>.

The sum of vectors a and b is found by adding the corresponding components of each vector. In this circumstance, the vectors a and b are given as follows:

a = <3, 2>

b = <-1, 4>

The sum of a and b is simply:

a + b = <3, 2> + <-1, 4> = <3 + (-1), 2 + 4> = <2, 6>

So, the sum of vectors a and b is equal to the vector <2, 6>.

Geometrically, the sum of vectors a and b can be represented as the displacement from the tail of vector a to the head of vector b or vice versa. This displacement is the result of adding the vectors, and it starts at the tail of either vector and ends at the head of the other vector.

The sum of vectors is a fundamental concept in linear algebra and is used to describe changes in position, velocity, or any other physical quantities that can be represented as vectors.

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Answer:The perimeter of a trapezoid is equal to the sum of the lengths of all its sides.

Let's call the height of the trapezoid "h".

Then, the length of the two sides equal (78 - 30 - 14) / 2 = 22 ft.

The area of the trapezoid is equal to (30 + 14) / 2 * h = 22 * h.

We can now find h using the formula:

h = (2 * area) / (30 + 14) = (2 * area) / 44.

We don't know the area, so we cannot find h.

Step-by-step explanation:

The required solutions of the situation of the function are  x = 3/4  or x = -1/4.

A unique connection where each input only produces one output. A common notation for it is "f(x)," where x denotes the input value. as in f(x) = x/2 ("f of x equals x divided by 2") Since there is only one output ("x/2") for each input ("x"), it is a function: • f(2) = 1.

We have,

f(x)=2x-1

g(x)=3/8x

To find the value of x for f(x) = g(x).

f(x) = g(x)

2x-1 = 3/8x

16x² - 8x = 3

16x² - 8x - 3 = 0

16x² - 12x + 4x - 3 = 0

4x(4x - 3) + (4x - 3) = 0

(4x - 3)(4x + 1) = 0

x = 3/4  or x = -1/4

Thus, required solution are  x = 3/4  or x = -1/4.

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The standard deviation is a useful tool in understanding the spread of data and making inferences about the population mean based on sample means.

The standard deviation is a measure of how spread out the data is from the mean.

In this case, the mean recycled material collected per household in other local authorities is 40kg with a standard deviation of 15kg.

In part a, the probability of a randomly selected household recycling less than 30kg in a given week is calculated. In part b, the probability of the sample mean recycled material collected in a given week being less than 30kg for a sample of 100 households is calculated.

In part c, it is explained that finding a sample mean of 30kg for 100 households on a single street does not necessarily mean that the recycling scheme of local authority A is not working.

Finally, in part d, a 95% confidence interval for the mean amount of recycled material per household in local authority A is constructed after an advertising campaign.

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The final answer is 886.1485297964202.

To evaluate the integral, we can use partial integration.

Let u = 17z³

dv = e^z dz

Then du = 51z² dz

v = e^z

Using partial integration, we have:

∫ 17z³ e^z dz = 17z³ e^z - ∫ 51z² e^z dz

= 17z³ e^z - 51 * ∫ z² e^z dz

= 17z³ e^z - 51 * (z² e^z - 2 * ∫ z e^z dz)

= 17z³ e^z - 51z² e^z + 102 * ∫ e^z dz

= (17z³ - 51z² + 102) * e^z + c

So the definite integral from 0 to 2 is equal to

∫_0² 17z³ e^z dz = (17 * 2³ - 51 * 2² + 102) * e^2 - (17 * 0³ - 51 * 0² + 102) * e^0

= (17 * 8 - 51 * 4 + 102) * e²

= (17 * 8 - 51 * 4 + 102) * 7.389056098930649

= (17 * 8 - 51 * 4 + 102) * 7.389056098930649

= 886.1485297964202

Therefore, the answer is 886.1485297964202.

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The average probability of earning $2 for any given roll of the dice is 8/9, or approximately 0.889.

Probability is an area of mathematics that deals with numerical representations of how probable an event is to occur or how likely a statement is to be true. The probability of an occurrence is a number between 0 and 1, where 0 denotes the event's impossibility and 1 represents certainty.

There are a total of 36 possible combinations when rolling two six-sided dice (6 possible outcomes for the first die and 6 possible outcomes for the second die).

The number of combinations that will result in a win (earning $2) are 32: (1,1), (1,2), (1,3), ..., (6,5), and (6,6).

The number of combinations that will result in a loss (losing $4) are 4: (2,2), (4,4), and (10,10).

The probability of earning $2 is the number of favorable outcomes (earning $2) divided by the total number of possible outcomes:

P(earning $2) = 32/36 = 8/9

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Answer:

f(x-1) = [tex]\sqrt{-3x+3}+7[/tex]

Step-by-step explanation:

if f(x) = [tex]\sqrt{x-3}[/tex] + 3

then f(x-1) = [tex]\sqrt{(x-1)-3}+7[/tex]

after expanding its f(x-1) = [tex]\sqrt{-3x+3}+7[/tex]

The square root of a negative number is not defined in the real numbers, r cannot be found. This means that y = x^r is not a solution for any real r.

What is square root ?

A square root of a number x is a number y such that y² = x; in other words, a number y whose square is x.

x(r^2 x^(r-2)) - rx^(r-1) = 0

Expanding the powers of x and collecting terms, we have:

r^2 x^r - rx^r + x^r = 0

Dividing through by x^r, we have:

r^2 - r + 1 = 0

This is a quadratic equation that can be solved using the quadratic formula:

r = (-b ± √(b^2 - 4ac)) / 2a

where a = 1, b = -1, and c = 1.

