Consider the letters in the word SAMPLE. In how many ways can you arrange 5 of the letters?

Expressing the expansion of the logarithm in terms of log y, we can write: log(8x^2) = log(8) + 2*log(y), where y = x

Using the logarithmic identity that states log(a^b) = b*log(a), we can expand log(8x^2) as:

log(8x^2) = log(8) + log(x^2)

Note that a  logarithmic identity is a mathematical relationship or equation that involves logarithms.

We can further simplify log(x^2) as:

log(x^2) = 2*log(x)

Therefore, we can write:

log(8x^2) = log(8) + 2*log(x)

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Answer:

24.6%

Step-by-step explanation:

denote B: bad mood; H: healthy; D: periodontal disease

P(B|D)

=P(B,D) / P(D)

=P(B,D) / [P(D,B)+P(D,H)]          ### law of total probability ###

=P(D|B)P(B) / [P(D|B)P(B) + P(D|H)P(H)]

=(0.85)*(0.1)/[(0.85)*(0.1) + (0.29)*(0.9)]

=0.245664739

Answer:

A) m∠3 + m∠6 = 90

Step-by-step explanation:

Angles 3 and 6 are complementary angles. This means they add up to 90 degrees. Since you know the angle that is a right angle due to the box, you know these angles add up to 90 degrees

The equation that models the amount of water in the tank after x minutes is:

y = -3x + 9

We know that the tank contains 9 liters of water, and it loses water at a constant rate of 3 liters per minute.

Then we can model the amount of water y in the tank after x minutes with a linear equation where the y-intercept is 9 and the rate of change is -3 (the amount it loses per unit of x)

Then the equation is:

y = -3x + 9

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The area of the training field that has the shape of a rectangle and two semicircle would be = 9,755.46m²

To calculate the area of the training field, the area of the rectangle and the two semicircles should be determined and added together.

The area of rectangle = length×width

where:

length = 96m

width = 66m

area of rectangle = 66×96 = 6,336m²

Area of the two semicircle = Area of a circle = πr²

radius = diameter/2 = 66/2 = 33m

area of a circle= 3.14×33×33 = 3419.46m²

Therefore the area of the training field = 6,336m²+3419.46m² = 9,755.46m²

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The length of the Green Route in miles is 64 miles.

Miles is a unit of linear measurement. It is traditionally defined as 5,280 feet, or 1,760 yards, and is equivalent to 1.609 kilometers. Miles are commonly used to measure long distances, such as the distance between two cities. They are also used to measure the lengths of roads and other paths. Miles are one of the most commonly used units of measurement in the United States.

The length of the Green Route in miles can be calculated by subtracting the total mileage traveled on the Blue Route from the total mileage traveled on both routes.

Total miles traveled on Monday = 38 miles
Total miles traveled on Tuesday = 80 miles

Total miles traveled on the Blue Route = 5 + 11 = 16 miles

Therefore, the total miles traveled on the Green Route = 80 - 16 = 64 miles

Therefore, the length of the Green Route in miles is 64 miles.

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The value of the function g(f(-4)) = 536. Option D

From the information given, we have that the functions are;

f(x)= 4x+5

g(x) = 4x^2 -5x - 3

To determine the composite function,  g(f(-4)), we need to first determine the value of the function f(x) at -4

Now, substitute the value, we have;

f(-4) = 4(-4) + 5

expand the bracket, we get;

f(-4) = - 16 + 5

Add the values

f(-4) = -11

Now, substitute the value in the function g(x), we have that;

g(f(-4)) = 4(-11)² - 5(-11) - 3

find the square and expand the bracket

g(f(-4)) =484 + 55 - 3

Subtract the values, we have;

g(f(-4)) = 536

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The x-intercepts of the given function y = -3x(4-x) are x = 0 and x = 4.

As we know that the x-intercept is defined as an intercept that is located at the x-axis of the plane, is the location or coordinate from where the line crosses

To determine the x-intercepts of a function, we need to set y=0 and solve for x.

In this case, we have:

y = -3x(4-x)

0 = -3x(4-x)

0 = 3x(x-4)

Therefore, the x-intercepts occur when either 3x=0 or x-4=0.

Solving for x, we get:

3x = 0, x = 0

x - 4 = 0, x = 4

Therefore, the x-intercepts of y=-3x(4-x) are x=0 and x=4.

