In the above triangle, if x = , y = 1, and z = 2, then which of the following is equal to tan(60°)?
A. [tex]\sqrt{3}[/tex]
B. [tex]\frac{\sqrt{2} }{2}[/tex]
C. [tex]\frac{\sqrt{3} }{2}[/tex]
D. [tex]\frac{\sqrt{3} }{3}[/tex]

Answer:

10/3 mph

Step-by-step explanation:

Obviously, (time going) + (time returning) = (total time spent en route) = 54 min.  Since time = distance / rate,

                           3 miles                3 miles

                    ----------------------- + --------------------- = 54 min

                     downhill speed      uphill speed

Let u = uphill speed and d = downhill speed; then d = u + 5 (all in mph)

Then we have:

                           3 miles                3 miles

                    ----------------------- + --------------------- = 54 min

                           u + 5                        u

and our task here is to determine the uphill speed, u.

The LCD is u(u + 5).  Thus we have:

                           3u                     3(u + 5) miles

                    ----------------------- + -----------------------  = 54 min = 0.9 hr

                           u(u + 5)                  u(u + 5)

so that:

             6u + 15

      -----------------------  = 0.9 hr     or      6u + 15 = 0.9(u)(u + 5), or

             u(u + 5)

             6u + 15 = 0.9u² + 4.5u

Combining the u terms, we get:

                      15 = 0.9u² + 4.5u,            or          0.9u² + 1.5u - 15 = 0

Eliminating the fractions, we get                         9u² + 15u - 150, or

                                                                               3u^2 + 5u - 50 = 0

This factors into (3u - 10)(u + 5) = 0.  The only positive root is u = 10/3.

Your average rate on the return trip (uphill) is 10/3 mph (3 1/3 mph).

R is approximately 6 mph.

Let's denote the average rate (speed) on the return trip as
r (in miles per hour, mph). Since it is given that the speed going to campus is 5 mph faster, the speed while going downhill would be r + 5 mph. We need to find the values of r.

The total distance for the round trip is 3 miles to campus and 3 miles back, adding up to 6 miles. The total time for the trip is given as 54 minutes, which we will convert to hours by dividing by 60, giving us 0.9 hours.

The time taken to go to campus is the distance divided by the speed, which is 3 / (r + 5) hours. The time taken for the return trip is 3 / r hours. Since both times add up to 0.9 hours, we can write the equation:

3 / (r + 5) + 3 / r = 0.9

Now we need to solve this equation for r. We find a common denominator and solve:

r(r + 5)(3 / (r + 5) + 3 / r) = r(r + 5)(0.9)

3r + 3(r + 5) = 0.9r(r + 5)

3r + 3r + 15 = 0.9r^2 + 4.5r

6r + 15 = 0.9r^2 + 4.5r

0.9r^2 - 1.5r - 15 = 0

Using the quadratic formula or factoring, we can find the root for r. The root that makes sense in this context (positive speed) gives us the average rate on the return trip.

After solving, we find that r is approximately 6 mph, which is the average speed of the student on their return trip.

Answer:

x-3y-7=0

Step-by-step explanation:

Given

m=1/3

The standard form of point slop form is:

y=mx+b

To find the value of b, we will put the point in the standard form

So,

[tex]-2=\frac{1}{3}(1)+b[/tex]

Solving for b[tex]-2=\frac{1}{3}+b\\-2-\frac{1}{3}=b\\\frac{-6-1}{3}=b\\b=\frac{-7}{3}[/tex]

Putting the values of b and m in standard form:

[tex]y=\frac{1}{3}x+\frac{-7}{3}\\y=\frac{1}{3}x-\frac{7}{3}Multiplying\ both\ sides\ by\ 3\\3y=x-7\\-x+3y+7=0\\Can\ also\ be\ written\ as\\x-3y-7=0[/tex]

Answer:

The answer is :

A. (7x + 4)(7x - 4)

Answer:

The correct answer is first option

(7x + 4)(7x − 4)

Step-by-step explanation:

Points to remember

Identities

(a + b)(a - b) = a² - b²

It is given an expression,

49x² - 16

To factorize  the expression 49x² - 16

We know that 7² = 49 and 4² = 16

Therefore we can write the given expression as,

49x² - 16 = (7x)² - 4²

It is in the form of the above identity,

(7x)² - 4² = (7x + 4)(7x - 4)

The correct answer is first option

Answer:

  A) 255,358.46

  E)  273,353.92

Step-by-step explanation:

The formula for future value of principal P at interest rate r per year compounded n times per year for t years is ...

