Drag the tiles to the correct boxes to complete the pairs.

The owner of a real estate agency is looking over the monthly commission earned by the top three brokers in the past month. She has calculated the mean and standard deviation, SD, of the their commissions, in dollars, as shown below.


Amanda Benjamin Cathy

Mean = 75,463.12

SD = 3,839.02 Mean = 74,124.87

SD = 4,062.50 Mean = 76,095.71

SD = 4,227.54


Match each broker with the correct description.


Amanda

Benjamin

Cathy

Which broker had the highest average monthly

commission during the past year?

arrowBoth

For which broker does 95% of the data lie

between $67,785.08 and $83,141.16

(2 SD from the mean)?

arrowBoth

For which broker does 68% of the data lie

between $70,062.37 and $78,187.37

(1 SD from the mean)?

arrowBoth

Answer:

B

Step-by-step explanation:

Conditional probability is:

P(A given B) = P(A and B) / P(B)

Here, P(A and B) = 0.052 and P(B) = 0.17:

P(A given B) = 0.052 / 0.17

P(A given B) = 0.306

Answer:

The vertex is: [tex](6, 8)[/tex]

Step-by-step explanation:

First solve the equation for the variable y

[tex]x^2-4y-12x+68=0[/tex]

Add 4y on both sides of the equation

[tex]4y=x^2-4y+4y-12x+68[/tex]

[tex]4y=x^2-12x+68[/tex]

Notice that now the equation has the general form of a parabola

[tex]ax^2 +bx +c[/tex]

In this case

[tex]a=1\\b=-12\\c=68[/tex]

Add [tex](\frac{b}{2}) ^ 2[/tex] and subtract [tex](\frac{b}{2}) ^ 2[/tex] on the right side of the equation

[tex](\frac{b}{2}) ^ 2=(\frac{-12}{2}) ^ 2\\\\(\frac{b}{2}) ^ 2=(-6) ^ 2\\\\(\frac{b}{2}) ^ 2=36[/tex]

[tex]4y=(x^2-12x+36)-36+68[/tex]

Factor the expression that is inside the parentheses

[tex]4y=(x-6)^2+32[/tex]

Divide both sides of the equality between 4

[tex]\frac{4}{4}y=\frac{1}{4}(x-6)^2+\frac{32}{4}[/tex]

[tex]y=\frac{1}{4}(x-6)^2+8[/tex]

For an equation of the form

[tex]y=a(x-h)^2 +k[/tex]

the vertex is: (h, k)

In this case

[tex]h=6\\k =8[/tex]

the vertex is: [tex](6, 8)[/tex]

Answer: 6, 8

Step-by-step explanation:

The approximate difference in the maximum height achieved by the two projectiles is 5.4 ft. (Option C).

How to calculate the difference between two maximum heights?

The approximate difference in the maximum height achieved by the two projectiles is calculated as follows;

The given function of one of the projectile;

f(x) = -16x² + 42x + 12

The function of the second projectile shown in the table, shows that the maximum of the function, g is 33

g(1) = 33 ft (maximum height)

The maximum height attained by the projectile with f(x) function occurs at x = 1

f(1) = -16(1)² + 42(1) + 12

f(1) = 38 ft

The difference between two maximum heights;

Δh = f(1) - g(1)

Δh = 38 ft - 33 ft

Δh = 5 ft

The option that is approximately 5 ft is option C (5.4 ft).

Answer:

[tex]900\ cm^2[/tex]

Step-by-step explanation:

We can notice that the the prism provided is a rectangular prism.

By definition, The lateral  area of a rectangular prism can be calculated by multiplying the  perimeter of its base by its height.

The height is:

[tex]height=15\ cm[/tex]

Then, the perimeter of the base is:

[tex]Perimeter=17\ cm+17\ cm+13\ cm+13\ cm=60\ cm[/tex]

Then the lateral area is:

[tex]LA=60\ cm*15\ cm\\\\LA=900\ cm^2[/tex]

The statement that explains why the squares are similar is

Option C. Translations and dilations preserve betweenness of points; therefore, the corresponding sides of squares T and T″ are proportional.

