Brookfield Street and Bloomington Avenue intersect. If Brookfield Street is 5 meters wide and Bloomington Avenue is 6 meters wide, what is the distance between two opposite corners of the intersection? If necessary, round to the nearest tenth.
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Simplified form of the algebraic expression (x^2 - x - 12) / (x^2 - 4) = (x + 3) / (x - 2)

To simplify the given algebraic expression (x^2 - x - 12) / (x^2 - 4), we first need to factor both the numerator and denominator as much as possible.

We can factor the numerator using the product-sum method or the quadratic formula, which yields:

x^2 - x - 12 = (x - 4)(x + 3)

Similarly, we can factor the denominator as a difference of squares, which gives:

x^2 - 4 = (x - 2)(x + 2)

Now, we can substitute these factorizations into the original expression:

(x^2 - x - 12) / (x^2 - 4) = [(x - 4)(x + 3)] / [(x - 2)(x + 2)]

At this point, we can simplify the expression by canceling out the factors that appear in both the numerator and denominator. Specifically, we can see that (x - 4) and (x + 2) appear in both the numerator and denominator, so they cancel out:

[(x - 4)(x + 3)] / [(x - 2)(x + 2)] = (x + 3) / (x - 2)

So the simplified expression is (x + 3) / (x - 2).

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Step-by-step explanation:

Let x be the number of suede coats and y be the number of cotton coats. Then, the system of inequalities representing the situation is:

150x + 69y ≥ 6200 (the total cost of the women's coats must be at least $6200)

x ≥ 0, y ≥ 0 (the number of coats cannot be negative)

Note that this system assumes that the store only sells suede and cotton coats for women, and that there are no other costs associated with these coats (such as shipping or storage costs).

Answer:

34

Step-by-step explanation:

In all right triangles, the two shortest sides squared equals the longest side squared, given by the equation [tex]a^2+b^2=c^2[/tex].

We know the two shortest sides, so we can plug in

[tex]16^2+30^2=c^2\\256+900=c^2\\1156=c^2\\c=\sqrt{1156} \\c=34[/tex]

The sequence is 3, 9, 15, 21, 27, and the recursive formula for this sequence is a_1 = 3, a_n = a_{n-1} + 6, and the next term is 33.

The sequence is an arithmetic sequence with a common difference of 6, starting at 3. A recursive formula for this sequence can be written as:

a_1 = 3

a_n = a_{n-1} + 6, for n > 1

This formula means that the first term in the sequence is 3, and every subsequent term is found by adding 6 to the previous term.

To find the next term in the sequence, we can use this formula to compute a_6:

a_6 = a_5 + 6

a_6 = 27 + 6

a_6 = 33

Therefore, the next term in the sequence is 33.

The given sequence is an arithmetic sequence, where each term is 6 more than the previous term, starting at 3.

A recursive formula is a mathematical formula that is used to define a sequence in terms of its previous terms. In this case, we can use the recursive formula a_1 = 3 and a_n = a_{n-1} + 6 to define the given sequence. The formula says that the first term in the sequence is 3, and each subsequent term can be found by adding 6 to the previous term.

Using this recursive formula, we can find the next term in the sequence, which is 33. We can continue to apply the formula to find any term in the sequence.

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Given:

Vince has 1/2 ton of gravel.

It is to be spread equally in 8 square yards.

To find:

How many tons of gravel will be spread in each square yard?

Solution:

Total gravel = 1/2 ton

To be spread equally in 8 square yards.

No. of tons of gravel will be spread in each square yard = (1/2) / 8 = 1/16 tons.

Therefore, 1/16 tons of gravel will be spread in each square yard.

Answer:

Given:

Vince has 1/2 ton of gravel.

It is to be spread equally in 8 square yards.

To find:

How many tons of gravel will be spread in each square yard?

Solution:

Total gravel = 1/2 ton

To be spread equally in 8 square yards.

No. of tons of gravel will be spread in each square yard = (1/2) / 8 = 1/16 tons.

Therefore, 1/16 tons of gravel will be spread in each square yard.

Step-by-step explanation:

167.7 feet is the length of the zipline.

The angle of inclination is the angle formed between a horizontal line and a line or surface that is sloping or inclined. It is a measure of the steepness or slope of the line or surface and is typically expressed in degrees or as a trigonometric ratio.

We have a zipline that drops 30 feet from one treetop to a second treetop.

If the angle of inclination from the shorter tree to the taller tree is 10 degrees.

