8₁
6) A triangular roof is built so that its height
is half its base, If the base of the roof is 32
feet long, what is the area of the roof? Show some work

(A)  the annual rate of change between 1992 and 2006 was 0.0804

(B) r = 0.0804 * 100% = 8.04%

(C) value in 2009 = $11,650

What is the rate of change?

The rate of change is a mathematical concept that measures how much one quantity changes with respect to a change in another quantity. It is the ratio of the change in the output value of a function to the change in the input value of the function. It describes how fast or slow a variable is changing over time or distance.

A) The initial value is $44,000 and the final value is $15,000. The time elapsed is 2006 - 1992 = 14 years.

Using the formula for an annual rate of change (r):

final value = initial value * [tex](1 - r)^t[/tex]

where t is the number of years and r is the annual rate of change expressed as a decimal.

Substituting the given values, we get:

$15,000 = $44,000 * (1 - r)¹⁴

Solving for r, we get:

r = 0.0804

So, the annual rate of change between 1992 and 2006 was 0.0804 or approximately 0.0804.

B) To express the rate of change in percentage form, we need to multiply by 100 and add a percent sign:

r = 0.0804 * 100% = 8.04%

C) Assuming the car value continues to drop by the same percentage, we can use the same formula as before to find the value in the year 2009. The time elapsed from 2006 to 2009 is 3 years.

Substituting the known values, we get:

value in 2009 = $15,000 * (1 - 0.0804)³

value in 2009 = $11,628.40

Rounding to the nearest $50, we get:

value in 2009 = $11,650

Hence, (A)  the annual rate of change between 1992 and 2006 was 0.0804

(B) r = 0.0804 * 100% = 8.04%

(C) value in 2009 = $11,650

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The equatiοn οf the circle is (x + 6)(x+6) + (y + 2)(y+2) = 9.

A circle is a twο-dimensiοnal geοmetric figure that cοnsists οf all pοints that are equidistant frοm a single fixed pοint called the center. A circle can alsο be defined as the lοcus οf a pοint that mοves in a plane in such a way that its distance frοm a fixed pοint is always cοnstant.

Tο find the equatiοn οf a circle, we need the cοοrdinates οf the center and the radius.

The center οf the circle is given as (-6, -2), sο the cοοrdinates οf the center are (h, k) = (-6, -2).

The pοint (-9, -2) is οn the circle, sο its distance frοm the centre is equal tο the radius. We can use the distance fοrmula tο find the radius:

[tex]\rm r = \sqrt{((x_2 - x_1)\times (x_2 - x1) + (y_2 - y_1)\times(y_2 - y_1))}[/tex]

[tex]= \sqrt{((-3)^2 + 0^2)[/tex]

= 3

Therefοre, the radius οf the circle is 3.

Nοw we can use the standard fοrm οf the equatiοn οf a circle, which is:

(x - h)(x-h)+ (y - k)(y-k)= r*r

Substituting the values we fοund, we get:

Simplifying:

(x + 6)(x+6) + (y + 2)(y+2) = 9

Therefοre, The equatiοn οf the circle is [tex](x + 6)(x+6) + (y + 2)(y+2) = 9[/tex].

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The probability of selecting a divisor of 6 is 66.7%.

It is expressed as a number between 0 and 1, where 0 represents an impossible event (it will never occur) and 1 represents a certain event (it will always occur).

1, 2, 3, and 6 can be divided by 6.

To find the probability (p) of selecting a divisor of 6, we need to divide the number of divisors of 6 by the total number of possible outcomes, which is also 6 (since there are 6 positive integers from 1 to 6).

So, p(divisor of 6) = number of divisors of 6 / total number of outcomes

= 4 / 6

= 2 / 3

We can multiply this fraction by 100 to get the percentage:

p(divisor of 6) = 2 / 3 * 100

= 66.7%

Rounded to the nearest tenth, the answer is 66.7%. Therefore, the probability of selecting a divisor of 6 is 66.7%.

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Answer:

1025

Step-by-step explanation

The formula to find the sum of the first n terms of an arithmetic sequence is

Sn = n/2 * [2a1 + (n-1)d]

Where

a1 = the first term of the sequence

d = the common difference between consecutive terms

n = the number of terms we want to sum

Substituting the given values,  we get

a1 = 5

d = 3

n = 25

S25 = 25/2 * [2(5) + (25-1)3]

= 25/2 * [10 + 72]

= 25/2 * 82

= 25 * 41

= 1025

Answer:

33 - 24x

Step-by-step explanation:

a. He made mistake here

9+(4)(5x+-6)

b.