There are several types of transformations:

Let us now tackle the problem!

[tex]\texttt{ }[/tex]

This problem is about Translation and Dilation.

Properties of Translation of the images compared to pre-images:

[tex]\texttt{ }[/tex]

Properties of Dilation of the images compared to pre-images:

[tex]\texttt{ }[/tex]

From the information above, we can conclude that:

Option A is not true because Dilations do not preserve side length.

Option B is not true because Dilations do not preserve orientation.

Option C is true because Translations and Dilations preserve betweenness of points.

Option D is not true. Although Translation and Dilations preserve collinearity but it cannot be related to the corresponding angles are congruent.

[tex]\texttt{ }[/tex]

Grade: High School

Subject: Mathematics

Chapter: Transformation

Keywords: Function , Trigonometric , Linear , Quadratic , Translation , Reflection , Rotation , Dilation , Graph , Vertex , Vertices , Triangle

Answer:

[tex]\dfrac{\sqrt[12]{55296}}{2}[/tex]

Step-by-step explanation:

Rationalize the denominator, then use a common root for the numerator.

[tex]\dfrac{\sqrt[4]{6}}{\sqrt[3]{2}}=\dfrac{(2\cdot 3)^{\frac{1}{4}}}{2^{\frac{1}{3}}}\\\\=\dfrac{(2\cdot 3)^{\frac{1}{4}}}{2^{\frac{1}{3}}}\cdot\dfrac{2^{\frac{2}{3}}}{2^{\frac{2}{3}}}=\dfrac{2^{\frac{1}{4}+\frac{2}{3}}3^{\frac{1}{4}}}{2}\\\\=\dfrac{2^{\frac{11}{12}}3^{\frac{3}{12}}}{2}=\dfrac{\sqrt[12]{2^{11}3^{3}}}{2}\\\\=\dfrac{\sqrt[12]{55296}}{2}[/tex]

Answer:

  72%

Step-by-step explanation:

QT has length 42-24 = 18.

ST has length 42-29 = 13.

The length ST is 13/18 ≈ 72.2% of the length of QT.

Answer:

Probability = 72.2%

Step-by-step explanation:

A number line contains points Q, R, S, and T with coordinated 24, 28, 29, and 42 respectively.

Now if a point lies on QT then the length of QT= coordinate of T - coordinate of Q

= 42 - 24

= 18

If a point lies on ST then the length of ST = coordinate of T - coordinate of S

= 42 - 29

= 13

Now we know Probability of an event = [tex]\frac{\text{Favorable event}}{\text{Total possible events}}\times 100[/tex]

Probability = [tex]\frac{13}{18}\times 100[/tex]

                  = 72.2%

Therefore, probability that a point chosen on QT will lie on ST will be 72.2%

Answer:

d = 13.8 feet

Step-by-step explanation:

Because we are talking about cubic feet of water, we need the formula for the VOLUME of a cylinder.  That formula is

[tex]V=\pi r^2h[/tex]

We will use 3.141592654 for pi; if the tank HALF filled with water is at 20 feet, then the height of the tank is 40 feet, so h = 40; and the volume it can hold in total is 6000 cubic feet.  Filling in then gives us:

[tex]6000=(3.141592654)(r^2)(40)[/tex]

Simplify on the right to get

[tex]6000=125.6637061r^2[/tex]

Divide both sides by 125.6637061 to get that

[tex]r^2=47.74648294[/tex]

Taking the square root of both sides gives you

r = 6.90988299

But the diameter is twice the radius, so multiply that r value by 2 to get that the diameter to the nearest tenth of a foot is 13.8

Answer:

9 + 15 = 3(3 + 5)

Step-by-step explanation:

The greatest common factor of 9 and 15 is 3.

Factor 3 out of both 9 and 15.

9 + 15 = 3(3 + 5)

Answer:

Measure 8 is 135°

Step-by-step explanation:

Measure 1 and Measure 8 are alternate exterior angles.