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first off, let's find out how much he's making on the 2nd year, so since he's getting slashed by 20%, that means his new commission is 100% - 20% = 80%, so 80% of 60000, how much is that?

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{80\% of 60000}}{\left( \cfrac{80}{100} \right)60000}\implies 48000[/tex]

now, if we want to go up to 57600, that means we need to increase his commission by 57600 - 48000 = 9600.

So, if we take 48000(origin amount) to be the 100%, what's 9600 off of it in percentage?

[tex]\begin{array}{ccll} Amount&\%\\ \cline{1-2} 48000 & 100\\ 9600& x \end{array} \implies \cfrac{48000}{9600}~~=~~\cfrac{100}{x} \\\\\\ 5 ~~=~~ \cfrac{ 100 }{ x }\implies 5x=100\implies x=\cfrac{100}{5}\implies \boxed{x=20}[/tex]

The expression 0 + 10 = 10 is an examle of the identity property of 0

From the question, we have the following parameters that can be used in our computation:

The list of options

The identity property of 0 states that

A number added to 0 equals to the number

Mathematically, we have

a + 0 = 0

The options are not clear

So, I will give another example of this property

The expression 0 + 10 = 10 is an examle of the identity property of 0

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The x-coordinate of this point is -1, so cos(5π) = cos(900 degrees) = -1.

The y-coordinate of this point is -1, so sin(-9π/2) = sin(-810 degrees) = -1.

The y-coordinate of this point is 1, so sin(183π/2) = sin(16,470 degrees) = 1.

It should be noted that go find cos(5π), we first need to convert 5π to degrees. We know that π radians is equal to 180 degrees, so 5π radians is equal to 5π × (180/π) = 900 degrees. The x-coordinate of this point is -1, so cos(5π) = cos(900 degrees) = -1.

Also, to find sin(-9π/2), we first need to convert -9π/2 to degrees. We know that π radians is equal to 180 degrees, so -9π/2 radians is equal to -9π/2 × (180/π) = -810 degrees. The y-coordinate of this point is -1, so sin(-9π/2) = sin(-810 degrees) = -1.

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Answer:

[tex]x^2 + 10x+25[/tex]

Step-by-step explanation:

(x + 5)^2 is the same as (x + 5)(x + 5), which is what you call a binomial expression, since two terms (x + 5) are being multiplied by each other.  You can multiply any binomial expression using the FOIL method, where you multiply the first (F), outer (O), inner (I), and last terms and simplify at the end:

First terms:  x * x

Outer terms:  x * 5

Inner terms:  5 * x

Last terms:  5 * 5

Simplifying:

[tex](x * x) + (x * 5) + (5 * x) + (5 * 5)\\x^2+5x+5x+25\\x^2+10x+25[/tex]

Answer:

The heights and shadows of the objects are legs of similar right triangles. We can use these similar triangles to set up a proportion to solve for the height of the building.

2.9 / 1.26 = x / 47.25

x = 108.75 m

The range of the monthly rainfall data for the cities is approximately 7 centimeters.

To find the range of the data from the given box plot, we need to determine the highest and lowest values of the dataset. Looking at the box plot, we can see that the "H" represents the high outlier and the "□" represents the low outlier. The vertical line at the bottom of the box represents the minimum value and the vertical line at the top of the box represents the maximum value of the dataset within the interquartile range.

Using the scale on the x-axis, we can estimate the minimum value to be around 0.5 centimeters and the maximum value to be around 7.5 centimeters. The high outlier is located at around 8 centimeters and the low outlier is located at around 0 centimeters.

Therefore, the range of the data is the difference between the highest and lowest values, which is:

range = maximum value - minimum value = 7.5 - 0.5 = 7 centimeters

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Answer:

Step-by-step explanation:To solve this problem, we need to work backwards from the final price that Gina paid and determine the discounted price before tax was added.

Let x be the original price of the laptop. After a 25% discount, the discounted price is:

Discounted price = x - 0.25x = 0.75x

Next, we add 7% tax to the discounted price:

Final price = (0.75x) + 0.07(0.75x) = 0.8025x

We know that the final price that Gina paid was $963, so we can set up an equation:

0.8025x = 963

Solving for x, we get:

x = 1200

Therefore, the original price of the laptop was $1200, and the discounted price before tax was:

Discounted price = 0.75x = 0.75(1200) = $900

To check, we can add the 7% tax to the discounted price:

Final price = $900 + 0.07($900) = $963

This matches the price that Gina paid, so we can be confident in our answer.