9 - 4(5x - 6)

= 9 + (- 4)(5x - 6)

= 9 + (- 4)(5x) - (- 4)(6)

= 9 + (- 20x) - (- 24)

= 9 - 20x + 24

= 9 + 24 - 24x

= 33 - 24x

P(X >= 15) ≈ 0.271 is an unusual event would be one that has a very low probability of occurring (e.g., less than 5%).

What is probability?

Probability is a numerical measurement that represents the likelihood or chance of an event happening. The value of probability always lies between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain to occur.

a) To find the probability that at least 15 out of 18 U.S. college graduates expect to stay at their first employer for three or more years, we can also use the CDF of the binomial distribution:

P(X >= 15) = 1 - P(X < 15)

Using a calculator or statistical software, we find:

P(X >= 15) ≈ 0.271

An unusual event would be one that has a very low probability of occurring (e.g., less than 5%). Therefore, these outcomes could be considered unusual.

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Answer:

We can start by simplifying the expression inside the parentheses using the identity:

cos 2x = 2 cos² x - 1

Substituting this in, we get:

1 – 3 cos 2x + 3 cos² 2x − cos³ 2x

= 1 – 3(2 cos² x - 1) + 3(2 cos² x - 1)² − (2 cos² x - 1)³

= 1 – 6 cos² x + 9 cos⁴ x - 4 cos⁶ x

Therefore, we can rewrite f(x) as:

f(x) = 4 cos x (1 – 6 cos² x + 9 cos⁴ x - 4 cos⁶ x)

Next, we can use the trigonometric identity:

sin 2x = 2 cos x sin x

to express cos x in terms of sin x:

cos x = √(1 - sin² x)

Substituting this in, we get:

f(x) = 4 sin x cos³ x (1 – 6 cos² x + 9 cos⁴ x - 4 cos⁶ x)

= 4 sin x (√(1 - sin² x))³ (1 – 6 (2 sin² x - 1) + 9 (2 sin² x - 1)² - 4 (2 sin² x - 1)³)

= 4 sin x (1 - sin² x)^(3/2) (16 sin⁶ x - 48 sin⁴ x + 36 sin² x - 8)

Next, we can use the substitution u = 1 - sin² x, du = -2 sin x cos x dx, to obtain:

f(x) dx = -2 du (u^(3/2)) (16 - 48u + 36u² - 8u³)

Integrating, we get:

f(x) dx = 2/3 (1 - sin² x)^(5/2) (8 - 36(1 - sin² x) + 36(1 - sin² x)² - 8(1 - sin² x)³) + C

Now, we can use the trigonometric identity:

sin² x = (1 - cos 2x)/2

to simplify the expression inside the parentheses. After some algebra, we obtain:

f(x) dx = 3/2 sin 7x + C

where C is the constant of integration. Since m is a positive real constant, we can set:

7x = m

and solve for x:

x = m/7

Substituting this in, we get:

f(x) dx = 3/2 sin(7m/7) = 3/2 sin m

Therefore, we have shown that:

f(x) dx = 3/2 sin m, where m is a positive real constant.

Answer:

Step-by-step explanation:

Given the figure with triangle ABC similar to triangle EDC and AB=8 mi, ED=2 mi, BD = 7.5 mi, you want the measures of CD and AE.

Similar triangles will have corresponding sides proportional. That means ...

  ED/CD = AB/CB

  2/CD = 8/(7.5 -CD)

Inverting the ratios and multiplying by 8 gives ...

  4·CD = 7.5 -CD

  5·CD = 7.5 . . . . . . . add CD

  CD = 1.5 . . . . . . . . . divide by 5

The distance AE is the hypotenuse of a right triangle with side lengths 7.5 and (8+2) = 10. The Pythagorean theorem can be used to find AE:

  AE² = 7.5² +10² = 56.25 +100 = 156.25

  AE = √156.25 = 12.5

AE = 12.5 miles, the distance to the mall.

__

Additional comment

You may recognize these triangles are 3-4-5 triangles. ABC has a scale factor of 2, so has side lengths 6-8-10. EDC has a scale factor of 1/2, so has side lengths 1.5, 2, 2.5. The triangle with AE as its hypotenuse is the sum of these, so has a scale factor of 2.5 (miles).

