a clock has an hour, minute, and second hand, all of length $1$. let $t$ be the triangle formed by the ends of these hands. a time of day is chosen uniformly at random. what is the expected value of the area of $t$?

Answer:

the answer is 2

Step-by-step explanation:

this answer will be 200⁰0000000000⁰00000⁸⁰643367897⁶43677443⁵=5.0

Answer: Yes, the Mean Value Theorem can be applied to f(x) = 9x^3 on the closed interval [1, 2].

To find all values of c in the open interval (1, 2) such that f'(c) = (f(b) - f(a))/(b - a), we first find the derivative of f(x):

f'(x) = 27x^2

Then, we can use the Mean Value Theorem to find a value c in the open interval (1, 2) such that:

f'(c) = (f(2) - f(1))/(2 - 1)

27c^2 = 9(2^3 - 1^3)

27c^2 = 45

c^2 = 5/3

c = +/- sqrt(5/3)

Therefore, the values of c in the open interval (1, 2) such that f'(c) = (f(b) - f(a))/(b - a) are:

c = sqrt(5/3), -sqrt(5/3)

Note that these values are not in the closed interval [1, 2], as they are not between 1 and 2, but they are in the open interval (1, 2).

Step-by-step explanation:

Answer:

Jack's company needs to sell 3540 cards this year in order to make their target profit.

Step-by-step explanation:

Given the average profit per card last year was £1.50, and the average profit per card is 22% lower this year, this year's average profit per card will be:

[tex]\begin{aligned}\implies \sf Average\;profit\;per\;card&= \£1.50 - (22\% \;\text{of}\; \£1.50)\\&=\£1.50-0.22 \times \£1.50\\&=\£1.50-\£0.33\\&=\£1.17\end{aligned}[/tex]

Given the total profit Jack is aiming for this year is 18% more than last year's profit of £3510, this year's target profit is:

[tex]\begin{aligned}\implies \sf Target\;profit&=\£3510 + (18\%\;\text{of}\;\£3510)\\&=\£3510 + 0.18 \times \£3510\\&=\£3510 + \£631.80\\&=\£4141.80\end{aligned}[/tex]

To calculate how many cards Jack's company needs to sell to make this target profit, divide the total target profit by the average profit per card:

[tex]\begin{aligned}\implies \sf Number\;of\;cards&=\dfrac{4141.80}{1.17}\\\\&=3540\end{aligned}[/tex]

Therefore, Jack's company needs to sell 3540 cards this year in order to make their target profit.

The balance after 3 years with daily compounding at an APR of 3% is $23,284.94.

To calculate the balance after 3 years with daily compounding, we need to use the formula for compound interest,

A = P(1 + r/n)^(nt)

Where,

A = the balance after 3 years

P = the initial investment, which is $21,000 in this case

r = the annual percentage rate, which is 3%

n = the number of times the interest is compounded per year. In this case, since it's daily compounding, n = 365 (the number of days in a year).

t = the number of years, which is 3 years.

Substituting the given values in the formula, we get

A = 21000(1 + 0.03/365)^(365×3)

A = $23,284.94

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The statement that gives the most reasonable description of the distribution of quiz scores is "A typical score is within 1 point of the mean." The correct answer is Option D.

The standard deviation (SD) is a measure of the variability of data in a population. Standard deviation is a measure of how much each value differs from the mean (average) value of the data set.

The difference between the highest and lowest values in a dataset is known as the range. It's a quick way to see the data's spread. If the range is big, it implies that the data is more diverse, while if it's small, it implies that the data is more consistent.

The first quartile (Q1) is the value that splits the lowest 25% of a data set from the rest of the data set. If we order the dataset from smallest to largest, the first quartile is the value at the 25th percentile.

The third quartile (Q3) is the value that splits the highest 25% of a data set from the rest of the data set. If we order the dataset from smallest to largest, the third quartile is the value at the 75th percentile.

The sum of all values in a dataset divided by the total number of values in the dataset is known as the mean. The mean, often known as the arithmetic mean, is one of the most basic measures of central tendency in statistics.