Answer:0,2,4,6,8,10

Step-by-step explanation:

take it, spread it, find the bars.

The solution is,

there are children = 12, teenagers= 66, adults = 102,

so, adults = 102 is the largest group represented

Here, we have,

given that,

The ratio of children: teenagers: adults in a survey sample are 2: 11:17.

now, we have,

If there are 180 people surveyed

now, let , we have,

children = 2x

teenagers= 11x

adults = 17x

now, we have,

2x + 11x + 17x = 180

or, 30x = 180

or. x = 6

Hence, The solution is,

there are children = 12, teenagers= 66, adults = 102,

so, adults = 102 is the largest group represented.

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When a number is selected at random, the probability that a number that is not a prime number would be selected would be = 3/7.

Prime numbers are defined as those numbers that are greater than one that cannot be divided by a whole number apart from themselves without a remainder.

To calculate the probability of non prime numbers from a game of cards the formula for probability should be used. That is;

Probability = possible outcome/sample space

The possible outcome = 1,4,6 = 3

The sample space = 7

Therefore probability = 3/7

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The sum expression [tex][2(4 + \frac9n)^4](\frac9n) + ... + [2(4 + \frac{9n}n)^4](\frac9n)[/tex] using the sigma notation is [tex]\sum\limits^{n}_{i=1} [2(4 + \frac{9i}n)^4](\frac9n)[/tex]

From the question, we have the following parameters that can be used in our computation:

[tex][2(4 + \frac9n)^4](\frac9n) + ... + [2(4 + \frac{9n}n)^4](\frac9n)[/tex]

From the above expression, we can see that the different expression in each term is

9/n

So, we introduce a variable

Assuming the variable is i, the i-th term of the expression would be[tex]t(i) = [2(4 + \frac{9i}n)^4](\frac9n)[/tex]

When represented using the sigma notation, we have

[tex]\sum\limits^{n}_{i=1} [2(4 + \frac{9i}n)^4](\frac9n)[/tex]

Hence, the sum expression using the sigma notation is [tex]\sum\limits^{n}_{i=1} [2(4 + \frac{9i}n)^4](\frac9n)[/tex]

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Roger and Trish can complete 11/30 of the math homework in 1 hour if they work together.

What is an expression?

An expression is a sentence that has at least two numbers/variables and at least one math operation.

According to the given information:

Let the total amount of work in the math homework is 1.

In one hour, Roger can finish 1/6 of the work, and Trish can finish 1/5 of the work.

If they work together for 1 hour, then the total amount of homework they can finish is the sum of the parts each of them can finish in one hour:

1/6 + 1/5 = 5/30 + 6/30 = 11/30

So together they can finish 11/30 of the homework in one hour.

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The value of the "trigonometric-expressions' are :

(a) tan(5π/2) is undefined,

(b) sec(315°) is √2.

Part(a) : The expression "tan(5π/2)" is undefined because the tangent function is undefined at odd multiples of π/2, which includes 5π/2.

Part(b) : To find the value of sec(315°), we use the identity: sec(θ) = 1/cos(θ),

First, we convert 315° to radians:

Which is : 315° = (7π/4) radians;

Next, we find the value of cos(7π/4):

⇒ cos(7π/4) = cos(π/4) = (√2)/2;

Finally, we substitute in the identity for sec(θ);

So, Sec(7π/4) = 1/cos(7π/4) = 1/[(√2)/2] = √2.

Therefore, value of sec(315°) is √2.

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The given question is incomplete, the complete question is

Find the value of the trigonometric expressions :

(a) tan(5π/2)

(b) sec(315°)

Answer:

f'(x) = 2x, so f'(-2) = 2(-2) = -4.

Answer:

First problem: x = 5.7√2 = (57√2)/10

Second problem: x = 10√3

Answer:

  (x -2)(x -4)

Step-by-step explanation:

You want to factor x² - 4x -2x +8.

We recognize that the product of the coefficients of the two linear terms is equal to the contant, so this is more easily factored by not combining like terms prior to factoring.

