The bell tower of the cathedral in Pisa, Italy, leans 5.6° from the vertical. A tourist stands 105 m from its base, with the tower leaning directly toward her. She measures the angle of elevation to the top of a the tower to be 29.2°. Find the length of the tower to the nearest meter.

It takes 1,95,000 seconds longer for light from the sun to reach Pluto as to reach Earth.

It is given to us that light from the sun takes 5×10² seconds to reach Earth and it takes 2×10⁴ seconds to reach Pluto.

We need to find the difference in the time it takes for the light to reach Pluto from the sun than it takes to reach the earth

First, lets convert the time it takes for the light to reach Pluto from the sun:

2×10⁴ seconds to reach Pluto

= 2 x 100000 = 200000 seconds

now lets convert the time it takes for the light to reach earth from the sun:

5×10² seconds to reach Earth

= 5 x 1000 = 5000 seconds

therefore the difference = time taken to reach Pluto - time taken to reach Earth

= 2,00,000 - 5,000 = 1,95,000

Therefore, it takes 1,95,000 seconds longer for light from the sun to reach Pluto as to reach Earth.

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We can use trigonometry to solve this problem. Let the length of the rope be x. Then, we can draw a right triangle where the height of the tree is the opposite side, the length of the rope is the hypotenuse, and the angle of elevation is 60 degrees. The adjacent side can be found using the trigonometric function cosine:

cos 60° = adjacent / hypotenuse

1/2 = adjacent / x

Multiplying both sides by x, we get:

x/2 = adjacent

To find the length of the hypotenuse, we can use the Pythagorean theorem:

hypotenuse^2 = adjacent^2 + opposite^2

Substituting in the values we know, we get:

x^2 = (x/2)^2 + (9√5)^2

Simplifying the right side, we get:

x^2 = x^2/4 + 405

Multiplying both sides by 4, we get:

4x^2 = x^2 + 1620

Subtracting x^2 from both sides, we get:

3x^2 = 1620

Dividing both sides by 3, we get:

x^2 = 540

Taking the square root of both sides, we get:

x = √540 = 6√60

Therefore, the length of the rope that Rachel needs to hang the piñata is 6√60 feet. Rounded to the nearest tenth, this is approximately 74.1 feet.

Answer:

  43.4 ft

Step-by-step explanation:

You want to know the length of rope needed to hang a piñata from the top of a tree that is 9√5 feet tall, if the rope makes an angle of 60° with the ground.

The hypotenuse of the triangle can be found by considering the sine of the 60° angle. The sine of the angle is given by ...

  Sin = Opposite/Hypotenuse

Then the hypotenuse of the triangle can be found from ...

  sin(60°) = tree height / hypotenuse

  hypotenuse = tree height/sin(60°) = 9√5/sin(60°)

We presume the length of rope Rachel wants is the total of the lengths from the anchor point to the tree top and back to the ground. That will be ...

  9√5 + 9√5/sin(60°) = 9√5·(1 +1/sin(60°)) ≈ 43.4 . . . . feet

Rachel needs about 43.4 feet of rope to hang the piñata.

The solution set for that absolute value inequality is -19/2 < y < 13/2.

The total cost of the water piping is $217.12 in the field.

Perimeter is the total length of the boundary or the outer edge of a two-dimensional geometric shape. It is the distance around the shape or the sum of the lengths of all its sides. The perimeter is usually measured in units such as centimeters, meters, feet, or inches, depending on the system of measurement used.

In the given question,

To calculate the total cost of the water piping, we need to first calculate the total length of the edges of the field.

For the rectangle, the length of its four edges can be calculated as:

2 x length + 2 x breadth = 2 x 68m + 2 x 64m = 136m + 128m = 264m

For the square extension, it has four edges each with length 26m, so the total length of its edges is:

4 x 26m = 104m

Therefore, the total length of the edges of the field is:

264m + 104m = 368m

To find the total cost of the water piping, we can multiply the total length of the edges by the cost per metre:

368m x $0.59/m = $217.12

So, the total cost of the water piping is $217.12.

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The coordinates of the pre-image point A are (-4, 8).Hence, the correct answer is (-4, 8).

The given image point of A after a translation right 5 units and down 3 units is the point B(-3, -11).

To determine the coordinates of the pre-image point A, we need to move 5 units to the left and 3 units up.

Let the pre-image point A be (x, y).

Then, the image point B is obtained from A by translation of 5 units to the right and 3 units down.

