A fire fighter sees a woman trapped in the building 81 feet up from the bottom floor. If the firetruck is parked 41 feet away from the bottom of the building, at what angle of elevation, to the nearest degree, shut the fire fighter extend the ladder to reach the woman?

Answer:

Step-by-step explanation:

Several trig identities are involved in the proof of this. This is the order in which they are used.

Starting with the left side, we can transform it into the right side.

  [tex]2\cos(2x)=2(\cos^2(x)-\sin^2(x)) = 2(\cos^2(x)-(1-\cos^2(x)))\\\\=2(2\cos^2(x)-1)=2\left(\dfrac{2}{\sec^2(x)}-\dfrac{\sec^2(x)}{\sec^2(x)}\right)\\\\=\boxed{\dfrac{4-2\sec^2(x)}{\sec^2(x)}}[/tex]

Answer:

3

Step-by-step explanation:

we can think of the slope of a graph as [tex]\frac{rise}{run}[/tex]

the "rise" is the difference in y-values between two points (the change in y)

the "run" is the difference in x-values between two points (the change in x)

when we divide these (rise/run), we find the slope

you can choose any two well-defined points (with clear x and y values), here are the two we will be using:

(remember, points are written as (x, y)  )  

(0 , 2)

(-1 , -1)

for the "rise", we have risen 3 (difference between 2 and -1 = 3 [2- -1 = 3] )

for the "run" we have ran 1 (difference between 0 and -1 = 1) (0 - - 1 = 1)

so, the slope for this graph is: 3/1 ; which simplifies to 3

So, the slope of our graph is 3

--------------------------------------------------------------

technical layout:

[tex]m=\frac{y_{2} -y_{1}}{x_{2}-x_{1}}[/tex]               (note: slope is called "m" in the format of y=mx + b)

[tex]y_{1}[/tex] means one y-value, [tex]y_{2}[/tex] means other y-value

[tex]x_{1}[/tex] means one y-value, [tex]x_{2}[/tex] means other x-value

{note: it is important to keep the y1 and x1 from the same point, and the y2 and x2 from a shared point}

our equation in this layout:

[tex]m=\frac{2 -(-1)}{0-(-1)}[/tex] = [tex]m=\frac{3}{1}[/tex]

                     m = 3

Answer:

x<7

Step-by-step explanation:

Consider the number x (as mentioned in the question). Now carefully pay attention to the given condition

⇒"The sum of a number and 3 is less than 10"

Remember the notation of  the term "less than" in mathematical context is indicated by "<" sign. Hence our inequality is

⇒ x+3<10

adding or subtracting any number from the both sides of an inequality DOESN'T change its direction. Thus, if we cancel out 3 from both sides it won't swap the inequality. Therefore

⇒ x+3-3<10-3

⇒ [tex]x<7[/tex]

In terms of  interval notation

[tex]\implies x\in( -\infty,7)[/tex]

In conclusion

The answer is D) [tex]x < 7[/tex]

Let the number be x

The sum of number and 3

Is less than 10

Lets solve it

Hence

Option D is correct

y=2x+3

to find inverse swap x and y

x=2y+3

x-3=2y

y=(x-3)/2

y=(x/2)-(3/2)

The inverse is f^-1(x)=(x/2)-(3/2)

Hope it helps!

The value of angle ∠DEC  = 60 degree.

The complete question is

In the given the figure above, m∠BAC = 64° and m∠CBA = 56°.

Part I: Find the m∠DEC.

Part II: Explain the steps you took to arrive at your answer. Make sure to justify your answer by identifying any theorems, postulates, or definitions used.

A triangle is a polygon with three sides and three vertices and three angles.

As it is given that

AB || CD

Let BC be the transversal , then

∠CBA = ∠DCB (alternate opposite angles)

∠DCB = 56 degree

As the sum of opposite interior angle sis equal to exterior angle therefore ∠ ECB = 120 degree

∠DCE = 120-56 = 64 degree

BC || DE

∠EDC = ∠DCB = 56 degree (alternate opposite angles)

Sum of all the interior angles of a triangle is equal to 180 degree

56 +64+∠DEC = 180

∠DEC  = 60 degree.

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Explanation:

Use the rational root theorem to determine this list of possible rational roots: 1, -1, 1/2, -1/2

Plug each possible root one at a time into the original expression given. If the simplified result is 0, then that possible root is an actual root.

If we tried say a = -1, then,

2a³-a²+a-2​ = 2(-1)³-(-1)²+(-1)-2​ = -6

The result is not zero, so a = -1 is not an actual root.

But if we tried say a = 1, then,

2a³-a²+a-2​ = 2(1)³-1²+1-2​ = 0

We get 0 so a = 1 is an actual root. I'll let you try the other values, but you should find that a = 1 is the only rational root.