Let AB be the distance between two trees and BC be the drop in height from A to C. Then,

We have BC/AB = tan(θ)

Where θ = 10 degrees

We know BC = 30 feet.

So,

AB = BC/tan(θ) = 30/tan(10°) = 167.7 feet (rounded to one decimal place)

Therefore, the length of the zipline is 167.7 feet.

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For the second order differential equation, we find that Height = 1333.33 meters, Velocity = 240 meters/second.

The acceleration of a rocket fired vertically upwards t seconds after launch is given by a = 20 + 4t m/s². The second order differential equation for the height of the rocket is given by ℎ′′ = a.

The initial conditions for the rocket are:

ℎ(0) = 0 (the rocket starts from the ground) and ℎ′(0) = 0 (the rocket is not moving initially). For the differential equation, we integrate the acceleration once to obtain the velocity, and then integrate the velocity to obtain the height.

Integrating a = 20 + 4t gives v = 20t + 2t² + C1, where C1 is a constant of integration. Using the initial condition v(0) = 0, we get C1 = 0. Integrating v = 20t + 2t² gives ℎ = 10t² + 2/3 t³ + C2, where C2 is another constant of integration.  Using the initial condition ℎ(0) = 0, we getC2 = 0.

Therefore, the general solution for the height of the rocket is ℎ = 10t² + 2/3 t³.The velocity of the rocket is given by v = ℎ′.

At t = 10 s, the height of the rocket is ℎ(10) = 10 × 100 + 2/3 × 1000 = 1333.33 m. The velocity of the rocket at t = 10 s is v(10) = ℎ′(10) = 20 × 10 + 2 × 10² = 240 m/s.

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Answer:

Step-by-step explanation:

Answer:

5h

Step-by-step explanation:

300÷60=5

km÷km/h=h

=5h

Answer: a. The equation for the total expenses E for n bikers is:

E = 30n + 125n + 700

b. The equation for the total income I for n bikers is:

I = 350n

c. The equation for the profit P for n bikers is:

P = I - E = 350n - (30n + 125n + 700) = 195n - 700

Answer:

The square root of 45 is approximately 6.708203932.

a) The probability is equal to the proportion of the range that lies below 17 minutes.P(X< 17) = (17 - 16) / (23 - 16) = 1/7

b) The probability is equal to the proportion of the range that lies above 22 minutes.P(X > 22) = (23 - 22) / (23 - 16) = 1/7

c)  The probability is equal to the proportion of the range that lies between 18 and 20 minutes.P(18 ≤ X ≤ 20) = (20 - 18) / (23 - 16) = 2/7

d) The mean and standard deviation of the download times are 19.5 and 1.4 minutes, respectively.

The probability that the download time will be less than 17 minutes.The probability is equal to the proportion of the range that lies below 17 minutes.P(X< 17) = (17 - 16) / (23 - 16) = 1/7

The probability that the download time will be more than 22 minutes.The probability is equal to the proportion of the range that lies above 22 minutes.P(X > 22) = (23 - 22) / (23 - 16) = 1/7

The probability that the download time will be between 18 and 20 minutes.The probability is equal to the proportion of the range that lies between 18 and 20 minutes.P(18 ≤ X ≤ 20) = (20 - 18) / (23 - 16) = 2/7

The mean and standard deviation of the download times.Using the formula for the mean and standard deviation for a uniform distribution with a range of [a, b],μ = (a + b) / 2 = (16 + 23) / 2 = 19.5σ = (b - a) / sqrt(12) = (23 - 16) / sqrt(12) ≈ 1.4 Therefore, the mean and standard deviation of the download times are 19.5 and 1.4 minutes, respectively.

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First vector v = [- 2 - 2 2] is in W and the second vector v = [6 - 1 - 3] is not in W.

W be the set of all vectors [x y x + y] with x and y real. To find: Whether each of the following vectors is in W. Let's check each vector whether it is in W or not: v = [- 2 - 2 2] To check the given vector is in W or not, we need to find the values of x and y such that the third component equals 2.So, x + y = 2 ⇒ y = 2 - x

The given vector can be written as v = [x y x + y]= [x, 2 - x, 2] Thus, given vector v is in W. v = [6 - 1 - 3]. To check the given vector is in W or not, we need to find the values of x and y such that the third component equals -3. So, x + y = -3 ⇒ y = -3 - x The given vector can be written as v = [x y x + y]= [x, -3 - x, -3]Thus, given vector v is not in W.