So, we have the following equations:

y - (-11) = -3x - (-3)y + 11 = -3x + 9On

solving the above equations, we get

x = -4 and y = 8.

So, the coordinates of the pre-image point A are (-4, 8).

Hence, the correct answer is (-4, 8).

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Let x be pounds of hamburger and y be pounds of hot dogs.

The equation is: 8x + 6y ≤ 75.

An equation is a mathematical statement that uses the equal sign (=) to show that two expressions are equal. Equations can include variables, which are symbols that represent unknown or changing values. Solving an equation means finding the value of the variable that makes the equation true. Equations are commonly used in many areas of mathematics, science, and engineering, as well as in everyday life.

Let x be the pounds of hamburger purchased and y be the pounds of hot dogs purchased.

The cost of hamburger at $8 per pound would be 8x dollars.

The cost of hot dogs at $6 per pound would be 6y dollars.

The total cost of the purchases should not exceed $75, so we can write:

8x + 6y ≤ 75

This is the equation that represents Breanna's situation, where x represents the pounds of hamburger purchased and y represents the pounds of hot dogs purchased, and the total cost of the purchases should not exceed $75.

Therefore, Let x be pounds of hamburger and y be pounds of hot dogs.

The equation is: 8x + 6y ≤ 75.

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The tour guide should row towards the point on the coast closest to point Q, which is the point of perpendicular intersection between the coast and the line passing through points Q and the initial position of the tour guide.

To reach point Q in the least time, the tour guide should row towards the point on the coast that is closest to point Q. Let's call this point P.

We can use the Pythagorean theorem to find the distance from point P to point Q

distance PQ = sqrt((3 miles)^2 + (1 mile)^2) = sqrt(10) miles

Now, let's consider two scenarios

Scenario 1: The tour guide rows directly towards point Q.

The distance the tour guide needs to row is 2 miles + sqrt(10) miles. This will take the tour guide

time = (2 miles + sqrt(10) miles) / (3 miles/hour) = (2/3) + (1/3)*sqrt(10) hours

Scenario 2: The tour guide rows towards point P, then walks the rest of the way to point Q.

The distance the tour guide needs to row is 2 miles - x miles, where x is the distance from point P to the nearest point on the coast. By similar triangles, we know that

x / 3 miles = 1 mile / sqrt(10) miles

Solving for x, we get

x = 3/sqrt(10) miles

So the distance the tour guide needs to row is

2 miles - 3/sqrt(10) miles = 2 - (3/sqrt(10)) miles

The distance the tour guide needs to walk is

sqrt(10) miles - 1 mile = sqrt(10) - 1 miles

The time it takes the tour guide to row and walk is:

time = (2 - (3/sqrt(10))) / (3 miles/hour) + (sqrt(10) - 1) / (3 miles/hour)

Simplifying this expression, we get

time = (2/3) - (1/3)*sqrt(10) + (sqrt(10)/3) - (1/3) = (1/3) + (1/3)*sqrt(10) hours

Comparing the two scenarios, we see that scenario 2 takes less time, so the tour guide should row towards point P in order to reach point Q in the least time.

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Answer: 1. -8, 2. 12, 3.-4 , 4.-7 , 5.3 , 6.15 , 7. 1, 8. 12, 9. -15, 10.318 ,

Step-by-step explanation:

The first terms are a_1=  -4  a_2= -8/3  a_3= -16/9  a_4= -32/27

The series converges as a result  the series' sum, if it exists, is equal to -12.

What exactly are geometric series?

A geometric sequence is a set of numbers where each phrase following the first is obtained by multiplying the previous term by the common ratio , a constant. Consider the order 2, 4, 8, 16, 32, etc. as an example. Given that each term after the first is derived by multiplying the previous term by 2, this geometric series has a common ratio of 2.

The following is the formula for determining the nth term in a geometric sequence:

a n = arⁿ⁻¹

where "a" denotes the initial word and "r" denotes the common ratio 2.