  FV = P(1 +r/n)^(nt)

Filling in the numbers and doing the arithmetic, we have ...

A) FV = $34,500(1 + 0.069)^30 ≈ $255,358.46

__

E) FV = $34,500(1 + 0.069/365)^(365·30) ≈ $273,352.92

Answer:

(1,2) and (2,-1)

Step-by-step explanation:

we have

[tex]y< 5-2x[/tex] ----> inequality A

The solution of the inequality A is the shaded area below the dashed line [tex]y=5-2x[/tex]

[tex]x+5y>-7[/tex] ----> inequality B

The solution of the inequality B is the shaded area above the dashed line [tex]x+5y=-7[/tex]

The solution of the system of inequalities is the shaded area between the two dashed lines

If a ordered pair is a solution of the system of inequalities, then the ordered pair must lie on the shaded area

Two points in the solution are

(1,2) and (2,-1)

see the attached figure

Answer:

Step-by-step explanation:

A.)Priscilla made a mistake when applying the horizontal shift.  That "x+2" indicates a horiz. shift to the LEFT, not to the right.

Answer:

Option A.

Step-by-step explanation:

Priscilla graphed function g(x), a transformation of the quadratic parent function f (x) = [tex]\frac{1}{2}[/tex] f ( x+2 ) - 1

(1) She graphed the vertical compression correctly

(2) Priscilla graphed the vertical shift of (-1) means 1 unit downwards on y-axis correctly.

(3) In the last she made a mistake because for f(x+2) the parent function should have been shifted left on x-axis by two units f (x) ⇒ f[ x- (-2)]

Therefore Option A will be the answer.

Answer:

  The series is divergent

Step-by-step explanation:

An infinite geometric series with a common ratio greater than 1 does not converge to a sum. The series is divergent.

Answer:

Step-by-step explanation:

Substitute g(x) for x in f(x) and evaluate:

  f(g(x)) = f(x^(1/3) -1)

  = ((x^(1/3) -1) +1)^3 -5

  = (x^(1/3))^3 -5

  = x^(3/3) -5

  f(g(x)) = x -5

This is confirmed by a graphing calculator. (See attached.)

If the functions were inverses, the value of f(g(x)) would be x. It is not, so the functions are not inverses.

Answer:

The Answer is:  -1 and 1

Step-by-step explanation:

Answer:

Formula of standard deviation = √Variance

Standard deviation = 1

Step-by-step explanation:

X-bar is the variance

Therefore, the answer would be

√X-bar

√1 = 1

!!

Answer:

C

Step-by-step explanation:

Answer:

[tex] -4 \sqrt{5} [/tex]

Step-by-step explanation:

Quadrant 2 means cosine is negative.

So [tex] \sin(\theta)=\frac{2}{3} =\frac{\text{ opp }}{\text{ hyp }} [/tex]

So the adjacent side is [tex] \sqrt{3^2-2^2}=\sqrt{9-4}=\sqrt{5} [/tex]

So [tex] \cos(\theta)=-\frac{\sqrt{5}}{3} [/tex]

Now to find [tex] \tan(2 \theta) [/tex]

[tex] \tan(2 \theta) =\frac{2\tan(\theta)}{1-\tan^2(\theta)}[/tex]

We will need [tex] \tan(\theta) [/tex] before proceeding.

[tex] \tan(\theta) =\frac{\sin(\theta)}{\cos(\theta)}=\frac{\frac{2}{3}}{\frac{-\sqrt{5}}{3}}=\frac{-2}{\sqrt{5} } [/tex]

Now plug it in and the rest is algebra.