Moreover, Vectors, in Maths, are objects which have both, magnitude and direction. Magnitude defines the size of the vector. It is represented by a line with an arrow, where the length of the line is the magnitude of the vector and the arrow shows the direction.

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The numbers 8 and 7, which have fixed values that never shift, are the constants.

A coefficient in mathematics is an integer or symbol that multiplies a variable or a variable product. In algebraic formulas, equations, and polynomials, coefficients are used. For instance, the coefficient of the variable x in the equation 3x + 5 is 3, and the coefficient of the constant term 5 is 1. The factors of x and y in the equation 2x + 3y = 7 are 2 and 3, respectively. Coefficients can be whole integers, fractions, or decimals and can have a positive, negative, or zero sign. They aid in the simplification of mathematical expressions and equations and serve to illustrate the relationship between variables.

given

3 coefficients

x is a variable.

8 and 7 are constants.

The coefficient in the equation 3x +  8 = 7 multiplies the variable x by the number 3.

We are looking for an unknown number, represented by the variable x. The numbers 8 and 7, which have fixed values that never shift, are the constants.

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Answer: 8

Step-by-step explanation:

Using SOHCAHTOA, we need TOA as we have the opposite (1) and the adjacent (7)

θ = [tex]tan^{-1}(\frac{1}{7} )[/tex] = 8.13 = 8

Answer:

6)180-99=81

81+43=124

180-124=56

b=56

Answer:

3% of 600 is (3/100) x 600 = 18.

So, a number that is 3% larger than 600 would be:

600 + 18 = 618.

Therefore, the number that is 3% larger than 600 is 618

Answer:

618

Step-by-step explanation:

Let us first see what is 3% of the 600

3/100 x 600 = 18

a number that is 3% larger than 600 is 600+18 = 618.

The solution to the equation is (c) No, because there is only one real solution and the value is x = 4

The given equation is

(6 – 2x)(3 – 2x)x = 40

Next, we answer the question from the numbers given from the list of options

In option (c), we have

x = 4

By substitution, the equation becomes

(6 - 2 * 4)(3 - 2 * 4) * 4 = 40

Evaluate the product expression

40 = 40

The above equation is true

Hence, the solution is (c)

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By answering the presented question, we may conclude that As a result, the probability of picking a clown performer from the circus is 1/3, or 0.33. (rounded to two decimal places).

Probabilistic theory is a branch of mathematics that calculates the chance that an event or statement will occur or be true. A risk is a number between 0 and 1, where 1 denotes certainty and 0 indicates how probable an event is to occur. Probability is a mathematical term for the likelihood of a certain event occurring. Probabilities can also be expressed as integers between 0 and 1 or as percentages ranging from 0% to 100%. In relation to all other outcomes, the ratio of occurrences among equally likely alternatives that result in a certain event.

The likelihood of picking a clown performer from the circus is proportional to the number of clowns to the total number of circus artists.

The probability of picking a clown performer is the number of clowns divided by the total number of performers.

So,

The likelihood of picking a clown performer is 5/15.

By dividing the fraction's numerator and denominator by 5, we get:

The likelihood of picking a clown performer is 1/3.

As a result, the likelihood of picking a clown performer from the circus is 1/3, or 0.33. (rounded to two decimal places).

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The volume of small pyramid with base length 2, base width 2, and height 2. Its volume was 8/3π. We then scaled it by a factor of 2 and verified the formula V' = V × k3 holds true.

To create the smallest pyramid possible, we will use a tool such as a ruler or protractor to measure and construct the pyramid. Let's assume that we are using a ruler and that the smallest pyramid we can construct has a base length of 2 units, a base width of 2 units, and a height of 2 units.

To calculate the volume of the pyramid, we use the formula:

V = (1/3) × base area × height

The base area of the pyramid is:

A = base length × base width = 2 × 2 = 4 square units

Therefore, the volume of the pyramid is:

V = (1/3) × 4 × 2 = 8/3 cubic units (in terms of π, this is 8/3π cubic units)

Now, let's scale the original pyramid by a factor of k = 2. To find the new dimensions of the scaled pyramid, we multiply each dimension of the original pyramid by the scale factor k:

Base length = 2 × 2 = 4 units

Base width = 2 × 2 = 4 units

Height = 2 × 2 = 4 units

The base area of the scaled pyramid is:

A' = base length × base width = 4 × 4 = 16 square units

The volume of the scaled pyramid is:

V' = (1/3) × A' × height = (1/3) × 16 × 4 = 64/3 cubic units (in terms of π, this is 64/3π cubic units)

Now, we can verify that the formula V' = V × k3 holds true for the scaled pyramid:

V' = 64/3 cubic units

V = 8/3 cubic units

k = 2

V' = V × k3

64/3 = (8/3) × 23

64/3 = 8/3 × 8

64/3 = 64/3

Therefore, the formula V' = V × k3 holds true for a pyramid, and we have successfully verified it using the scaled pyramid.