A geometric sequence of infinite nature is the sum of an infinite geometric series. ¹. The following formula is used to calculate the sum of an infinite geometric series:

sum = a/ (1-r)

where "r" is the common ratio and "a" is the first term.

n=1, 2, 3, and 4 can be added to the formula for the nth term of a geometric sequence to find the first four terms of the series:

a n = arⁿ⁻¹

where r is the common ratio and an is the first term. As a result, we have:

n= 1

= -4(2/3)⁽¹⁻¹⁾   =  -4

n= 2

= -4(2/3)⁽²⁻¹⁾   = -8/3

n=3

= -4(2/3)⁽³⁻¹⁾  =   -16/9

n=4

= -4(2/3)⁽⁴⁻¹⁾  = -32/27

b) b) We must locate the common ratio "r" in order to establish whether the series converges or diverges. In this instance, r=2/3, satisfying |r|<1 ⁵ The series converges as a result.

c) The formula for the sum of an infinite geometric series can be used to determine whether the series has a sum.

sum = a/ (1-r) (1-r)

where "r" is the common ratio and "a" is the first term. In this formula, substituting a=-4 and r=2/3 results in:

sum = -4 / (1-2/3)

= -12

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The evaluation of the function gives:

f(-4) = 1, f(1) = 2, f(2) = 6, f(5) = 29/3

A function is an expression that shows the relationship between the independent variable and the dependent variable.  A function is usually denoted by letters such as f, g, etc.

The evaluation is as follow:

For f(-4):

since x = -4. This means x ≤ -4. Thus, we use the function |2x + 7|. That is:

f(-4) = |2*(-4) + 7|

f(-4) = |-8 + 7|

f(-4) = | -1 |

f(-4) = 1

For f(1):

x = 1. Thus, -4< x ≤ 1. Use 1 + x²

f(1) = 1 + x²

f(1) = 1 + 1²

f(1) = 2

For f(2):

x = 2. Thus, 1< x < 3. Use the value 6

f(2) = 6

For f(5):

x = 5. Thus, x ≥ 3. Use (1/3)x + 8.

f(5) = (1/3)*5 + 8

f(5) = 29/3

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Evaluating the function at various values of x over different intervals,

f(-4) = 1, f(1) = 2, f(2) = 6, f(5) = 29/3

To evaluate the function f(x) at various values of x, we need to use the definition of the function for each interval of x.

For x ≤ -4:

f(x) = |2x + 7|

f(-4) = |2(-4) + 7| = |-1| = 1

For -4 < x ≤ 1:

f(x) = 1 + x²

f(1) = 1 + 1² = 2

For 1 < x < 3:

f(x) = 6

f(2) = 6

For x ≥ 3:

f(x) = (1/3)x + 8

f(5) lies in the interval x ≥ 3, so we use the definition of f(x) for this interval:

f(5) = (1/3)(5) + 8 = 29/3

Therefore,

f(-4) = 1

f(1) = 2

f(2) = 6

f(5) = 29/3

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After answering the presented question, we can conclude that r is the equation distance from the origin and is the angle measured anticlockwise from a reference direction.

An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "[tex]2x + 3 = 9[/tex]" asserts that the phrase "[tex]2x + 3[/tex]" equals the value "9." The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. The variable x is raised to the second power in the equation "[tex]x2 + 2x - 3 = 0[/tex]." Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.

We may use two alternative reference points or coordinate systems to find the end of the hiking trail. As an example:

Using a Cartesian coordinate system: Assume the origin is the trailhead and the x-y axis is standard. If the trail's end is 5 units to the right and 7 units up from the trailhead, we may write it as (5, 7).

Using a polar coordinate system: Assume that the origin is defined as the centre of a circular trail and that we use polar coordinates (r,), where r is the distance from the origin and is the angle measured anticlockwise from a reference direction

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Answer:

Step-by-step explanation:

two circles: 2(πr²)

1 rectangle:Lw

3.14(7)(7)

3.14(49)

2(153.86)=307.72

307.72(14)

4308.08

For the given linear equation, the solution is (3,3) i.e. B.

What is a linear equation, exactly?

A linear equation is a mathematical equation in which the highest power of the variable(s) is one. In other words, it is an equation of the form:

y = mx + b

where y and x are variables, m is the slope or gradient of the line, and b is the y-intercept (the point at which the line intersects the y-axis). The slope represents the rate of change of the line, or how steep it is, and the y-intercept represents the value of y when x is zero.

Now,

We can solve this problem by plugging in the x and y values of each ordered pair and seeing if they make the equation true.

Let's start with option A, which is (2,4):

-3x+5y = 2x+3y (original equation)

-3(2) + 5(4) = 2(2) + 3(4) (substituting x=2 and y=4)

-6 + 20 = 4 + 12

14 = 16

The left side of the equation is not equal to the right side, so (2,4) is not a solution.

Now let's move on to option B, which is (3,3):

-3x+5y = 2x+3y (original equation)

-3(3) + 5(3) = 2(3) + 3(3) (substituting x=3 and y=3)

-9 + 15 = 6

6 = 6

The left side of the equation is equal to the right side, so (3,3) is a solution.

