A rectangle has a width that is 6 inches less than its length. The perimeter of the rectangle is 72 inches. What is the length?

Answer:

The answer is 15 nickels and 13 quarters\

Step-by-step explanation:

Answer:

[tex]tan(x)=\frac{4}{3}[/tex]

Step-by-step explanation:

In the unit circle,

- [tex]cos(a)=\frac{x}{r}[/tex] where [tex]a[/tex] is the degree measure, [tex]x[/tex] is the x-coordinate of the triangle, and [tex]r[/tex] is the radius of the circle

- [tex]sin(a)=\frac{y}{r}[/tex] where [tex]a[/tex] is the degree measure, [tex]y[/tex] is the y-coordinate of the triangle, and [tex]r[/tex] is the radius of the circle

Thus, since tangent is equal to sine over cosine, we can simplify our knowledge to:  [tex]tan(a)=\frac{sin(a)}{cos(a)}=\frac{y}{x}[/tex]

In this problem, [tex]sin(x)=\frac{4}{5}[/tex]. We can conclude from our previous knowledge that [tex]y=4[/tex] and the radius is 5.

Similarly, [tex]cos(x)=\frac{3}{5}[/tex], which means [tex]x=3[/tex] and the radius is the same, at 5.

Since we know that [tex]x=3[/tex] and [tex]y=4[/tex], we can find the value of  [tex]tan(x)[/tex] by using the formula [tex]tan(x)=\frac{y}{x}[/tex] and plug in the numbers.

Therefore, [tex]tan(x)=\frac{4}{3}[/tex].

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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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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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As a result, there is a 26% chance that two red marbles will be chosen at random, or around 0.26.

The area of mathematics known as probability is concerned with analysing the results of random events. It represents a probability or likelihood that a specific occurrence will occur. A number in 0 and 1 is used to represent probability, with 0 denoting an event's impossibility and 1 denoting its certainty. In order to produce predictions and guide decision-making, probability is employed in a variety of disciplines, such science, finance, economics, architecture, and statistics.

given

Given that there are 12 red marbles and a total of 24 marbles in the jar, the likelihood of choosing the first red marble is 12/24.

There are 11 red marbles and a total of 23 marbles in the jar after choosing the first red marble.

As a result, the likelihood of choosing a second red marble is 11/23.

We compound the probabilities to determine the likelihood of both outcomes occurring simultaneously (i.e., choosing two red marbles):

P(choosing 2 red marbles) = (12/24) x (11/23) = 0.2609, which is roughly 0.26.

As a result, there is a 26% chance that two red marbles will be chosen at random, or around 0.26.

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Answer:

y > x² – 2x – 3

y > 3x + 4

Step-by-step explanation:

I took the test;. hoped this help.

(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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Answer:

{0, 1, 2}

Step-by-step explanation:

4x<8x+2

-4x<2

x<-1/2

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

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.

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

The polynomial expression (−4b² + 8b) + (−4b³ + 5b² – 8b) when evaluated is −4b³ + b²

We can start by combining like terms.

The first set of parentheses has two terms: -4b² and 8b. The second set of parentheses also has three terms: -4b³, 5b², and -8b.

So we can first combine the like terms in the set of parentheses:

(−4b² + 8b) + (−4b³ + 5b² – 8b) = −4b³ + b²

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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:

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

Check the picture below.

[tex]\cfrac{3^2}{4^2}=\cfrac{27}{A}\implies \cfrac{9}{16}=\cfrac{27}{A}\implies 9A=432\implies A=\cfrac{432}{9}\implies A=48[/tex]

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: 16.5 gallons of water.

Step-by-step explanation:

If it was me. I would be setting up as a table to keep my work organized.

So first we find how much 1 minute is.

15g : 10m

15/10 : 10m/10

1.5g : 1m

Then I multiply how many minutes there are.

1.5g x 11 : 1m x 1

16.5g : 11m

And there we find the answer of 16.5 gallons.

