In addition to the trigonometric ratios, what other methods can be used to solve a right triangle? Give situations for when you might want to use them

The equations to solve for [tex]\( |2x-3|=x+6 \)[/tex] are [tex]\( 2x - 3 = x + 6 \)[/tex] and [tex]\( 2x - 3 = -(x + 6) \).[/tex]

Let's solve the absolute value equation |2x-3|=x+6 step by step.

We'll start by writing down the equation:

|2x - 3| = x + 6

Since the absolute value expression |2x - 3| can be either positive or negative, we'll split the equation into two cases:

  Case 1: 2x - 3 = x + 6

  Case 2: 2x - 3 = -(x + 6)

Let's solve Case 1:

  2x - 3 = x + 6

  Subtract x from both sides:

  2x - x - 3 = 6

  Simplify:

  x - 3 = 6

  Add 3 to both sides:

  x = 9

Now, let's solve Case 2:

  2x - 3 = -(x + 6)

  Distribute the negative sign:

  2x - 3 = -x - 6

  Add x to both sides:

  2x + x - 3 = -6

  Simplify:

  3x - 3 = -6

  Add 3 to both sides:

  3x = -3  

  Divide both sides by 3:

  x = -1

So, the solutions to the equation |2x-3|=x+6 are x = 9 and x = -1.

The distance of the plane from the base of the tower is:

                    Option: D

                    D.  98 feet  

Let x denotes the distance of the plane from the base of the tower.

Now, in the right angled triangle we will need to use trignometric identity corresponding to 63°.

Based on the figure we have:

[tex]\tan 63=\dfrac{x}{50}\\\\\\x=50\times \tan 63[/tex]

[tex]x=50\times 1.9626\\\\\\x=98.13\ feet[/tex]

which to the nearest feet is:

                        98  feet

Answer:

7 times during the first 15 minutes

Step-by-step explanation:

Remember that

[tex]1\ min=60\ sec[/tex]

so

[tex]15\ min=15(60)=900\ sec[/tex]

Decompose the numbers 18 and 42 in prime factors

we know that

[tex]18=(2)(3^2)[/tex]

[tex]42=(2)(3)(7)[/tex]

Find the least common multiple (LCM)

The LCM is

[tex](2)(3^2)(7)=126\ sec[/tex]

we need to find all multiples of 126 that are less than or equal 900.

[tex]126*1=126\ sec\\126*2=252\ sec\\126*3=378\ sec\\126*4=504\ sec\\126*5=630\ sec\\126*6=756\ sec\\126*7=882\ sec[/tex]

therefore

7 times during the first 15 minutes

Answer: The all possible rational roots are [tex]x=\pm1,\pm2,\pm3,\pm4,\pm6,\pm12,\pm\frac{1}{2},\pm\frac{3}{2},\pm\frac{1}{4},\pm\frac{3}{4},\pm\frac{1}{10},\pm\frac{1}{5},\pm\frac{3}{5}\pm\frac{3}{10},\pm\frac{2}{5},\pm\frac{6}{5},\pm\frac{1}{20},\pm\frac{3}{20},\pm\frac{4}{5},\pm\frac{12}{5}[/tex].

Explanation:

The given polynomial is,

[tex]f(x)=20x^4+x^3+8x^2+x-12[/tex]

The Rational Root Theorem states that the all possible roots of a polynomial are in the form of a rational number,

[tex]x=\frac{p}{q}[/tex]

Where p is a factor of constant term and q is the factor of coefficient of leading term.

In the given polynomial the constant is -12 and the leading coefficient is 20.

All possible factor of -12 are [tex]\pm1,\pm2,\pm3,\pm4,\pm6,\pm12[/tex].

All possible factor of 20 are [tex]\pm1,\pm2,\pm4,\pm5,\pm10,\pm20[/tex].