Plugging in the values, we get:

r = (-(-1) ± √((-1)^2 - 4 * 1 * 1)) / 2 * 1

r = (1 ± √(1^2 - 4 * 1 * 1)) / 2 * 1

r = (1 ± √(1 - 4)) / 2

r = (1 ± √(-3)) / 2

The square root of a negative number is not defined in the real numbers, r cannot be found. This means that y = x^r is not a solution for any real r.

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The number of gallons in  64 fluid ounces would be 0.5.

While fluid ounce, denoted by the sign "fl oz," is an imperial measurement system in the US systems volume unit, gallon is a volume unit in the US Customary measurement systems. Thirty-eight fluid ounces make up a gallon of water. It takes 128 fluid ounces to make up a fluid gallon.

Despite the fact that they are very different, ounces and fluid ounces both serve as units of measurement. As the name suggests, fluid ounces are designed to measure volume (generally of liquid ingredients like water), whereas ounces measure weight, typically of solid materials like all-purpose flour.

In a gallon, there are 128 fluid ounces. However, it can change depending on whether the unit system is in use for liquid or dry gallons. There are 148.94545 ounces in a dry US gallon. There are 153 fluid ounces in an Imperial gallon if you measure using the British Imperial method.

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The selling price of the article that makes 15% of the profit will be Rs 13,800.

The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100. The phrase % translates for one hundred percent. It is symbolized by the character '%'.

The percentage is given as,

Percentage (P) = [Final value - Initial value] / Initial value x 100

A loss of 15% is incurred by selling an article at Rs 10,200. The cost price of the article is given as,

⇒ Rs 10,200 / (1 - 0.15)

⇒ Rs 10,200 / 0.85

⇒ Rs 12,000

The selling price of the article that makes 15% of profit is given as,

⇒ Rs 12,000 x (1 + 0.15)

⇒ Rs 12,000 x 1.15

⇒ Rs 13,800

The selling price of the article that makes 15% of the profit will be Rs 13,800.

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In one minute 0.6 sq meters of land will be cut with machete.

The unitary approach is a strategy for problem-solving that involves first determining the value of a single unit, then multiplying that value to determine the required value.

Direct Variation

When there is a direct variation, an increase or decrease in one quantity will result in an equivalent rise or fall in the other. For instance, a rise in the quantity of commodities will result in higher costs.

Additionally, a guy working alone will accomplish less than a man working with a group of men. Therefore, if we add more men, there will be more labor.

Indirect Variation

It is the direct variation's opposite. If we enhance one quantity, another item's value decreases. For instance, if we speed up, it will take us less time to travel a distance. Therefore, the distance traveled will be shorter as the speed increases.

In 90 minutes 54 meters of land is cut with machete.

In one minute 54/90 = 0.6 sq meters  of land will be cut with machete.

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The data that is considered univariate is, Shoe size, salary, height, sports played in high school, years of experience, favorite color.

Data that just has one variable is said to be univariate data. Since there is only one variable that varies, univariate data analysis is the most straightforward type of analysis. The analysis's primary goal is to explain the data and identify any patterns in it; it does not deal with causes or correlations. Height is an example of a univariate data.

We know that univariate data only has one variable thus the data that can be represented with a single variable is:

Shoe size, salary, height, sports played in high school, years of experience, favorite color.

Hence, the data that is considered univariate is, Shoe size, salary, height, sports played in high school, years of experience, favorite color.

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Jose's height is 1 standard deviation above the mean height for 7-year-olds and Jamal's height is 0.67 standard deviations above the mean height for 12-year-olds.

Standard deviation is a mathematical term used to measure how far away a data point is from the mean of a set of data.

To determine the number of standard deviations, we use the following formula:

(Jose's height - mean height for 7-year-olds) / standard deviation

(130 - 120) / 10 = 1

So, Jose's height is 1 standard deviation above the mean height for 7-year-olds.

Now, let's look at Jamal. Let's assume that the mean height for 12-year-olds is 150 cm and the standard deviation is 15 cm. Jamal's height is 160 cm.

Using the same formula,

(Jamal's height - mean height for 12-year-olds) / standard deviation

(160 - 150) / 15 = 0.67

So, Jamal's height is 0.67 standard deviations above the mean height for 12-year-olds.

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The letter of protest from the Standing Rock Sioux Tribe to the federal government is likely to concern the construction of the Dakota Access Pipeline, which threatened the Tribe's water supply and cultural sites.

The Standing Rock Sioux Tribe is a Native American tribe located primarily in North and South Dakota. The tribe is known for its opposition to the Dakota Access Pipeline, a controversial oil pipeline that would cross the tribe's reservation and potentially impact its water supply and cultural sites. In 2016, the Standing Rock Sioux Tribe and its supporters held a series of protests and demonstrations against the pipeline, attracting national attention and leading to its temporary halt.

The Dakota Access Pipeline (DAPL) is a

n oil pipeline that runs from the Bakken oil fields in North Dakota to Illinois. The tribe and its supporters held protests, leading to confrontations with law enforcement. In 2016, the US Army Corps of Engineers declined to grant the pipeline an easement, but this decision was later reversed by the Trump administration.

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