Grouping the terms in pairs, we find we can factor each pair:

  (x² -4x) + (-2x +8)

  = x(x -4) -2(x -4) . . . . . these terms have a common factor of (x -4)

  = (x -2)(x -4) . . . . . . . factored form of the expression

The maximum horsepower of the ice cream truck engine is 827.025.

For quadratic equation y = ax² + bx + c, the maximum value of y is given by:

y(max) =  c- b²/4a

Since the horsepower generated by the ice cream truck engine power curve is approximated by Y = -X²/5000 + 19x/23 -26

In this case, a = -1/5000, b = 19/23 and c = -26

Thus, the maximum horsepower will be:

maximum horsepower =  -26 - (19/23)²/[4*(-1/5000)]

maximum horsepower = 827.025 horsepower

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The surface area of the cube is 384 mm².

A cube is a solid three-dimensional figure, which has 6 square faces, 8 vertices and 12 edges.

Since there are six square that makes up a cube, to find the surface area of the cube, we find the area of one of the square and multiply by six

Where:

From the question,

Given:

Substitute this value into equation 1

Hence the right option is D. 384 mm².

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The experimental probability of spinning a 2 on Olivia's spinner is 0.1667.

The experimental probability of spinning a 2 on Jackie's spinner is 0.02 or 2%.

Olivia and Jackie are playing a game with a spinner. Olivia's spinner has 12 equal sections, two of which are marked as "funny 2". Jackie's spinner has 100 equal sections, and two of them are marked as "fun 2".

To find the experimental probability of spinning a 2 in Olivia's spinner, we first need to determine the total number of possible outcomes, which is the number of sections on the spinner. In this case, the spinner has 12 sections. Out of these, two sections are marked as "funny 2". Therefore, the number of favorable outcomes is 2.

Experimental probability of spinning a 2 in Olivia's spinner can be calculated as Experimental probability = Number of favorable outcomes / Total number of possible outcomes

= 2 / 12

= 1 / 6

= 0.1667

Therefore, the experimental probability of spinning a 2 on Olivia's spinner is 0.1667.

To find the experimental probability of spinning a 2 in Jackie's spinner, the total number of possible outcomes is the number of sections on the spinner, which is 100. Out of these, two sections are marked as "fun 2". Therefore, the number of favorable outcomes is 2.

Experimental probability of spinning a 2 in Jackie's spinner can be calculated as Experimental probability = Number of favorable outcomes / Total number of possible outcomes

= 2 / 100

= 0.02

Therefore, the experimental probability of spinning a 2 on Jackie's spinner is 0.02 or 2%.

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Answer:

[tex]f(t) = 8e^{0.223144t}[/tex]

If you double the length and the width, you are quadrupling the area.

This is because 2 × 2 = 4.

Perimeter of new rectangle=   2 × perimeter of initial rectangle.

The perimeter will be double of if we double the length and breadth of rectangle

→ When doubling the dimensions of a rectangle, why the area is quadruple:

We know that :

Area of rectangle = l × b

where, l is length and b is breadth

A = L × W

This means area equals length times width.

if you double the length and you double the width, the formula becomes:

A = 2 × L × 2 × W

A = 2 × 2 × L × W

A = 4 × L × W

if you double the length and the width, you are quadrupling the area.

This is because 2 × 2 = 4.

→When doubling the dimensions of a rectangle, why the perimeter is doubles?

Let length be equal to ′l′

And breadth be equal to ′b′

Perimeter of rectangle = 2 (l + b) ____(equation i )

The perimeter if the length and breadth of the rectangle will increase by two, or will it double? So new parameters of rectangle when its length and breadth will get double on initial one.

Double length of rectangle of initial one= 2 × l=2l

Double breadth of rectangle of initial one = 2×b=2b

So the perimeter of new rectangle whose length and breadth are double of initial one are:

As we know that:

Perimeter of rectangle = 2 (l + b)

So,

Perimeter of new rectangle= 2(2l+2b),  (by taking 2 common)

Perimeter of new rectangle = 2×2(l + b) ____  equation (ii)

As from equation (i) perimeter of initial rectangle is = 2(l + b)

So from equation (i) and (ii)

Perimeter of new rectangle= 2 × 2(l + b)

Perimeter of new rectangle=   2 × perimeter of initial rectangle.

Hence, the perimeter will be double of if we double the length and breadth of rectangle.

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