So, the all possible rational roots of the given polynomial are,

[tex]x=\pm1,\pm2,\pm3,\pm4,\pm6,\pm12,\pm\frac{1}{2},\pm\frac{3}{2},\pm\frac{1}{4},\pm\frac{3}{4},\pm\frac{1}{10},\pm\frac{1}{5},\pm\frac{3}{5}\pm\frac{3}{10},\pm\frac{2}{5},\pm\frac{6}{5},\pm\frac{1}{20},\pm\frac{3}{20},\pm\frac{4}{5},\pm\frac{12}{5}[/tex]

Answer:

A.) -4/5 and 3/4

Step-by-step explanation:

The correct option is A. [tex]m=-4[/tex]. By substituting the second equation into the first one in the system of equations, the value of m is -4.

The value of m in the system of equations can be found by substitution or elimination. Starting with the given equations:

[tex]m - 2n = 8 (1)[/tex]

[tex]n = m - 2 (2)[/tex]

Using the second equation (2) to substitute for n in the first equation (1):

[tex]m - 2(m - 2) = 8[/tex]

[tex]m - 2m + 4 = 8[/tex]

[tex]-m + 4 = 8[/tex]

[tex]-m = 8 - 4[/tex]

[tex]-m = 4[/tex]

[tex]m = -4[/tex]

This means that the value of m is -4, which corresponds to option A.

The slope of a line perpendicular to x = -3 is 0. Option B is correct answer.

Perpendicular line slopes are the negative reciprocals of one another. In other words, if one line has a slope of m, a line perpendicular to it has a slope of -1/m. The definition of perpendicular lines, which states that the angles created by the lines are 90 degrees, may be used to demonstrate this relationship. The product of the slopes of perpendicular lines must be -1 in order to meet the requirement that the tangent of a 90 degree angle be specified.

Given the line x = -3.

Any line perpendicular to this line will have slopes of negative reciprocals of each other.

The slope of a horizontal line is 0, since the line does not change in the y-direction.

Hence, the slope of a line perpendicular to x = -3 is 0.

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x=  standard

y = high quality

x +y = 1070

y=1070-x

2.8x + 4.9y =3521

2.8x +4.9(1070-x) = 3521

2.8x+5243-4.9x =3521

-2.1x=-1722

x=-1722/-2.1 = 820

820*2.8 =2296

1070-820 =250

250*4.9 = 1225

1225+2296 = 3521

there were 250 high quality downloads

Answer:

Scientific notation is a method for expressing a given quantity as a number having significant digits necessary for a specified degree of accuracy, multiplied by 10 to the appropriate power, as 1385.62 written as [tex]1.3862*10^{4}[/tex].

Step-by-step explanation:

         The answer is:

Sigma 960 (1/4)^i-1 ;    the sum is 1280

We are given the first term of the geometric sequence as:

[tex]a_1=960[/tex]

Also, the common ratio of the terms in geometric sequence is: [tex]\dfrac{1}{4}[/tex]

We know that if the series is a geometric series than the sum of the terms is given by:

[tex]a+ar+ar^2+.....\\\\=a(1+r+r^2+....)\\\\=\sum ar^{n-1}[/tex]

where a is the first term of the series and r is the common difference.

Here a=960

and r=1/4

Hence,

The sum of the series is:

[tex]=\sum 960(\dfrac{1}{4})^{i-1}[/tex]

Now we know that the sum of the infinite geometric series is given by:

[tex]S=\dfrac{a}{1-r}[/tex]

where S is the sum of the series.

Hence, here the sum of the series is calculated by:

[tex]S=\dfrac{960}{1-\dfrac{1}{4}}\\\\\\S=\dfrac{960}{\dfrac{3}{4}}\\\\\\S=\dfrac{960\times 4}{3}\\\\\\S=1280[/tex]

Hence, the sum is:   1280

Final answer:

The probability of drawing two yellow marbles in succession without replacement from a bag of 10 marbles, where 2 are yellow, is 1/45.

Explanation:

The student is asking about the probability of drawing two yellow marbles successively without replacement from a bag containing a total of 10 marbles with different colors. To solve this problem, we use conditional probability. The probability of drawing the first yellow marble is 2 out of 10 since there are 2 yellow marbles among 10 total marbles. This can be written as P(Y1) = 2/10 or 1/5. After the first yellow marble is drawn, there is only 1 yellow left among 9 total marbles, so the probability of drawing a second yellow marble is P(Y2|Y1) = 1/9.