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At the 1% level of significance, both the number of security guards (g) and the unemployment rate (u) have a significant effect on the dollar loss from theft each week (t). This is indicated by the p-values for both variables, which are both less than 0.01 (0.0012 for g and 0.0096 for u).

This means that there is less than a 1% chance that the observed relationship between these variables and the dependent variable (t) is due to chance. Therefore, we can reject the null hypothesis that there is no relationship between these variables and the dependent variable, and conclude that they have a significant effect on the dollar loss from theft each week.

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The length of the ramp is approximately 16 meters.

Trigonometric ratios are ratios of the sides of a right triangle that relate the angles of the triangle to its sides.

There are 3 basic trigonometric ratios are given by

sin θ = opposite side /hypotenuse.

cos θ = adjacent side /hypotenuse.

tan θ  = opposite side /adjacent.

Here we have

The angle of the incline of the ramp is 32 degrees.

The height of the ramp is 10m.

Represent the following data as a right-angled triangle  

where the height of the ramp will be opposite side to the angle of the incline and the length of the ramp will be adjacent side

Let's assume the length of the ramp is "x".  

From the trigonometric ratios,

tan(32°) = 10/x

Multiply both sides by x:

x × tan(32°) = 10

Divide both sides by tan(32°):

x = 10 / tan(32°)

x = 10/0.62

x = 16.0033 meters

x = 16 meters

Therefore,

The length of the ramp is approximately 16 meters.

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You would need to deposit approximately $47.83 now to be able to withdraw $650 at the end of each year for 20 years, assuming an annual interest rate of 11% compounded annually.

It is typically expressed as an annual percentage rate (APR). Interest rates can be fixed, meaning they remain the same for the entire term of the loan, or variable, meaning they can change over time based on market conditions or other factors. The interest rate is an important factor to consider when borrowing money, as it affects the overall cost of the loan. It is also a key factor in investment decisions, as it determines the return on an investment.

To calculate the present value of an investment that will yield a future value of $650 for 20 years at 11% annual interest, we can use the formula for present value of an annuity:

PV = FV / (1 + r)ⁿ

where PV is the present value, FV is the future value, r is the interest rate per period, and n is the number of periods.

In this case, we want to find the present value of a 20-year annuity that pays $650 at the end of each year, with an annual interest rate of 11%. Plugging in the values, we get:

PV = $650 / (1 + 0.11)²⁰

PV = $650 / 13.584

PV = $47.83 (rounded to the nearest cent)

Therefore, you would need to deposit approximately $47.83 now to be able to withdraw $650 at the end of each year for 20 years, assuming an annual interest rate of 11% compounded annually.

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Answer:

its bar graph

Step-by-step explanation:

thanks for the question

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The respοnses fοr favοurite vacatiοn spοt can be described best by a pie chart.

One sοrt οf graph that illustrates the infοrmatiοn in the circular graph is a pie chart. It is a sοrt οf graphical representatiοn οf data where the slices οf pie depict the relative sizes οf the data. A list οf numerical and categοrical variables is necessary fοr a pie chart. Pie in this cοntext refers tο the entire thing, and slices tο its cοmpοnent pοrtiοns.

If yοu asked "What is yοur favοrite vacatiοn spοt?" tο a grοup οf peοple, the mοst apprοpriate type οf graph tο represent the respοnses wοuld be a pie chart.

A pie chart is a circular chart that is divided intο slices tο represent the prοpοrtiοn οf each categοry in a dataset.

In this case, each slice οf the pie chart wοuld represent a different vacatiοn spοt, and the size οf each slice wοuld cοrrespοnd tο the prοpοrtiοn οf respοndents whο selected that vacatiοn spοt as their favοrite.

Pie charts are useful fοr displaying categοrical data, where the categοries are mutually exclusive and add up tο 100%.

They are easy tο read and understand, and can quickly shοw the distributiοn οf respοnses amοng the different categοries.

On the οther hand, a bar graph οr a line graph wοuld nοt be apprοpriate fοr this type οf data since the respοnses are nοt numerical οr cοntinuοus.

A line plοt wοuld alsο nοt be apprοpriate since it is used tο display a series οf data pοints, and in this case, there is οnly οne data pοint per categοry.