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Check the picture below.

so let's get the chord using the pythagorean theorem hmmm using sine

[tex]\sin(35^o )=\cfrac{\stackrel{opposite}{x}}{\underset{hypotenuse}{5}}\implies 5\sin(35^o )=x\implies 2.87\approx x~\hfill \underset{ PQ }{\stackrel{ 2.87+2.87 }{\approx \text{\LARGE 5.74}}}[/tex]

now let's get the arc

[tex]\textit{arc's length}\\\\ s = \cfrac{\theta \pi r}{180} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ \theta =70\\ r=5 \end{cases}\implies s=\cfrac{(70)\pi (5)}{180}\implies s\approx \text{\LARGE 6.11}[/tex]

and the perimeters, keeping in mind that for the sector is just the arc plus the radii, and for the segment is simply the arc plus the chord.

[tex]\stackrel{ \textit{sector's perimeter} }{5+5+6.11 ~~ \approx ~~} \text{\LARGE 16.11}\hspace{5em}\stackrel{ \textit{segment's perimeter} }{5.74+6.11 ~~ \approx ~~} \text{\LARGE 11.85}[/tex]

The area is 49 square yards.

By using differentials to estimate the maximum error when measuring the volume of the sphere if the possible error in measuring the radius is 0.5. It will be 32,572.03 in³. Which is option (d).

The possible error in measuring the radius of the sphere is 0.5 in

The formula for the volume of a sphere is given by V(r) = 4/3πr³

The volume of the sphere when r=72 in is given by V(72) = 4/3π(72)³

When r= 72 + 0.5 in= 72.5 in, the volume of the sphere can be calculated using the formula:

V(72.5) = 4/3π(72.5)³

The difference between these two volumes, V(72) and V(72.5), gives us the maximum error while measuring the volume of a sphere. It can be calculated as follows:

V(72.5) - V(72) = 4/3π(72.5)³ - 4/3π(72)³= 4/3π [ (72.5)³ - (72)³ ]= 4/3π [ (72 + 0.5)³ - 72³ ]= (4/3)π [ 3(72²)(0.5) + 3(72)(0.5²) + 0.5³ ]≈ (4/3)π [ 777.5 ]= 3.28 × 10⁴ in³

Therefore, the maximum error while measuring the volume of a sphere with a radius of 72 in, where the possible error in measuring the radius is 0.5 in, is approximately 3.28 × 10⁴ in³ or 32,572.03 in³. Therefore coorect option is (D).

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To solve the question asked, you can say:  Therefore, the final answers expressions are: E(X) = 5.8; E(X²) = 59.8; V(X) = 21.16 and E(3X + 2) = 20.4

In mathematics, an expression is a set of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation that represent quantities or values. Expressions can be as simple as "3 + 4" or as complex as they can contain functions like "sin(x)" or "log(y)" . Expressions can be evaluated by substituting values ​​for variables and performing mathematical operations in the order specified. For example, if x = 2, the expression "3x + 5" is 3(2) + 5 = 11. In mathematics, formulas are often used to describe real-world situations, create equations, and simplify complex math problems. 

To calculate these values, we first need to compute the mean (expected value) and variance of X, which are given by:

E(X) = ∑[x * p(x)]

= 1 * 0.05 + 2 * 0.10 + 4 * 0.35 + 8 * 0.40 + 16 * 0.10

= 5.8

E(X²) = ∑[x² * p(x)]

= 1² * 0.05 + 2² * 0.10 + 4² * 0.35 + 8² * 0.40 + 16² * 0.10

= 59.8

V(X) = E(X²) - [E(X)]²

= 59.8 - 5.8²

= 21.16

E(3X + 2) = 3E(X) + 2

= 3(5.8) + 2

= 20.4

E(3X² + 2) = 3E(X²) + 2

= 3(59.8) + 2

= 179.4

V(3X + 2) = V(3X)

= 9V(X)

= 9(21.16)

= 190.44

E(X + 1) = E(X) + 1

= 5.8 + 1

= 6.8

V(X + 1) = V(X)

= 21.16

Therefore, the final answers are:

E(X) = 5.8

E(X²) = 59.8

V(X) = 21.16

E(3X + 2) = 20.4

E(3X² + 2) = 179.4

V(3X + 2) = 190.44

E(X + 1) = 6.8

V(X + 1) = 21.16

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Answer:

x = -1

Step-by-step explanation:

The axis of symmetry is a line that divides the two sides of a parabola through the vertex.

a. Calculate the expected arrival time:

Given: Time range for arrival of elevator during the next 4.7 minutes is equally likely. The expected value of a discrete random variable is calculated by multiplying each possible value by its probability and adding up the products. So, we can calculate the expected value of the elevator arrival time by integrating the value of the probability density function (which is a straight line in this case) over the given interval. The area under the curve of the probability density function over the entire interval of possible values is 1. The expected arrival time (E) of the elevator is given by: E = (1/4.7) ∫(0 to 4.7) tdt= (1/4.7) [t²/2] [from 0 to 4.7]= 2.3596 minutes or 2.36 minutes (rounded to 2 decimal places)Therefore, the expected arrival time is 2.36 minutes.

b. Probability that an elevator arrives in less than 1.8 minutes:

To calculate the probability of an event happening, we need to find the area under the probability density function (pdf) over the given interval (in this case, less than 1.8 minutes). The pdf is a straight line with a slope of 1/4.7, so the equation of the line is: f(t) = (1/4.7) t. The probability of the elevator arriving in less than 1.8 minutes is: P(T < 1.8) = ∫(0 to 1.8) f(t) dt= ∫(0 to 1.8) (1/4.7) t dt= (1/4.7) [t²/2] [from 0 to 1.8]= 0.56765 (rounded to 4 decimal places)Therefore, the probability that an elevator arrives in less than 1.8 minutes is 0.568 (rounded to 3 decimal places).

c. Probability that the wait for an elevator is more than 1.8 minutes: The probability that the wait for an elevator is more than 1.8 minutes is the complement of the probability that it arrives in less than 1.8 minutes. P(T > 1.8) = 1 - P(T < 1.8) = 1 - 0.56765= 0.43235 (rounded to 3 decimal places)Therefore, the probability that the wait for an elevator is more than 1.8 minutes is 0.432 (rounded to 3 decimal places).

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In response to the stated question, we may state that To accommodate scatter plot the provided data on the scatter graph, the grid must be at least 65 squares wide and 62 squares height.

"Scatter plots are graphs that show the association of two variables in a data collection. It is a two-dimensional plane or a Cartesian system that represents data points. The X-axis represents the independent variable or characteristic, while the Y-axis represents the dependent variable. These plots are sometimes referred to as scatter graphs or scatter diagrams."

To plot the supplied data on a scatter graph, we must ensure that the distance and height values are both within the grid.

The given distances are 37, 6, 71, and 28. As a result, the distance axis's minimum and maximum values are 6 and 71, respectively.

Height values are as follows: 61, 32, 94, 48. As a result, the lowest and maximum height axis values are 32 and 94, respectively. To ensure that all of the height values fit on the graph, we need a grid at least 94-32 = 62 squares tall.

To accommodate the provided data on the scatter graph, the grid must be at least 65 squares wide and 62 squares height.

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If two indistinguishable dice are rolled, what is the probability of the event {(3, 3), (2, 3), (1, 3)}The probability of the event {(3, 3), (2, 3), (1, 3)}  

If two indistinguishable dice are rolled, it is 3/36 or 1/12.

Explanation: Indistinguishable dice are dice that appear identical to one another but do not have unique markings. As a result, indistinguishable dice will have the same number of faces, but the values on each face will be identical.

The total number of possible outcomes is 6 * 6 = 36 because there are six possible outcomes for each roll of a single die.

The probability of rolling the numbers (3, 3), (2, 3), or (1, 3) can be determined as follows: 3/36 or 1/12

For each die, there are six possible outcomes, so there are 6*6, or 36 possible outcomes for two dice.

Because (3, 3), (2, 3), and (1, 3) are the only possible ways to obtain a 3 on one of the dice and a 3, 2, or 1 on the other, the probability is 3/36 or 1/12.

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The estate dealer must sell 13 houses to reach a total commission of GHc6500

Number is a mathematical object used to count, measure, and label. It is an abstract concept that is used in many different contexts. Numbers can be used to represent a variety of different things, including quantities, values, and relationships. They are also used to represent abstract concepts such as time and money. In mathematics, numbers are used to represent sets, operations, and relationships between elements. Numbers play a crucial role in almost all areas of mathematics, from the simple counting to the study of complex equations.

Arithmetic progression is a mathematical process
which involves adding a constant number to a sequence of numbers. In this case, the constant
number is GHc500, and the sequence of numbers is the commission of GHc3750 for the first house
sold.

To find the total number of houses that must be sold to reach a commission of GHc6500, we
need to use arithmetic progression. To do this, we need to calculate the arithmetic mean of the
two numbers GHc3750 and GHc6500. This is done by adding the two numbers and dividing by two.
The arithmetic mean is GHc5125.

We then subtract the initial commission of GHc3750 from the arithmetic mean to find the
increment in the commission for each additional house sold. This gives us the amount of GHc500
for each additional house sold.