[tex] \tan(2 \theta) =\frac{2\tan(\theta)}{1-\tan^2(\theta)} =\frac{2 (\frac{-2}{\sqrt{5}}}{1-\frac{4}{5}} [/tex]

Now the algebra, the simplifying.... We need to get rid of the compound fraction.  We will multiply top and bottom by [tex] 5 \sqrt{5} [/tex]

This will give us

[tex] \frac{-4(5)}{5 \sqrt{5}-4 \sqrt{5}} [/tex]

[tex] \frac{-20}{\sqrt{5}} [/tex]

Multiply top and bottom by [tex] \sqrt{5} [/tex]

[tex] \frac{-20 \sqrt{5}}{5} [/tex]

The answer reduces to

[tex] -4 \sqrt{5} [/tex]

Final answer:

To find tan2θ when sinθ = 2/3 and θ is in Quadrant II, we use the Pythagorean identity to find cosθ and the double angle identity for tangent. After calculation, tan2θ = -4√5.

Explanation:

If sinθ = 2/3 and θ is located in Quadrant II, we first have to find cosθ and then use the double angle identity for tangent to find tan2θ. Since sinθ is positive in Quadrant II and cosθ must be negative (as the x-values are negative in Quadrant II), we can use the Pythagorean identity sin2θ + cos2θ = 1 to find cosθ. Thus, cosθ = -√(1 - sin2θ) = -√(1 - (2/3)2) = -√(1 - 4/9) = -√(5/9) = -√5/3.

The double angle identity for tangent is tan2θ = 2tanθ / (1 - tan2θ). But first, we find tanθ = sinθ/cosθ = (2/3) / (-√5/3) = -2/√5. Then, tan2θ = 2(-2/√5) / (1 - (-2/√5)2) = -4/√5 / (1 - 4/5) = -4/√5 / (1/5) = -20/√5. Simplifying further, we multiply by √5/√5 to rationalize the denominator, which gives us tan2θ = -20√5/5 = -4√5.

Answer: [tex]\dfrac{15}{32}[/tex]

Step-by-step explanation:

Given : The number of cherry lollipop = 5

The total number of lollipop = 8

the number of lollipops other than grape =6

The probability of selecting a cherry lollipop is given by :_

[tex]\text{P(Cherry)}=\dfrac{5}{8}[/tex]

The probability of selecting a lollipop other than grape is given by :_

[tex]\text{P(Other than grape)}=\dfrac{6}{8}[/tex]

Since, there is replacement , then the events are independent of each other.

Now, the probability that Julie will select a cherry lollipop and then a lollipop other than grape is given by :-

[tex]\text{P(Cherry and other than grape)}=\dfrac{5}{8}\times\dfrac{6}{8}=\dfrac{15}{32}[/tex]

Hence, the required probability =[tex]\dfrac{15}{32}[/tex]

Answer: The answer is A, because if you subtract 1.3 from 9.2, you would get 7.9, and after that you would divide 7.9 by 4. After doing that, you would get an answer of 1.975, which rounds up to 2.

Step-by-step explanation:

1. Subtract 1.3 from 9.2

2. Take the Answer from Step 1, and divide it by 4

3. Round up your answer.

Answer:

c. 68, 112

Step-by-step explanation:

The angle is our unknown, so we will call it x.  If the angle is x, then its supplement is 180 - x (supplementary angles add up to equal 180).  The word "is" means "equals", so putting the equation together looks like this:

180 - x = 2x - 24.  Add x to both sides and at the same time add 24 to both sides (combining like terms, in other words):

204 = 3x so

x = 68

x is the angle measure, so the angle is 68 and its supplement is 180 - 68 = 112

The angle will be 68 and its supplement will be 112.

It is given that supplement of an angle is 24 less than twice the measure of the angle.

We have to find out the measure of the angle and its supplement.

What are the supplementary angles ?

The supplementary angles are the angles whose sum is equal to 180° i.e., sum of angles 150° and 30° equal to 180°.

The angle is unknown. Let's assume angle be x.  

We know that , supplementary angles add up to equal 180.

If the angle is x, then its supplement will be :

180 - x ----------- Equation 1

Also ;

angle's supplement is equal to ;

2 × x - 24 ----------- Equation 2

Keeping both the equations equal ;

180 - x = 2x - 24.  

180 + 24 = 3x

204 = 3x

x = 68

Supplement will be ;  180-x = 180 - 68 = 112

Thus , x is the angle measure, so the angle is 68 and its supplement is 112.

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Answer:

Step-by-step explanation:

Alright. Looking at the graph, angle CGE is equal to angle FGD.