Therefore, the answer is option B, which is "Only (3,3)".

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The compositions of the given functions are as follows:

hf(x) = 3/(x² - 1)

fh(x) = (9 - x²)/x²

A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X.

The sets X and Y are collectively referred to as the function's domain and codomain, respectively.

A mathematical phrase, rule, or law establishes the link between an independent variable and a dependent variable (the dependent variable).

So, we have:

h(x) =3/x, x and f(x) = x^2 - 1

Now, solve as follows:
h(x) = 3/x

f(x) = x² - 1

h(f(x)) = 3/f(x)

hf(x) = 3/(x² - 1)

f(h(x)) = (h(x))² - 1

= (3/x)² - 1

= 9/x² - 1

fh(x) = (9 - x²)/x²

Therefore, the compositions of the given functions are as follows:

hf(x) = 3/(x² - 1)

fh(x) = (9 - x²)/x²

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Correct question:

Suppose that the functions h and f are defined as follows. h(x) =3/x, x notequalto 0 f(x) = x^2 - 1 Find the compositions h composite function h and f composite function f Simplify your answers as much as possible. (Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain.)

The probability of getting P(sum > 9) when a cube is rolled twice is 6/36.

Mathematical study of random occurrences and their results is the subject of probability theory. It entails analysing and quantifying risk, chance, and uncertainty. For modelling and studying complex systems and events in disciplines like physics, engineering, economics, and finance, probability theory offers a framework. Concepts like random variables, probability distributions, expected values, variance, and covariance are all part of the theory. In many aspects of daily life, including forecasting weather patterns, data analysis for scientific study, and calculating the likelihood of certain outcomes in games of chance, probability theory is applied.

The total number of outcomes for a 6 sided cube rolled twice is 6 x 6 = 36.

For a sum greater than 9 the possibilities are:

(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6) = 6 possibilities.

Thus,

P(sum > 9) = 6 / 36.

Hence, the probability of getting P(sum > 9) when a cube is rolled twice is 6/36.

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answer = (3x - 5) (3x + 3).

simplified = 3(x + 1) (3x - 5).

An equation that shoes the total amount paid y if they played x number of games is y = 2x + 5

In mathematics, an equation is-a-statement that shows the equality between two-expressions. An equation typically has one or more unknowns that need to be solved for in order to satisfy the equality.

Here, y = total amount paid

x = number of games

Each game costs $2

So total cost of all games played = $2x

But to enter the game they need to rent shoes $5.

So, total amount paid = $5 + $2x

Thus, An equation that shoes the total amount paid y if they played x number of games is y = 2x + 5

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Answer:

the GCF is 5y and the factored form is 5y(2y^4 + 6y^2 - 3).

Step-by-step explanation:

The greatest common factor (GCF) of the terms 10y^5, 30y^3, and -15y is 5y.

To factor out the GCF, we can divide each term by 5y:

10y^5 / (5y) = 2y^4

30y^3 / (5y) = 6y^2

-15y / (5y) = -3

So we have:

10y^5 + 30y^3 - 15y = 5y(2y^4 + 6y^2 - 3)

Therefore, the GCF is 5y and the factored form is 5y(2y^4 + 6y^2 - 3).

Answer: GCF=5

Factored form= 5y(2y^4+6y^2-3

Step-by-step explanation:

Answer:

  145°

Step-by-step explanation:

You want the exterior angle of a triangle that has remote interior angles with measures 115° and 30°.

The measure of an exterior angle of a triangle is equal to the sum of the remote interior angles:

  ∠1 = 115° +30°

  ∠1 = 145°

__

Additional comment

You can see how this works if you consider the unmarked angle in the triangle to be "u", then its measure satisfies two sums: one is the sum of the interior angles; the other is the sum of angles of a linear pair.

  115° +30° +u = 180*

  ∠1 +u = 180°

Combining these equations, we have ...

  ∠1 +u = 115° +30° +u

If we subtract u from this equation, we have ...