Since a = 1 is a root, this makes (a-1) to be a factor.

From here, use either synthetic or polynomial long division to determine the other factor. Refer to the diagram below for each method.

Regardless of which method you pick, the quotient is 2a² + a + 2 which is the other factor needed. The remainder of 0 tells us we have (a-1) as a factor. For more information, check out the remainder theorem.

Based on the data given on the number pattern, 4 + 5 = 180

7 + 9 = 254016

3 + 7 = 7779240

5 + 8 = 2496000

7 + 9 = (7 × 9)^|7-9| × (7 + 9) × |7 - 9)

= 69² × 16 × 2

= 4761 × 16 × 2

= 254,016

3 + 7 = (3×7)^|3-7| × (3+7) × |3-7|

= 21⁴ × 10 × 4

= 194,481 × 10 × 4

= 7,779,240

So,

4 + 5 = (4×5)^|4-5| × (4+5) × |4-5|

= 20¹ × 9 × 1

4 + 5 = 180

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Answer:

75%

Step-by-step explanation:

48x=36

x=36/48

x=3/4

x=0.75=75%

The point which lies on the graph of the given function is; Choice D; (0, 2).

The point which lies on the graph of the given function; y = 2*3^x is it's y-intercept. The y-intercept which corresponds to the value of y when variable X is set to zero.

Hence, y = 2*3^0;

y = 2 ×1 = 2.

Hence, the point is; (0,2).

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The marginal cost when x = 25 and when x = 100 are $0.1 and $0.05 respectively.

A marginal production cost is a derivative of a cost function. From the information given, the total cost is:

The marginal production cost can be expressed as:

[tex]\mathbf{C'(x) = \dfrac{d}{dx}(\sqrt{x})}[/tex]

[tex]\mathbf{C'(x) = \dfrac{1}{2\sqrt{x}}}[/tex]

However, when the cost is 25, the marginal production is:

[tex]\mathbf{C'(25) = \dfrac{1}{2\sqrt{25} }}[/tex]

[tex]\mathbf{C'(25) = \dfrac{1}{2\times5}}[/tex]

[tex]\mathbf{C'(25) = \dfrac{1}{10}}[/tex]

C' (25) = $0.1

Also, when the cost is 100, the marginal production is:

[tex]\mathbf{C'(100) = \dfrac{1}{2\sqrt{100} }}[/tex]

[tex]\mathbf{C'(100) = \dfrac{1}{2\times10}}[/tex]

[tex]\mathbf{C'(100) = \dfrac{1}{20}}[/tex]

C' (100) = $0.05

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Answer:

D.

because 2 times 120 reduced.

The student did not make any mistake, the solution is correct. Then the correct option is A.

It is a method for dividing a polynomial into pieces that will be multiplied together. At this moment, the polynomial's value will be zero.

A student perform the following steps to find the solution of the equation x² – 2x – 15 = 0.

The factor of the equation will be

Step 1 :  x² – 2x – 15 = (x – 5)(x + 3)

Step 2 : (x – 5) = 0 and (x + 3) = 0

Step 3 : x = 5 and – 3

The student did not make any mistake, the solution is correct.

Then the correct option is A.

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The slope of the linear function is 3 and the y-intercept is -3 if the linear function contains the following points: (0,-3) and (6,15)

The ratio that y increase as x increases is the slope of a line. The slope of a line reflects how steep it is, but how much y increases as x increases. Anywhere on the line, the slope stays unchanged (the same).

[tex]\rm m =\dfrac{y_2-y_1}{x_2-x_1}[/tex]

We have:

A linear function contains the following points: (0,-3) and (6,15)

The slope:

[tex]\rm m =\dfrac{15-(-3)}{6-0}[/tex]

m = 18/6

m = 3

The equation of a line:

y = mx + c

Plug x = 0, y = -3

-3 = c

Thus, the slope of the linear function is 3 and the y-intercept is -3 if the linear function contains the following points: (0,-3) and (6,15)

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Step-by-step explanation:

option 2 is correct

here the question ask for value of X

to find it we multiple the matrix X with the given matrix in question and equate the answer.

Answer: 30 and 7

Step-by-step explanation: 30 plus 7 is 37, 30 times 7 is 210

Answer:

87.92

Step-by-step explanation:

We are given this expression:

3.14(a² + ab)

And we want to evaluate it if a is 4 and b is 3.

To solve this, we need to follow the order of operations, usually known as PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction) in the United States, although other countries have their own variations on the acronym.

Despite this, let's start with P, which would be parentheses.

Because we have the a² + ab in parentheses, we should evaluate this first.