Answer:

B) 135°

Step-by-step explanation:

<1 = <4 vertical angles

<4 = <5 alternate interior angles

<5 = <8 vertical angles

Therefore <1 = <8

Since <1 = 135, we know <8 = 135

Answer:

Topmost option

Step-by-step explanation:

(see attached)

Step-by-step explanation:

You forgot to include the formula, but it has to be either a sine wave or cosine wave:

h = A sin(ωt + φ) + B

The coefficient A is called the amplitude.  The diameter of the ferris wheel is double the amplitude.

d = 2A

Let a = car's age in years and v = value of car.

v = 25000(1 - 0.17)^a 

v = 25000(0.83)^a 

v = 25000(0.83)^a 

We need to find a.

Let v = 10,000

10,000 = 25000(0.83)^a

The value of a is about 4.91758.

Round off to the nearest whole number we get 5.

Answer is after more than 4 years but less than 6 and 8.

Answer:

(x-3,y+2)

Step-by-step explanation:

Answer:  The correct option is

(B) [tex]T^{-1}(x,y)=(x-3,y+2).[/tex]

Step-by-step explanation:  Given that under T, the point (0,2) gets mapped to (3,0).

We are to find the expression for [tex]T^{-1}(x,y).[/tex]

According to the given information, we have

[tex]T(0,2)=(3,0)=(0+3,2-2)\\\\\Rightarrow T(x,y)=(x+3,y-2)\\\\\Rightarrow T^{-1}(x+3,y-2)=(x,y)\\\\\Rightarrow T^{-1}(x+3-3,y-2+2)=(x-3,y+2)\\\\\Rightarrow T^{-1}(x,y)=(x-3,y+2).[/tex]

Thus, the required expression is [tex]T^{-1}(x,y)=(x-3,y+2).[/tex]

Option (B) is CORRECT.

Answer:

Third choice

Step-by-step explanation:

The standard form of a circle is

[tex](x-h)^2+(y-k)^2=r^2[/tex]

where h and k are coordinates of the center and r is the radius squared.  We have h = 2, k = 7, and r = 4 (we will have to square it to fit it into the equation properly).  Filling in accordingly:

[tex](x-2)^2+(y-7)^2=16[/tex]

The third choice is the one you want.

Answer:

  C)  a positive correlation

Step-by-step explanation:

More people ⇒ Longer time is a positive correlation between those variables. However, longer time is not the desired outcome.

Rather, shorter time is the desired outcome. The correlation between more people and shorter time is negative. In order to compute that correlation numerically, one would have to define a function that would give a numerical value for "shorter time" that would model the goodness of outcome as time gets shorter.

Answer:

true

The wording does not quite mean anything,

but what I think was meant to ask is

"if we use some parts of two triangles to prove they are congruent,

can we then use that to prove that

a pair of corresponding parts not used before are congruent?"

The answer is

Yes, of course,

Corresponding Parts of Congruent Triangles are Congruent,

which teachers usually abbreviate as CPCTC.

For example, if we find that

side AB is congruent with side DE,

side BC is congruent with side EF, and

angle ABC is congruent with angle DEF,

we can prove that triangles ABC and DEF are congruent

by Side-Angle-Side (SAS) congruence.

We then, by CPCTC, can conclude that other pairs of corresponding parts are congruent:

side AB is congruent with side DE,

angle BCA is congruent with angle EFD, and

angle CAB is congruent with angle FDE.

It was possible (by CPCTC) to prove those last 3 congruence statements,

after proving the triangles congruent.

The expected answer is FALSE.

Step-by-step explanation:

Answer:

  (a) see below for a graph

  (b) the vertex is (months, sales) = (12, 15); sales is $15M at 12 months

  (c) .375 million per month increasing and decreasing

Step-by-step explanation:

(a) You can read the vertex from the equation of the function. The equation is of an absolute value function translated so its vertex is at (12, 15), and vertically scaled by a factor of -0.375. The negative scale factor means the graph will open downward.

__

(b) Since graph is of sales in millions versus months, the meaning of the vertex at (12, 15) is that sales is $15 millions 12 months after the phone comes on the market.