Happy Solving

Answer:16.5

Step-by-step explanation:

Answer:

3r^2+10r+3

Step-by-step explanation:

Answer:

O = { -4,-3,-2,-1,0,-1 }

Answer:

a) To find the probability that X1 is less than or equal to 0.5 and X2 is greater than or equal to 0.25, we need to integrate the given density function over the region where X1 ≤ 0.5 and X2 ≥ 0.25.

P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫∫(x1,x2) f(x1,x2) dxdy

where the limits of integration are:

0.25 ≤ x2 ≤ 1

0 ≤ x1 ≤ 0.5

Substituting the given density function:

P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫0.25^1 ∫0^0.5 (x1 + x2) dx1 dx2

Evaluating the inner integral:

P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫0.25^1 [(x1^2/2) + x1x2] |0 to 0.5 dx2

Simplifying the expression:

P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫0.25^1 [(0.125 + 0.25x2)] dx2

Evaluating the upper and lower limits:

P(X1 ≤ 0.5, X2 ≥ 0.25) = [0.125x2 + 0.125x2^2] |0.25 to 1

Substituting the limits:

P(X1 ≤ 0.5, X2 ≥ 0.25) = [(0.125 + 0.125) - (0.03125 + 0.015625)]

Solving for the final answer:

P(X1 ≤ 0.5, X2 ≥ 0.25) = 21/64

Therefore, the probability that X1 is less than or equal to 0.5 and X2 is greater than or equal to 0.25 is 21/64.

b) To find the probability that X1 + X2 is less than or equal to 1, we need to integrate the given density function over the region where X1 + X2 ≤ 1.

P(X1 + X2 ≤ 1) = ∫∫(x1,x2) f(x1,x2) dxdy

where the limits of integration are:

0 ≤ x1 ≤ 1

0 ≤ x2 ≤ 1-x1

Substituting the given density function:

P(X1 + X2 ≤ 1) = ∫0^1 ∫0^(1-x1) (x1 + x2) dx2 dx1

Evaluating the inner integral:

P(X1 + X2 ≤ 1) = ∫0^1 [(x1x2 + 0.5x2^2)] |0 to (1-x1) dx1

Simplifying the expression:

P(X1 + X2 ≤ 1) = ∫0^1 [(x1 - x1^2)/2 + (1-x1)^3/6] dx1

Evaluating the integral:

P(X1 + X2 ≤ 1) = [x1^2/4 - x1^3/6 - (1-x1)^4/24] |0 to 1

Substituting the limits:

P(X1 + X2 ≤ 1) = (1/4 - 1/6 - 1/24) - (0/4 - 0/6 - 1/24)

Solving for the final answer:

P(X1 + X2 ≤ 1) = 1/8

Therefore, the probability that X1 + X2 is less than or equal to 1 is 1/8.

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]

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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Answer: Only x=7

Step-by-step explanation:

The probability that the mean of a sample of 90 computers would be greater than 117.13 months, if the quality control manager is correct, is approximately 0.9955 or 99.55%.

The sampling distribution of the sample mean follows a normal distribution with a mean of 120 and a standard deviation of 10/sqrt(90) = 1.0541 months (using the formula for the standard deviation of the sample mean).

To find the probability that the mean of a sample of 90 computers would be greater than 117.13 months, we can standardize the sample mean using the formula:

z = (sample mean - population mean) / (standard deviation of sample mean) = (117.13 - 120) / 1.0541 = -2.6089

Using a standard normal distribution table or calculator, we can find that the probability of obtaining a z-score greater than -2.6089 is approximately 0.9955.

Therefore, the probability that the mean of a sample of 90 computers would be greater than 117.13 months, if the quality control manager is correct, is approximately 0.9955 or 99.55%.

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Answer:

yes.

Step-by-step explanation:

cause yes.

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.

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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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:

136 units

Step-by-step explanation:

All sides are equal in a rectangle:

Value of b : 16-8 = 8 units

h = 13-7 = 6 units.

So Area of triangle= bh/2 = 8*6/2 = 24 units

Area of rectangle = lb = 16*7 = 112 units

So Area of figure= 112+24 units = 136 units