FGD = 90-73

FGD = 17

Therefore, as angle CGE is equal to angle FGD, angle CGE is equal to 17.

Answer: 17 degrees.

By applying angle relationships and recognizing the equality of angles, we determined that angle CGE measures 17 degrees, as it is equal to angle FGD in the context of the graph.

In geometry, analyzing angles within shapes and figures is fundamental. Here, we have a graph depicting intersecting lines and angles. Let's break down the steps and reasoning for finding the measurement of angle CGE:

1. Angle CGE and Angle FGD:

First, we observe the graph and notice that angle CGE is mentioned to be equal to angle FGD. This equivalence allows us to focus on finding the measurement of angle FGD, which, in turn, will be the measurement of angle CGE.

2. Calculation of Angle FGD:

We know that angle FGD is related to a right angle, as indicated by the angle symbol "90" next to it. To calculate the measurement of angle FGD, we use the fact that the sum of angles around a point is 360 degrees, and the angle opposite to a right angle is complementary, totaling 90 degrees.

So, we calculate angle FGD as follows:

Angle FGD = 90 degrees (right angle) - 73 degrees (given angle)

Angle FGD = 17 degrees

3. Angle CGE:

Since angle CGE is stated to be equal to angle FGD, we determine that angle CGE is also 17 degrees.

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An extraneous solution is a solution that arises during the algebraic process of solving a radical equation but does not actually satisfy the original equation. It is essential to substitute the solution back into the original equation to determine its validity.

An extraneous solution to a radical equation refers to a solution that emerges from the process of solving the equation, but is not a valid solution to the original equation. When we solve radical equations, we often have to square both sides to remove the radical. This process can introduce solutions that aren't true for the original equation. To determine whether a solution is extraneous, we must always substitute it back into the original equation to verify its validity.

If during your process you encounter coefficient terms that correspond to variable-dependent outcomes in function theory, be mindful that the extraneous solutions can impact the interpretation of the solution set. The goal is always to reduce the equation to a state that is readily solvable—via algebraic normal equations in the case of more complex equations like the quintic equation, which sometimes involves elliptic functions for their solution, not to be confused with solutions by radicals.

Equations that are solvable by radicals mean they can be reduced to pure equations using algebraic processes, eliminating the need for other non-algebraic methods. However, during the solution process, extraneous solutions may appear, and thus, it is essential to substitute any found solution into the original equation to ensure it was not introduced during the algebraic manipulations.

Answer:

5.75t=p

Step-by-step explanation:

22p=126.50/22

p=5.75 per hour

5.75t=p

The $5.75 is earned for every hour worked.

Answer:

[tex]y=10(2)^x[/tex]

Step-by-step explanation:

This is an exponential function.  Peter is not simply adding $10 a week to his savings, he's doubling each value each week.  The first case of adding $10 a week is linear.  

The standard form of an exponential equation is

[tex]y=a(b)^x[/tex]

where a is the initial amount and b is the growth or decay factor.  We know a = 10 because we are told he started with $10.  After the first week he would have $20 then because $10 doubled is $20.  That coordinate is (1, 20).  We will use that in place of x and y and solve for b:

[tex]20=10(b)^1[/tex]

b to the first is just b, so what we have essentially is 10b = 20, so b = 2.  The equation then is:

[tex]y=10(2)^x[/tex]

Answer:

x = -1

Step-by-step explanation:

[tex]\sqrt{1-3x}=x+3\\\\1-3x=x^2+6x+9 \qquad\text{square both sides}\\\\x^2+9x+8=0 \qquad\text{put in standard form}\\\\(x+8)(x+1)=0 \qquad\text{factor}\\\\\left \{ {{x=-8} \atop {x=-1}} \right. \qquad\text{values that make the factors zero}[/tex]

Only the solution x = -1 will work for this equation. The other solution is extraneous.