  AE = (2.5 miles) · 5 = 12.5 miles

When the x-values rise while the y-values fall, this is known as a declining pattern. So, as x increases from 3 to 6, the graph declines. When the point on the graph at the interval's left end is higher than the interval's right end, the average rate of change will be declining.

An indicator of how much the function changed on average per unit throughout that time is the graph's average rate. In the graph of the function, it is calculated from the slope of the straight line joining the interval's ends. So, by applying the average rate of change formula, the slope of a graphed function is calculated.

Hence divide the y-value change by the x-value change in order to determine the average rate of change. When analyzing changes in observable parameters like average speed or average velocity, finding the average rate of change is extremely helpful.

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The complete question is:

Choose the intervals where the graph has a decreasing average rate of change. The graph is attached below:

x = 0 to x = 13

x = 3 to x = 6

x = 4 to x = 8

x = 6 to x = 10

Answer:

x > 2

Hope this helps!

Step-by-step explanation:

3x - 4 > 2

3x - 4 ( + 4 ) > 2 ( + 4 )

3x > 6

3x ( ÷ 3 ) > 6 ( ÷ 3 )

x > 2

Answer:

[tex]\frac{13}{21}[/tex]

Hope this helps!

Step-by-step explanation:

[tex]\frac{39}{63}[/tex]  ( Simplify both numerator and denominator by 3 )

39 ÷ 3 / 63 ÷ 3

[tex]\frac{13}{21}[/tex]

1) Not a triangle as According to the triangle inequality theorem  ,2)Triangle. as According to the triangle inequality theorem    ,  3)Not a triangle. ,  4)Not a triangle., 5)Triangle. ,  6)Not a triangle.,  7) Not a triangle., 8)Equilateral triangle. 9)Not a triangle   10)  Triangle.

what is  triangle ?

A triangle is a two-dimensional geometric shape that has three sides and three angles. It is one of the basic shapes in geometry, and it is formed by connecting three non-collinear points. The sum of the angles in a triangle is always 180 degrees.

In the given question,

Not a triangle. (6 + 5 = 11 > 3)

Explanation: According to the triangle inequality theorem, the sum of any two sides of a triangle must be greater than the third side. However, in this case, 6 + 5 is equal to 11, which is not greater than the third side of length 3.

Triangle. (5 + 12 > 13)

Explanation: The sum of the two smaller sides (5 and 12) is greater than the largest side (13), satisfying the triangle inequality theorem. Therefore, a triangle can be formed with these side lengths.

Not a triangle. (2 + 2 = 4 > 3)

Explanation: Similar to the first case, the sum of the two smaller sides (2 and 2) is equal to 4, which is not greater than the third side of length 3.

Not a triangle. (1 + 2 = 3 > 4)

Explanation: Again, the sum of the two smaller sides (1 and 2) is equal to 3, which is not greater than the third side of length 4.

Triangle. (6 + 8 > 10)

Explanation: The sum of the two smaller sides (6 and 8) is greater than the largest side (10), satisfying the triangle inequality theorem. Therefore, a triangle can be formed with these side lengths.

Not a triangle. (1 + 1 = 2 > 2)

Explanation: Similar to cases 1 and 3, the sum of the two smaller sides (1 and 1) is equal to 2, which is not greater than the third side of length 2.

Not a triangle. (4 + 5 = 9 > 7)

Explanation: In this case, the sum of the two smaller sides (4 and 5) is greater than 7, but the difference between the two larger sides (7 - 5) is smaller than the smallest side (4), violating the triangle inequality theorem.

Equilateral triangle. (All sides are equal)

Explanation: All sides are equal, satisfying the criteria for an equilateral triangle.

Not a triangle. (1 + 3 = 4 > 5)

Explanation: The sum of the two smaller sides (1 and 3) is greater than the largest side (5), but the difference between the two larger sides (5 - 3) is smaller than the smallest side (1), violating the triangle inequality theorem.

Triangle. (3 + 4 > 5)

Explanation: The sum of the two smaller sides (3 and 4) is greater than the largest side (5), satisfying the triangle inequality theorem. Therefore, a triangle can be formed with these side lengths.