To find the total number of houses that must be sold to reach a commission of GHc6500, we
then need to divide the total commission of GHc6500 by the GHc500 increment. This gives us
13 houses. Therefore, the estate dealer must sell 13 houses to reach a total commission of GHc6500.

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a) With the following probability table F F, Let’s apply the formula for the intersection of events to solve the first part of the problem.

P(EF) = P(E) x P(F|E).We know that P(E) = 0.1 and that P(F|E) = 0.3. Therefore,P(EF) = P(E) x P(F|E) = 0.1 x 0.3 = 0.03.b) Two events E and F are independent if and only if their intersection is equal to the product of their individual probabilities.

P(EF) = P(E) x P(F) if and only if E and F are independent. We know that P(E) = 0.1 and that P(F) = 0.1 + 0.3 = 0.4. Therefore, P(EF) = 0.03, which is different from 0.1 x 0.4 = 0.04.

Since P(EF) is different from P(E) x P(F), it means that E and F are not independent.c) Two events E and F are mutually exclusive if and only if their intersection is the null set.P(EF) = ∅ if and only if E and F are mutually exclusive. We know that P(EF) = 0.03, which is not equal to the null set. Therefore, E and F are not mutually exclusive.

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The required perimeter is 25+√61 units.

We can find the distance between each pair of consecutive points and then add them up to get the perimeter of the polygon.

Using the distance formula, the distance between points A and B is:

[tex]$$AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(-4 - 4)^2 + (8 - 2)^2} = \sqrt{100} = 10$$[/tex]

Similarly, the distances between the other pairs of points are:

[tex]$$BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(-7 + 4)^2 + (4 - 8)^2} = 5$$[/tex]

[tex]$$CD = \sqrt{(x_D - x_C)^2 + (y_D - y_C)^2} = \sqrt{(-1 + 7)^2 + (-4 - 4)^2} = 10$$[/tex]

[tex]$$DA = \sqrt{(x_A - x_D)^2 + (y_A - y_D)^2} = \sqrt{(4 + 1)^2 + (2 + 4)^2} = \sqrt{61}$$[/tex]

Therefore, the perimeter of the polygon is:

[tex]$$AB + BC + CD + DA = 10 + 5 + 10 + \sqrt{61}$$[/tex]

= 25+√61

Thus, required perimeter is 25+√61.

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The sample correlation coefficient would be closest to -0.9.

Here's why:

Correlation Coefficient: The correlation coefficient is a statistical measure of the degree of correlation (linear relationship) between two variables. Pearson’s correlation coefficient is the most widely used correlation coefficient to assess the correlation between variables.

Pearson’s correlation coefficient (r) ranges from -1 to 1. A value of -1 denotes a perfect negative correlation, 1 denotes a perfect positive correlation, and 0 denotes no correlation. There is a negative correlation between speed and time. As the speed of the car increases, the time needed to traverse the segment decreases. So, the sample correlation coefficient would be negative.

Since the sample size is large enough, the sample correlation coefficient should be close to the population correlation coefficient. The population correlation coefficient between speed and time should be close to -1, which implies that the sample correlation coefficient should be close to -1.

Therefore, the sample correlation coefficient would be closest to -0.9.

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The Laplace transform is a mathematical technique used to solve differential equations and analyze signals and systems in engineering, physics, and other fields. It is named after the French mathematician Pierre-Simon Laplace.

The Laplace transform of the given initial value problem is given by:

Y(s) = (2s^2 + 16) / (s^2(s^2+16))

Inverting the Laplace transform to find y(t) gives us:

y(t) = e^(-8t) * (1-cos(4t)) + 2sin(4t) + u2(t)

Where u2(t) is the Heaviside function that turns on at t = 2.

To find the Laplace transform of y(t), we first take the Laplace transform of both sides of the differential equation:

L(y''(t)) + 16L(y(t)) = L(e^(-2t)u_2(t))

Using the property L(y''(t)) = s^2Y(s) - sy(0) - y'(0) and noting that y(0) = 0 and y'(0) = 0, we can simplify to get:

s^2Y(s) + 16Y(s) = L(e^(-2t)u_2(t))

Using the property L(e^(-at)u_c(t)) = 1/(s + a) * e^(-cs), we can substitute to get:

s^2Y(s) + 16Y(s) = 1/(s + 2)^2

Now we can solve for Y(s):