Answer:

[tex]6\ polyphonic\ ringtones[/tex]  and [tex]1\ standard\ ringtone[/tex]

Step-by-step explanation:

Let

x -----> the number of polyphonic ringtones

y ----> the number of standard ringtones

we know that

[tex]x+y=7[/tex]

[tex]x=7-y[/tex] -----> equation A

[tex]3.25x+1.50y=21[/tex] -----> equation B

Solve the system of equations by substitution

Substitute equation A in equation B and solve for y

[tex]3.25(7-y)+1.50y=21[/tex]

[tex]22.75-3.25y+1.50y=21[/tex]

[tex]3.25y-1.50y=22.75-21[/tex]

[tex]1.75y=1.75[/tex]

[tex]y=1\ standard\ ringtones[/tex]

Find the value of x

[tex]x=7-1=6\ polyphonic\ ringtones[/tex]

Answer:

6 polyphonic, 1 standard

Step-by-step explanation:

Answer:

4280 passes were sold

Step-by-step explanation:

The unknown we are looking for the number of passes that were sold.  We don't know how MANY were sold, but we do know that the cost of ONE is 12.50.  We represent the number of passes at that price per pass as

12.50x

We also know that money earned from the sale of this unknown number of passes came to 53,500.  Our equation then becomes

12.50x = 53,500

Solving for x, we divide both sides by 12.50 to get that

x = 4280

Final answer:

To find the number of annual visitor passes sold at Woodland Mound Park, divide the total sales ($53,500) by the price per pass ($12.50), resulting in 4,280 passes sold.

Explanation:

The question asks how many annual visitor passes were sold at Woodland Mound Park last year if each pass costs $12.50 and the total amount raised from pass sales was $53,500. To find the number of passes sold, you'll need to divide the total sales by the price per pass. Here is the step-by-step calculation:

Answer:

  5 times as large

Step-by-step explanation:

The ratio of the two numbers tells you how many times as large one is as the other:

[tex]\dfrac{5\cdot 10^5}{1\cdot 10^5}=\dfrac{5}{1}\cdot\dfrac{10^5}{10^5}=5[/tex]

Answer: 5 is the correct answer!

Step-by-step explanation:

Answer:

$42,890

Step-by-step explanation:

The standard form for an exponential equation is

[tex]y=a(b)^x[/tex]

where a is the initial amount value and b is the growth rate or decay rate and t is the time in years.  Since we are dealing with money amounts AND this is a decay problem, we can rewrite accordingly:

[tex]A(t)=a(1-r)^t[/tex]

where A(t) is the amount after the depreciation occurs, r is the interest rate in decimal form, and t is the time in years.  We know the initial amount (70,000) and the interest rate (.04), but we need to figure out what t is.  If the car was bought in 2006 and we want its value in 2018, a total o 12 years has gone by.  Therefore, our equation becomes:

[tex]A(t)=70,000(1-.04)^{12}[/tex] or, after some simplification:

[tex]A(t)=70,000(.96)^{12}[/tex]

First rais .96 to the 12th power to get

A(t) = 70,000(.6127097573)

and then multiply.

A(t) = $42,890

A reflection is a transformation where the mirror image of a figure is shown directly opposite its line of reflection.

To find an image that has been reflected across the line y = x, switch the x- and y-coordinates.

Therefore, the rule for reflecting an image across the line y = x can be described as (x, y) → (y, x).

Now, apply rule to coordinates ABC:

A': (2, 3)

B': (1, -4)

C': (-2, -3)

Answer:

got it right on odyssey

A'(2,3)

B'(1,-4)

C'(-2,-3)

Step-by-step explanation:

[tex]n,n+1[/tex] - two consecutive integers

[tex]n+n+1=9\\2n=8\\n=4\\n+1=5[/tex]

4 and 5

Final answer:

To convert a whole number to a percent, multiply the number by 100. This process turns the whole number into a fraction with a denominator of 100, which can then be written as a percent. This simple multiplication can be done on most calculators.

Explanation:

Converting Whole Numbers to Percent

To convert a whole number to a percent, you simply need to think of the whole number as a fraction with a denominator of 1 and then convert it to an equivalent fraction with a denominator of 100. Once you have this fraction, you simply write it as a percent. Here is a step-by-step process using a calculator:

Since a whole number is over 1, multiply this number by 100. (5 * 100).

The result will be the whole number as a percent (e.g., 500%).

It's important to remember that calculating percents is essentially finding out how many parts out of a hundred the number represents. When using a calculator, this process may involve additional steps or functions, depending on the calculator's design.

For example, with some calculators, you can enter the number, press the multiplication key, enter 100, and then press the equals key to get the percent (e.g., 5 * 100 = 500%).