We should substitute the values  of a and b with 4 and 3 respectively, to get this (we can ignore the 3.14 for now):
(4)² + 4 * 3

First, let's go on to exponents.

We do have one exponent, which is the 4 being raised to the second power.

So let's evaluate that.

Raise 4 to the second power (multiply it by itself).

4 * 4 = 16

We can replace (4)² with 16.

16 + 4 * 3

Now, let's move on to multiplication.

We do have one multiplication operation, which is the 4 * 3, so let's evaluate that.

4 * 3 = 12

Replace 4 * 3 with 12.

16 + 12

We don't have any division in this, so we can skip that.

Moving on to addition:

We can add these two numbers together.

16 + 12 = 28

28 is the value of a² + ab.

We now multiply this by 3.14 to get the answer.

3.14 * 28 = 87.92

Topic: substituting values into an expression

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The value of expression 3.14([tex]a^{2}[/tex]+ab) when a =4 and b=3 is 87.92.

Given Expression :3.14([tex]a^{2}[/tex]+ab) and we have to find the value at a=4 and b=3

we know expressions are the combinations of numbers, symbols and coefficients and in the expression given the variable are a and b .Expression shows the relationship between the variables which are present in the expression. When we have to find the value of expression at a=4 and b=3 we have to just put the values of a and b in the expression .

[tex]a^{2}[/tex]=16 and ab=4*3=12

=3.14[tex]((4)^{2}+4*3)[/tex]

=3.14(16+12)

=3.14*28

=87.92

Hence the value of the expression 3.14(a2 + ab) at a=4 and b=3 is 87.92.

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Answer:

Step-by-step explanation:

56565656565656565656565

Using the cosine ratio, the value of cos(45) in the diagram is: B. 1/√2.

Cosine ratio is, cos ∅ = adjacent length/hypotenuse length of a right triangle.

The triangle is a 45-45-90 triangle, thus, the two legs will be the same length, which is: 9 units each.

Hypotenuse = √(9² + 9²) = √162 = 9√2 units

∅ = 45°

Adjacent = 9 units

cos 45 = 9/9√2 = 1/√2

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Answer:

2nd one

as the y is both 4, the points are on the same plane. As such, we care about the distance of x which is between 3 and 7

Answer:

The balance after 37 years is $3869.99

Step-by-step explanation:

Given:

The Amount (A),

A = P (1 + [tex]\frac{R}{100n}[/tex][tex])^{nt}[/tex]        if compound n times per year

Amount (A) at 37 years,

A = ($2,520) (1 + [tex]\frac{1.16}{100(12)}[/tex][tex])^{12 x 37}[/tex]

A = ($2,520) (1 + [tex]\frac{1.16}{1,200}[/tex][tex])^{444}[/tex]

A = ($2,520) (1.00096667 [tex])^{444}[/tex]

A = ($2,520) (1.5357)

A = $3869.9944

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Answer:

the first,second,forth and fifth.

Step-by-step explanation:

I got it off of edg.

The most misleading graph is graph B because the blue rectangle and the red rectangle do not have the same width when plotted on the same scale

From the graph, we have the median weekly earnings to be:

So, the ratio is:

Ratio = $750 : $1250

Simplify

Ratio = 3 : 5

Hence, the ratio of the median weekly earnings is 3 : 5

In (a), we have:

Ratio = 3 : 5

The scale on the horizontal axis is given as:

1 unit per grid mark

Both rectangles have a width of 1 unit.

So, we have:

Ratio = 3 * 1: 5 * 1

Simplify

Ratio = 3 : 5

Hence, the ratio of the area of the red rectangle to the blue rectangle in graph A is 3 : 5

In (a), we have:

Ratio = 3 : 5

The scale on the horizontal axis is given as:

1 unit per grid mark

The red rectangle has a width of 3 units, while the blue has 5 units as its width

So, we have:

Ratio = 3 * 3 : 5 * 5

Simplify

Ratio = 9 : 25

Hence, the ratio of the area of the red rectangle to the blue rectangle in graph B is 9 : 25

In (a), we have:

Ratio = 3 : 5

The scale on the horizontal axis is given as:

1 unit per grid mark

The red rectangle has a width of 3 units, while the blue has 5 units as its width.

Since the base are squares, we have:

Ratio = 3 * 3  * 3 : 5 * 5 * 5

Simplify

Ratio = 27 : 125

Hence, the ratio of the volume of the red cube to the blue cube in graph C is 27 : 125

The most misleading graph is graph B.

This is so because the blue rectangle and the red rectangle do not have the same width when plotted on the same scale

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The order of the graphs from largest to lowest correlation coefficients is:

Graph D, Graph A, graph C, graph B.