Therefοre, a pie chart is the best chοice.

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Answer:

The zeros of the polynomial function f(x) = x^4 - 2x^3 - 8x^2 + 10x + 15 are x = -1, x = 3 + √29/2, x = 3 - √29/2, and x = -0.4495 (approximately).

Step-by-step explanation:

To find all the zeros of the polynomial function f(x) = x^4 - 2x^3 - 8x^2 + 10x + 15, we can use the Rational Root Theorem and synthetic division.

Write the polynomial function in descending order of degree: f(x) = x^4 - 2x^3 - 8x^2 + 10x + 15.

Use the Rational Root Theorem to generate a list of possible rational zeros: ±1, ±3, ±5, ±15.

Use synthetic division to test each possible zero. We start with x = 1:

1 │ 1 -2 -8 10 15

│ 1 -1 -9 1

└───────────────

1 -1 -9 1 16

x = 1 is not a zero of the polynomial function.

We continue testing the remaining possible zeros:

-1 │ 1 -2 -8 10 15

│ -1 3 5 -15

└───────────────

1 -3 -3 15 0

Since the remainder is zero, we have found a zero of the polynomial function at x = -1.

We can use synthetic division to factor the polynomial function:

(x + 1)(x^3 - 3x^2 - 6x + 15)

Now we can solve for the remaining zeros of the polynomial function by factoring the cubic equation using the Rational Root Theorem and synthetic division:

3 │ 1 -3 -6 15

│ 3 0 -18

└─────────────

1 0 -6 -3

x = 3 is not a zero of the polynomial function.

-3 │ 1 -3 -6 15

│ -3 18 -36

└────────────

1 -6 12 -21

x = -3 is not a zero of the polynomial function.

The only remaining possible rational zeros are ±1/2 and ±5/2, but testing these values using synthetic division does not yield any more zeros.

However, we can see that the polynomial function can be factored as follows:

(x + 1)(x - 3)(x^2 - 3x - 5)

We can solve for the remaining zeros of the polynomial function by factoring the quadratic equation using the quadratic formula or factoring by grouping. Either way, we find that the remaining zeros are approximately x = (3 + √(29))/2 and x = (3 - √(29))/2.

Therefore, the zeros of the polynomial function f(x) = x^4 - 2x^3 - 8x^2 + 10x + 15 are x = -1, x = 3 + √29/2, x = 3 - √29/2, and x = -0.4495 (approximately).

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Mode: The mode of this data is 8 hours. This is because 8 hours was the most frequently reported amount of time that the students watched TV during a school week.

Amount is a quantitative expression of magnitude, size, or degree of a particular quantity. It is typically used to describe an object, person, or an event. Amount is used to quantify something, to describe its size, duration, or extent. It can be used to measure a wide range of physical and abstract entities, such as money, time, energy, resources, and emotions. For example, you might say “there was a large amount of people at the event” or “I have a certain amount of money saved up.”

Range: The range of this data is 8 hours. This is because the difference between the highest amount of time reported (16 hours) and the lowest amount of time reported (8 hours) is 8 hours.

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Answer:

First, let's simplify both sums by combining like terms:

3x + 5y - 2 + 2x - 3y + 1 = 5x + 2y - 1

4x - 8y + 3 - 5x + 6y + 7 = -x - 2y + 10

Now we can subtract the second sum from the first:

(5x + 2y - 1) - (-x - 2y + 10) = 5x + 2y - 1 + x + 2y - 10

Simplifying this expression, we get: 6x + 4y - 11

Answer:

3

Step-by-step explanation:

HCF(a, b) x LCM(a, b) = a x b

We are given that LCM(18, 21) = 126. Let's use this to find the HCF:

HCF(18, 21) x 126 = 18 x 21

HCF(18, 21) = (18 x 21) / 126

HCF(18, 21) = 3

Therefore, the HCF of 18 and 21 is 3.

There will be six possibilities in the sample space if a notebook is selected at random.

the shop sells two types of notebooks and each notebook is offered in red, blue or yellow.

each of the two notebook kinds is available in three different colors.

As a result, there are a total of the following possibilities in the sample space:2 types of notebooks x 3 colors each = 6

Hence, the sample space contains six distinct alternatives.

they are,

Red type 1 notepad

Blue Type 1 notepad

Yellow type 1 notepad

Red type 2 notepad

Blue Type 2 notepad

Yellow type 2 notepad

therefore, the sample space contains six distinct alternatives.

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