Remember that calculating percents can be further applied to situations where you have a 'part' and you want to find out what percentage that 'part' is of a 'total'. In such cases, you would divide the 'part' by the 'total' and multiply by 100 to get the percentage. For instance, if 13 out of 35 students in a class wear sandals, the percentage of students wearing sandals can be calculated as (13/35) * 100 which equals approximately 37%.

Answer:

  about 18.8°

Step-by-step explanation:

The depth of the tumor (1.8 cm) is the leg of the right triangle that is opposite the angle of interest. The offset distance (5.3 cm) is the adjacent leg of the triangle.

We know that ...

  tan(angle) = opposite/adjacent = (1.8 cm)/(5.3 cm) = 18/53

Then the angle is found from ...

  angle = arctan(18/53) ≈ 18.76°

_____

The arctangent, or inverse of the tangent function, is also written tan⁻¹. It may be a "second function" of your calculator's tan key.

The angle to aim the gamma ray source to hit the tumor 1.8cm beneath the skin without damaging the vital organ, while moving the source over 5.3cm, can be determined by calculating the inverse tangent (arctan) of the ratio 1.8cm / 5.3cm.

In physics, this problem involves the concept of triangles and trigonometry. Since a right triangle can be formed in this configuration with the distance beneath the patient's skin being one side (1.8cm), the distance over which the radiologist moves the gamma ray source being the second side (5.3cm), the angle required would be an inverse tangent (arctan) of the ratio between these two lengths.

The calculation is thus as follows:

Firstly, set up the ratio of the opposite over the adjacent side of the triangle. This equates to 1.8cm / 5.3cm.

Next, compute the inverse tangent (arctan) of this ratio. You can compute this using a scientific calculator. The answer will be in degrees.

Thus, the required angle to aim the gamma ray source to hit the tumor without damaging the vital organ will be the result of the above calculation.

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Answer:  (f·g)(x) = -5x³ - 31x² + 62x + 18

               (f·g)(-1) = -70

(fog)(x) = 25x² - 130x + 155

(fog)(-1) = 310

Step-by-step explanation:

f(x) = x² + 8x + 2          g(x) = -5x + 9

(f·g)(x) = (x² + 8x + 2)(-5x + 9)

          = -5x³ + 9x²

                    - 40x² + 72x

                               - 10x + 18

          = -5x³ - 31x² + 62x + 18

(f·g)(-1)= -5(-1)³ - 31(-1)² + 62(-1) + 18

         =  -5(-1)  - 31(1)    - 62      + 18

         =    5     -   31      - 62      + 18

         =  -70

****************************************************************************************

(fog)(x) = (-5x + 9)² + 8(-5x + 9) + 2

           = 25x² - 90x + 81

                       - 40x + 72

                                +   2

           = 25x² - 130x + 155

(fog)(-1) = 25(-1)² - 130(-1) + 155

            =   25    +  130    + 155

            = 310

It wasn't clear if you wanted multiplication or composition so I solved both.

The composition (f⋅g)(x) is 50, and the product of functions (f.g)(-1) is -20.

To find the composition of the functions f(x) and g(x), denoted as (f⋅g)(x), we first need to determine g(x). Given g(x) = -5 + 9, we simply add these numbers together to get g(x) = 4. With g(x) found, we can now substitute g(x) into f(x) to find the composition. To do this, wherever there is an x in f(x), we replace it with 4.

So, (f⋅g)(x) = f(g(x)) = f(4) = 4^2 + 8(4) + 2 = 16 + 32 + 2 = 50.

Next, to find (f.g)(-1), which represents the product of the two functions evaluated at x = -1, we evaluate both f(-1) and g(-1). We already know that g(x) is constant at 4, so g(-1) = 4. Now we find f(-1):

f(-1) = (-1)^2 + 8(-1) + 2 = 1 - 8 + 2 = -5.

Thus, the product at x = -1 is:

(f.g)(-1) = f(-1) ⋅ g(-1) = (-5) ⋅ 4 = -20.

Answer:

  A)  The population in the year that she was born.

Step-by-step explanation:

The multiplier of an exponential function is the value that function has when the exponent is zero -- the initial value. The initial value of population in this context is the population in the year Adriana was born.