Y(s) = 1/(s^2 + 16) * 1/(s + 2)^2

To find y(t), we need to take the inverse Laplace transform of Y(s). We can use partial fraction decomposition to simplify the expression:

Y(s) = A/(s^2 + 16) + B/(s + 2) + C/(s + 2)^2

Multiplying both sides by the denominator and solving for A, B, and C, we get:

A = 1/8

B = -1/4

C = 1/8

Substituting these values, we get:

Y(s) = 1/8 * 1/(s^2 + 16) - 1/4 * 1/(s + 2) + 1/8 * 1/(s + 2)^2

Taking the inverse Laplace transform of each term, we get:

y(t) = (1/8)sin(4t) - (1/4)e^(-2t) + (1/4)te^(-2t)

Therefore, the solution to the initial value problem y'' + 16y = e^(-2t)u_2(t), y(0) = 0, y'(0) = 0 is y(t) = (1/8)sin(4t) - (1/4)e^(-2t) + (1/4)te^(-2t).

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The error is the substitution of coordinates. Coordinates are ordered pairs of points that help us locate any point in a 2D plane or 3D space.

Cartesian coordinates, also known as the coordinates of a point in a 2D plane, are two integers, or occasionally a letter and a number, that identifies a specific point's precise location on a grid. This grid is referred to as a coordinate plane.

The distance between two points A(x₁, y₁) and B(x₂, y₂) is given by

[tex]AB = \sqrt{(x_{1} , x_{2})^{2} + (y_{1} - y_{2})^{2} }[/tex]

Observe that the x-coordinate of B is subtracted from the x-coordinate of A. This goes with the y-coordinates.

Therefore, the error is the substitution of coordinates.

The correct computation is

[tex]AB = \sqrt{(6-1)^{2} + [2 - (-4)]^{2} }[/tex]

[tex]= \sqrt{5^{2} + 6^{2} }[/tex]

[tex]= \sqrt{25 + 36} \\[/tex]

[tex]= \sqrt{61}[/tex]

≈ 7.81

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The complete question is as follows:

Describe and correct the error in finding the distance between A(6, 2) and B(1, -4). AB = √[(6 - 2)² + {2 - (-4)}²] = √(4² + 5²) = √(16 + 25) = √41 ≈ 6.4.

The context that could be modeled by a linear function is "snow was falling at a rate of 2 inches per hour."

In mathematics, a function is a relation between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. In other words, a function takes an input and produces a corresponding output. It is often represented as a mathematical equation or a graph. Functions are used to model real-world phenomena and are an important tool in many areas of mathematics, science, and engineering.

Here,

A linear function describes a constant rate of change, and in this context, the rate of snowfall is constant at 2 inches per hour. The other contexts involve exponential or percentage change, which cannot be modeled by a linear function.

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There are several predictions that the food service manager may make based on the data from the survey of 200 students regarding their preference for new lunch menu items. Let's examine some of these predictions and see if they are supported by the data.

The majority of students will like the new menu items.

The food service manager may predict that the majority of students in the school will like the new menu items, based on the positive responses from the 200 surveyed students. However, it's important to note that the sample size of 200 is relatively small compared to the total student population of 1,500. Therefore, it's possible that the preferences of the 200 surveyed students may not be representative of the preferences of the entire student population. To make a more accurate prediction, the manager may need to conduct a larger survey or pilot program to test the new menu items with a larger group of students.

Certain menu items will be more popular than others.

Based on the survey data, the food service manager may be able to identify which new menu items are more popular among the surveyed students. For example, if a majority of students indicate that they would like to see more vegetarian options, the manager may predict that introducing more vegetarian menu items will be popular among the broader student population. However, it's important to keep in mind that the preferences of the 200 surveyed students may not be representative of the preferences of the entire student population, so the manager may need to conduct additional research or testing to confirm these predictions.

The introduction of new menu items will increase overall satisfaction with the school lunch program.

If the survey data shows that a significant number of students are excited about the new menu items, the food service manager may predict that introducing these items will increase overall satisfaction with the school lunch program. However, it's important to note that satisfaction is a complex concept that can be influenced by many factors beyond just the menu items, such as the quality of service, cleanliness of the cafeteria, and overall atmosphere. Therefore, the manager may need to consider these other factors when predicting the impact of the new menu items on overall satisfaction with the lunch program.

In summary, while the data from the survey of 200 students can provide valuable insights into student preferences for new lunch menu items, it's important to interpret these results with caution and consider additional factors that may influence the broader student population. Conducting further research or testing can help to confirm these predictions and make more accurate decisions about the school lunch program.