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Answer:

98+x=154

x=154-98

x=56

x-4=20

x=20+4

x=24

x+25=-10

x=-10-25

x=-35

Answer:

[tex]\boxed{8-6i}[/tex]

Step-by-step explanation:

First, we developed the square binomial [tex](-3+\mathrm{i})^2[/tex].

[tex]\implies (-3+\mathrm{i})(-3+\mathrm{i})\\9-3\mathrm{i}-3\mathrm{i}+i^2\\9-6\mathrm{i}+\mathrm{i}^2[/tex]

Remember the next product:

[tex]i^2= \mathrm{i} \times \mathrm{i} = -1[/tex]

then:

[tex]9-6\mathrm{i}+ (-1)\\8-6i[/tex]

Hope it helps

[tex]\text{-B$\mathfrak{randon}$VN}[/tex]

Answer:

x-3

Step-by-step explanation:

Answer:

I'm sorry, I cannot create a labeled diagram of the circular fountain in the public park or find its map location in coordinates without more specific information about the park and fountain. However, I can provide some general information about circular fountains.

To find the map location in coordinates of the center of a circular fountain, you would need to know the specific location of the park and fountain. Once you have the location, you can use a mapping tool or website to find the coordinates of the center of the fountain.

To find the distance from the center of the fountain to its circumference, you would need to know the radius of the fountain. Once you have the radius, you can use the formula for the circumference of a circle, which is C = 2πr, where C is the circumference and r is the radius. The distance from the center of the fountain to its circumference is equal to the radius of the fountain.

I hope this information helps. If you have more specific information about the circular fountain in the public park, please let me know and I can try to provide more detailed information.

Answer:

Let Haley be represented as x

Now Mateo has 8 more coins than haley

Mateo = 8 + x

total number of coins is Mateo coins and Haley coins.

x + 8 + x

2x + 8

Answer:

The answer is 15 nickels and 13 quarters\

Step-by-step explanation:

Answer:

y > x² – 2x – 3

y > 3x + 4

Step-by-step explanation:

I took the test;. hoped this help.

Answer:

{0, 1, 2}

Step-by-step explanation:

4x<8x+2

-4x<2

x<-1/2

Only {0, 1, 2} meets the critera.

Our assumption that I U{—A} is satisfiable must be false. Hence, I U{—A} is not satisfiable if I = A.

A set of propositional formulae, sometimes referred to as a propositional theory, can be satisfiable in terms of propositional logic by having the quality of being true or untrue according to a certain interpretation or model. If there is at least one interpretation that makes all of a set of formulae true, the set is said to be satisfiable.

Using the proof by contradiction we have:

Assume that I U{—A} is satisfiable.

Then, by definition of satisfiability, every formula in the set I U{—A} is true in M.

Since I = A, every formula in I is also in A. Therefore, every formula in I is true in M, since A is true in M.

Consider the formula —A, which is in {—A}. Since M satisfies {—A}, —A is true in M.

But this contradicts the fact that A is true in M, since —A is the negation of A.

Therefore, our assumption that I U{—A} is satisfiable must be false. Hence, I U{—A} is not satisfiable if I = A.

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The answer is 0.351

To estimate the proportion p of American adults with monthly credit card debt, the sample proportion can be used as an estimate. The sample ratio is simply the number of people in the sample with monthly credit card debt divided by the total number of people in the sample.

p hat = 587/1669

p-hat = 0.3511 (rounded to four decimal places)

Therefore, based on this sample, the percentage of adult Americans with monthly credit card debt is estimated to be approximately 0.351. After rounding to three decimal places, its estimate is 0.351.

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The amount Angela earned this month is $2,618.

Percentage can be described as a fraction of an amount expressed as a number out of hundred.

Angela's earnings = percentage commission x worth of goods sold

[tex]11\% \times 23,800[/tex]

[tex]0.11 \times 23,800 = \bold{\$2618}[/tex]

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The length of the line of sight from the plane to the base of the water tower is approximately 19298 feet.

The length of the line of sight from the plane to the base of the water tower can be determined using trigonometry. We can use the tangent function, which relates the opposite side of a right triangle (in this case, the height of the water tower) to the adjacent side (the length of the line of sight), to find the length of the line of sight.