The two events are dependent since the outcome of the first draw affects the second draw. Therefore, to find the overall probability of both events happening, we multiply their probabilities: P(Y1 and Y2) = P(Y1) × P(Y2|Y1) = (1/5) × (1/9) = 1/45. So, the probability of drawing two yellow marbles successively without replacement is 1/45.

For the given probabilities P(A∩B) = 1150 and P(A|B) = 1120, using Bayes' theorem for conditional probability, the probability of event B, P(B), is found to be approximately 1.027.

The topic of this question is probability, specifically concurrent events in probability theory. Here, we use Bayes' theorem for conditional probability which states that P(A|B) = P(A ∩ B) / P(B).

Given that P(A∩B) = 1150 and P(A|B) = 1120, we can plug these values into the formula to find P(B). Hence, P(B) = P(A ∩ B) / P(A|B), substituting the given numbers, we get P(B) = 1150 / 1120 = 1.027 approximately.

So P(B), the probability of event B occurring, we would expect to be approximately 1.027 (or 102.7% assuming the probabilities were in decimal form to start with).

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Alright, let's solve the problem with these given values.

The formula for calculating the probability of B in the context of A and B both happening is:

`P(A and B) / P(A|B)`.

In this case, the probability of A and B occurring is 1150, while the likelihood of A given that B has occurred is 1120. We can substitute these values into the formula:

So, P(B) = P(A and B) / P(A|B)

Substitute the given into the formula:

P(B) = 1150 / 1120 = 1.0267857142857142

Therefore, the Probability of event B (P(B)) is approximately 1.027 when rounded to three decimal places.

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A functions B and D are valid functions because they meet the criteria of each x value having only one corresponding y value.

Function B: You mentioned that B is a function because each x value has only one corresponding y value. In mathematical terms, a function is defined as a relation between a set of inputs (x values) and a set of outputs (y values), where each input (x) maps to exactly one output (y). So, if B satisfies this definition, it is indeed a function.

Function D: Similar to B, if D also adheres to the definition of a function, where each x value corresponds to only one y value, then it is a function as well.

Function A: You mentioned that for A, the x value of 1 has 2 y values. In mathematical terms, a function cannot have one input (x) mapping to multiple outputs (y). If this is the case for A, it does not meet the definition of a function.

Function C: You mentioned that for C, the x value of 7 has 2 y values, 10 and 3. Again, if one input (x) is associated with multiple outputs (y) in C, it does not qualify as a function.

In summary, functions B and D are valid functions because they meet the criteria of each x value having only one corresponding y value. Functions A and C do not meet this criterion and are not functions in the mathematical sense.

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I’ve answered this before so here’s the complete given for this problem.

Danielle conducts an experiment by tossing a fair coin three times. She records the number of heads out of the three trials. The probabilities are given in the table below (X = number of heads out of three trials). 

Given:

x 0 1 2 3 
P(x)1/8 3/8 3/8 1/8 


 n = 3; p = 0.5; q = 0.5; P(x = 0) = nCx p^x 
x N(x) P(x) ΣP(x) 1- ΣP(x) x * P(x) 
--- ----- --------- --------- --------- --------- 
0 1 0.125 0.125 0.875 0 
1 3 0.375 0.5 0.5 0.375 
2 3 0.375 0.875 0.125 0.75 
3 1 0.125 1 0 0.375 

0+.375+.75+.375 = 1.5

Answer: C.The transfer rate of the USB stick is 200 times the transfer rate of the modem .

Length of arc VZX is 27.92 units.

The arc is a portion of the circumference of a circle.

Given,

Radius of circle = 8 units

∠VCX = 160°

Let length of arc VZX = x

x = area of circle×(360°- ∠VCX)/360°

x = 2π×8×(360-160)/360

x = 2π×8×(200)/360

x = 400π/45

x = 80π/9

x = 27.92 units

Hence, length of arc VZX is 27.92 units.

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