  ∠1 = 115° +30°

Answer:

145°

Step-by-step explanation:

? + 30 + 115 = 180

? = 180 - 145

? = 35

180 - 35 = 145

m< 1 = 145°

Answer:no

Step-by-step explanation:

for example Exponents are numbers that have been multiplied by themselves. For instance, 3 · 3 · 3 · 3 could be written as the exponent 34

For x-intercept, [tex]y=0[/tex]

[tex]\implies b-x=0[/tex]

[tex]\implies b=6[/tex]

For vertical intercept [tex]y\rightarrow \pm \infty[/tex]

[tex]\implies cx+d=0[/tex]

[tex]\implies-3x+d=0[/tex]

[tex]\implies d=3c[/tex]

Now, [tex]y=\dfrac{6-x}{cx+3c}[/tex]

[tex]\implies x(xy+1)=6-3cy[/tex]

[tex]\implies x=\dfrac{6-3xy}{cy+1}[/tex]

At horizontal asymptote, [tex]x\rightarrow\pm\infty\implies cy+1=0[/tex]

[tex]\implies c(-1)+1=0[/tex]

[tex]\implies c=1[/tex]

[tex]\implies d=3[/tex].

Answer:

-720

Step-by-step explanation:

Simplify the two negatives with positive:

120 * 6 * -1

Multiply 6 and -1 with -6:

120 * -6

= -720

The output of g(x) when x is -2 is 1/3.

How to find output?

To find the output of g(x) when x is -2, we simply substitute -2 into the formula for g(x):

g(-2) = 3⁻²/²

Simplifying the exponent, we get:

g(-2) = 3⁻¹

Now, we need to evaluate 3⁻¹. Recall that any number raised to a negative exponent can be written as its reciprocal raised to the corresponding positive exponent. That is:

a⁻ⁿ = 1/(aⁿ)

In our case, a is 3 and n is 1, so we have:

3⁻¹= 1/(3¹) = 1/3

Therefore, the output of g(x) when x is -2 is 1/3.

Note that 1/3 is already in its simplest form, so we do not need to simplify it any further. We also do not need to round it to two decimal places since it is a fraction.

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After multiplying the y-value of each point by 1/2, The new location of the blue point would be at x = 2, y = 1.

To apply the given transformation to the points in the graph using the cube root function, we would need to take the cube root of the y-value of each point and then multiply it by 1/2.

Then, we can drag the blue point to its new location based on the transformed coordinates.

For example, if we consider the point (2, 8) in the original graph, the transformed point would be:

(2, 8*(1/3) * 1/2) = (2, 1)

So, the new location of the blue point would be at x = 2, y = 1.

Similarly, we can apply this transformation to all the other points in the graph.

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the answer is that Sophia, Lexi, and Jason collected a total of 162 bananas.

In this problem, we are given that Sophia picked 18 bananas. We are also told that Lexi picked 6 more bananas than Sophia, so Lexi picked 18+6=24 bananas. Additionally, we are told that Jason picked 5 times as many bananas as Lexi, so we need to multiply Jason picked 5*24=120 bananas.

To find the total number of bananas picked by all three people, we need to add up the number of bananas picked by Sophia, Lexi, and Jason. So, we add 18 + 24 + 120 = 162 bananas.

Therefore, the answer is that Sophia, Lexi, and Jason collected a total of 162 bananas.

It's important to understand the relationships between the numbers given in the problem in order to solve it. Sophia is the starting point, with 18 bananas, and Lexi is picking 6 more bananas than her. Jason's number is more complex, being 5 times Lexi's amount. By breaking down each piece of information and understanding how they relate to each other, we can easily find the total number of bananas collected by all three.

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Answer:

Step-by-step explanation:

If you substitute x = 7, then the reasoning is correct, but if you think that x = 7 should turn out at the end (answer), then this is not correct, because it turns out [tex]\frac{19}{3}[/tex]

A(n) asymptote is a line that a graphed curve gets closer to but never touches as the value of a variable gets extremely small or extremely large. Asymptotes can be horizontal, vertical, or slanted, and they are often used to describe the behavior of functions or graphs as they approach certain limits or values.

Answer:

A(n) asymptote is a line that a graphed curve gets closer to but never touches as the value of a variable gets extremely small or extremely large. Asymptotes can be horizontal, vertical, or slanted, and they are often used to describe the behavior of functions or graphs as they approach certain limits or values.

Terry and Ricky traveled 35% of the total distance on the first day of their canoe trip.

Terry and Ricky went on a 60-mile canoe trip with their class, and on the first day, they traveled 21 miles. To find the percent of the total distance they traveled on the first day, follow these steps:
Divide the distance traveled on the first day (21 miles) by the total distance of the trip (60 miles).
Multiply the result by 100 to convert it into a percentage.
Step-by-step explanation:
Divide 21 miles by 60 miles:
  21 / 60 = 0.35
Convert the result into a percentage by multiplying by 100:
  0.35 x 100 = 35%

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