First, we can draw a diagram and label the relevant angles and sides:

       |\

       | \

12000 ft|  \ height of tower

       |   \

       |22°\

       -----

Let x be the length of the line of sight. Then, we can use the tangent function:

tan(22°) = height of tower / x

We know the height of the tower is not given, but we can set up a right triangle with the height of the tower as one of the legs and the distance from the tower to the point directly below the plane as the other leg. Since the angle of depression is 22 degrees, the angle between the two legs of the triangle is 90 - 22 = 68 degrees.

Using the trigonometric ratio for the tangent of 68 degrees, we get:

tan(68°) = height of tower/distance from the tower to point below the plane

Solving for the height of the tower, we get:

height of tower = distance from tower to point below the plane x tan(68°)

Substituting this into the first equation, we get:

x = height of tower / tan(22°) = (distance from tower to point below the plane x tan(68°)) / tan(22°)

We don't have any values for the distance or the height of the tower, but we can simplify the expression by noting that the distance from the tower to the point directly below the plane is equal to the length of the line of sight plus the height of the plane above the ground. Assuming the height of the plane is negligible compared to the distance from the tower, we can approximate the distance as just the length of the line of sight:

distance from the tower to the point below the plane ≈ x

Substituting this approximation into the expression for x, we get:

x = x tan(68°) / tan(22°)

Solving for x, we get:

x ≈ 19298 ft

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Therefore, the doubling time is approximately 10.83 years.

a) The exponential function that describes the amount in the account after time t, in years, is given by:

[tex]$A(t) = A_0 e^{rt}$[/tex]

where $A_0$ is the initial investment, $r$ is the annual interest rate as a decimal, and $t$ is the time in years. Since the interest is compounded continuously, we have $r = 0.064$.

Substituting the given values, we get:

[tex]$A(t) = 10,405 e^{0.064t}$[/tex]

b) To find the balance after 1 year, we plug in $t=1$ into the exponential function:

[tex]$A(1) = 10,405 e^{0.064(1)} \approx 11,069.79$[/tex]

Similarly, we can find the balance after 2, 5, and 10 years:

[tex]$A(2) = 10,405 e^{0.064(2)} \approx 11,778.79$[/tex]

[tex]$A(5) = 10,405 e^{0.064(5)} \approx 14,426.77$[/tex]

[tex]$A(10) = 10,405 e^{0.064(10)} \approx 19,682.08$[/tex]

c) The doubling time can be found using the formula:

[tex]$t_{double} = \frac{\ln 2}{r}$[/tex]

Substituting $r = 0.064$, we get:

[tex]$t_{double} = \frac{\ln 2}{0.064} \approx 10.83$ years[/tex]

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Answer:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of the two endpoints.

In this case, (x1, y1) = (4.25, 6.25) and (x2, y2) = (22, 6.25).

Plugging these values into the distance formula, we get:

distance = sqrt((22 - 4.25)^2 + (6.25 - 6.25)^2)

= sqrt(17.75^2 + 0^2)

= sqrt(315.0625)

= 17.75

Therefore, the length of the line segment is 17.75 units.

Answer: Only x=7

Step-by-step explanation:

Therefore, the probability that AT LEAST ONE of them has been vaccinated is 0.98693

If 58% = 58÷100 of the people have been vaccinated, then;

1 - (58/100) = 42% = 42÷100 of the people who have not been vaccinated.

Now:

Probability, P( > 1 ), that at least one of the selected four has been vaccinated is given by;

P( > 1 ) = 1 - P(0)                 -----------(1)

Where;

P(0) = probability that all of the four have not been vaccinated.

P(0) = P(1) x P(2) x P(3) x P(4)×p(5)

Where;

P(1) = Probability that the first out of the four has not been vaccinated

P(2) = Probability that the second out of the four has not been vaccinated

P(3) = Probability that the third out of the four has not been vaccinated

P(4) = Probability that the fourth out of the four has not been vaccinated

P(5)=Probability that the fifth out of the five has not been vaccinated

Remember that 42÷100 of the population has not been vaccinated. Therefore,

P(1) = 42÷100    

P(2) = 42÷100

P(3) = 42÷100

P(4) = 42÷100

P(5)=42÷100

P(0) = (42÷100) x (42÷100) x (42÷100) x (42÷100)×(42÷100)

P(0) = (42÷100)⁵

P(0) = (0.42)⁵

P(0) = 0.013067

Therefore, from equation (1);

P( > 1 ) = 1 -0.013067

P( > 1 ) = 0.98693

Therefore, the probability that AT LEAST ONE of them has been vaccinated is 0.98693

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