The correlation coefficient between two variables is a coefficient that tells us "how much" these variables relate.

So, in the case for linear correlation, as "more linear" the data appears to be, a large correlation is between the two variables.

With that in mind, we conclude that the order of the graphs (going for larger correlation coefficient to smaller correlation coefficient) is:

Graph D, Graph A, graph C, graph B.

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The only true statement is A:

"The data show a negative linear relationship."

On the graph, we can see how the car's vale decreases almost linearly with the age of the car.

Where the response variable would be the one on the y-axis, which is the car's value.

For that linear behavior, we know that there is a correlation coefficient different than zero. So options B, C, and D are false.

Finally, we already saw the linear behavior (decreasing, so the slope is negative). Then we conclude that the only true statement is A.

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Answer:

X=∑fX ∑f , where ¯X is the mean.

Mean = 115/4 = 28.75

Standard deviation = [tex]\sigma={\sqrt {\frac {\sum(x_{i}-{\mu})^{2}}{N}}}[/tex]     n = the number of items = 4

Standard deviation = √28.37/4 = √7.0925 = 2.66317

Step-by-step explanation:

Mean is the number which is obtained by adding the values of all the items of a series and dividing the total by the number of items.

Standard deviation is the square root of the arithmetic mean of the squares of deviations of the items from their mean values

So, Class intervals are given here.

We have to find the middle point of the class interval.

So,

Class interval= 25- 34

Middle point= [tex]\frac{34-25}{2}[/tex]= [tex]\frac{9}{2}= 4.5[/tex]

Let's add 4.5 to the lower interval,

=> 25 + 4.5= 29.5

This is the median point and do the same for the other intervals too

i.e 39.5,49.5 and then 59.5 then multiply the number with the middle point.

So midpoint times the number. So, we get

29.5 * 21.7 = 640.15 & so on we get 129,...

Then add these numbers together. So that becomes 5375.75. And what about the total number here, that is

=>21.7 + 32.8 + 36.8 + 21.7= 118.5

So here,

Mean value, that should be equal to the sum of mean times the number divided by the total number here,

Mean= [tex]\frac{5375.75}{118.5}[/tex] = 45.36

So, this is the mean value.

Standard deviation:

Standard deviation =  [tex]\sigma={\sqrt {\frac {\sum(x_{i}-{\mu})^{2}}{N}}}[/tex]

[tex]\sigma[/tex] = population standard deviation

N = the size of the population

[tex]x_{i}[/tex] = each value from the population

[tex]\mu[/tex] = the population mean

So here is what we have to do,

We have to subtract the mean value from the middle point that is,

=>45.36- 21.5 and squared plus 45.36 -39.5 Squared. And plus this is 45.36 -49.5 Squared plus 45.36 -59.5 squared. And then, this is 45.

Let me just inside the apprentices Inside the practices, this is 45 .36 29.5 and take the square of this one And then add another one Which is 45.36 39.5 and then take the square this one too. And plus inside parentheses, this is 45.36 minus Which is 490.5 And then squared. That is 45.36 And -59 0.5 so And then take the square, there's 12 if you just add all of them, so we will get five 47.1184.

So the standard deviation is the square of 547, divided by The total frequency here.

Standard deviation = √28.37/4 = √7.0925 = 2.66317

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Answer:

9a + 18

Step-by-step explanation:

so the first step is to evaluate the value of f(2).

[tex]f(2) = 9(2)^2-7\\f(2)=9(4)-7\\f(2)=36-7\\f(2)=29[/tex]

Now that we have this value we can use the slope formula (y2 - y1) / (x2 - x1) where y2 is f(a), y1 is 29, x2 is a, and x1 is 2

[tex]\frac{f(a)-29}{a-2}\\\\\frac{(9a^2-7)-29}{a-2}\\\frac{9a^2-36}{a-2}[/tex]

Now from here we can use long division to simplify it further

[tex]9a^2/a = 9a\\9a(a-2)=9a^2-18a\\(9a^2-36)-(9a^2-18a)\\18a-36\\\\18a/a=18\\18(a-2)=18a-36\\(18a-36)-(18a-36)\\0\\\\9a+18[/tex]

Answer:

it's 5 degrees celsius I think you must have entered a 4-32 instead of a 41-32 is what threw you off

Step-by-step explanation:

The correct representations of this solution are

The inequality is given as:

6 + 2n > 12

The solution is given as:

n > 3

When the above inequality is represented as an interval, we have:

(3, ∞)

Also, the inequality symbol > is represented using an open circle.

This means that the number line would use an open circle and the arrow would point right

Hence, the correct representations of this solution are (d) and (e)

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Answer:

25/36

Step-by-step explanation:

put 2500 over 3600 and remove the 0's to simplify it