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Answer: the answer is A

Step-by-step explanation:

Answer:

Step-by-step explanation:

We can simplify the expression by factoring out a common factor of 5^1999 from the numerator:

5^2000 + 5^1999

= 5^1999(5 + 1)

= 5^1999(6)

And we can also factor out a common factor of 5^1997 from the denominator:

5^1999 - 5^1997

= 5^1997(5^2 - 1)

= 5^1997(24)

So the entire expression simplifies to:

(5^2000 + 5^1999) / (5^1999 - 5^1997)

= (5^1999 * 6) / (5^1997 * 24)

= (6/24) * 5^2

= 5/2

Therefore, the value of the expression is 5/2.

Answer:

83219083219389238293829381293

Step-by-step explanation:

according to my calculations i am not smart

The values of x which satisfies the fractional inequality (x - 1) / (2 - x) ≥ 0 is x ≥ 1

(x - 1) / (2 - x) ≥ 0

This is a fractional inequality whose numerator is (x - 1) and the denominator is (2 - x)

(x - 1) / (2 - x) ≥ 0

cross product

(x - 1) ≥ 0 × (2 - x)

(x - 1) ≥ 0

x - 1 ≥ 0

Add 1 to both sides

x ≥ 1

Therefore, x ≥ 1 satisfies the inequality.

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2:3 is a simple ratio of the red triangle's vertical and horizontal lengths.

In order to put numbers into the proper perspective and so simplify difficulties, ratios are widely used in daily life.

A ratio is a tool used only to compare the sizes of two or much more numbers in relation to one another in mathematics. By making amounts easier to understand, ratios enable us to measure but also express quantities.

When translating from one currency to another, ratios are used.

For the red triangle:

horizontal length = 3 units

vertical length = 2 units

vertical length / horizontal length = 2/3

Thus, 2:3 is a simple ratio of the red triangle's vertical and horizontal lengths.

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Answer:

Let's call the number "x".

Then, we can write the equation:

7 + 2x = 35

To solve for x, we need to isolate x on one side of the equation.

Subtracting 7 from both sides:

2x = 28

Dividing both sides by 2:

x = 14

Therefore, the number is 14.

Step-by-step explanation:

The angle of elevation of the sun is approximately 22.6 degrees.

The partnerships between the sides and angles of triangles are the subject of the mathematical discipline of trigonometry. It is used exhaustively in fields such as physics, engineering, and assessing.

In a right triangle, the side opposite the right angle is called the hypotenuse, while the other two sides are called the legs.

Given that, 7.6 m flagpole casts an 18.2 m shadow.

Using trigonometric ratio we have:

tan(θ) = h / s

Substituting the values:

tan(θ) = 7.6 / 18.2

tan(θ) ≈ 0.417

θ ≈ 22.6°

Hence, the angle of elevation of the sun is approximately 22.6 degrees.

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The trip from Winston to Carver takes 8 min longer during rush hour, when the average speed is 75 km/h, than in off-peak hours when the average speed is 90 km/h.  the distance between the two towns is  60 km.

Considering the distance between the Winston and carver be "d", since here we got that during off-peak hours, the average speed is 90 km/h. so  by using the formula of  distance which is  distance =speed x time=> d = 90t.

Whereas in rush hour, the average speed is 75 km/h, we also know that the trip takes 8 minutes longer during rush hour.  calling the time it takes to travel during rush hour "t+8/60"( since 8 minutes is 8/60 of an hour). Now using the same formula as before:

d = 75(t + 8/60), since here we have  two equations for d we can equal  them to each other  then we get :

90t = 75(t + 8/60)

=>90t = 75t + 10

=>90t-75t=10

=>t = 2/3

Since during the off-peak hours, it takes 2/3 hours or 40 minutes to travel the distance between Winston and carver. now  using either equation to find the distance we get  

d = 90t = 90(2/3) = 60 km or d = 75(t + 8/60) = 75(2/3 + 8/60) = 60 km

To know more about speed refer to the  link  brainly.com/question/17661499

#SPJ4

Answer:

x = 45.

Step-by-step explanation:

We know the full angle of this is 180 degrees.

Given: (2x+45) + x = 180

First, collect like terms ( in this case 2x and x, 180 and 45 )

2x + x = 180 - 45

Then calculate:

3x = 135. ( Divide both sides by 3 )

x = 45