__

(c) If the function were written as a piecewise function, the coefficient of x would be +0.375 for x < 12 and -0.375 for x > 12. The "rate of change" is 0.375 millions per month both going up and coming down.

_____

Translation of a point on the graph of f(x) by "h" horizontal units and "k" vertical units changes the function to f(x -h) +k. That is, if you can identify the function f(x), you can read the translation from the expression f(x -h) +k. For the absolute value function |x|, the vertex is normally (0, 0). Translating it to (h, k) makes the expression be |x -h|+k, the form you see in this problem. It's not a mystery. It's just pattern matching.

Answer:

  see below

Step-by-step explanation:

The image is reflected across the origin and enlarged by a factor of 3.

___

The first choice shows some funny combination of translation, rotation, and dilation. The last choice has point N invariant, which means that is the center of the (horizontal only) dilation. Neither of these matches the problem description.

D. [tex]x[/tex] represents the number of level 1 tickets; [tex]y[/tex] represents the number of level 2 tickets

The costs of the tickets are represented as constants ([tex]50[/tex] and [tex]150[/tex]), and so they are not variables.

We know [tex]x[/tex] is level 1 tickets and [tex]y[/tex] is level 2 tickets because the price of a level 1 ticket is $50 and the second equation contains [tex]50x[/tex].  Similarly, the cost of a level 2 ticket is $150 and the equation contains [tex]150y[/tex].

Answer:

(D) X represents the number of level 1 tickets.

[tex]a^{\log_a b}=b\\\\12^{\log_{12}24}=24[/tex]

Answer:

The correct answer option is A) 24.

Step-by-step explanation:

We are given the following log expression and we are to simplify it:

[tex] 1 2 ^ { log _ { 1 2 } } ^ { 2 4 } [/tex]

Here, we are going to apply the rule for solving a log problem:

[tex]a^{log_a^{(b)}[/tex] [tex] = b[/tex]

So if [tex] 1 2 ^ { log _ { 1 2 } } ^ { 2 4 } [/tex], then it would be equal to 24.

1. Let [tex]a,b,c[/tex] be the three points of intersection, i.e. the solutions to [tex]f(x)=g(x)[/tex]. They are approximately

[tex]a\approx-3.638[/tex]

[tex]b\approx-1.862[/tex]

[tex]c\approx0.889[/tex]

Then the area [tex]R+S[/tex] is

[tex]\displaystyle\int_a^c|f(x)-g(x)|\,\mathrm dx=\int_a^b(g(x)-f(x))\,\mathrm dx+\int_b^c(f(x)-g(x))\,\mathrm dx[/tex]

since over the interval [tex][a,b][/tex] we have [tex]g(x)\ge f(x)[/tex], and over the interval [tex][b,c][/tex] we have [tex]g(x)\le f(x)[/tex].

[tex]\displaystyle\int_a^b\left(\dfrac{x+1}3-\cos x\right)\,\mathrm dx+\int_b^c\left(\cos x-\dfrac{x+1}3\right)\,\mathrm dx\approx\boxed{1.662}[/tex]

2. Using the washer method, we generate washers with inner radius [tex]r_{\rm in}(x)=2-\max\{f(x),g(x)\}[/tex] and outer radius [tex]r_{\rm out}(x)=2-\min\{f(x),g(x)\}[/tex]. Each washer has volume [tex]\pi({r_{\rm out}(x)}^2-{r_{\rm in}(x)}^2)[/tex], so that the volume is given by the integral

[tex]\displaystyle\pi\int_a^b\left((2-\cos x)^2-\left(2-\frac{x+1}3\right)^2\right)\,\mathrm dx+\pi\int_b^c\left(\left(2-\frac{x+1}3\right)^2-(2-\cos x)^2\right)\,\mathrm dx\approx\boxed{18.900}[/tex]

3. Each semicircular cross section has diameter [tex]g(x)-f(x)[/tex]. The area of a semicircle with diameter [tex]d[/tex] is [tex]\dfrac{\pi d^2}8[/tex], so the volume is

[tex]\displaystyle\frac\pi8\int_a^b\left(\frac{x+1}3-\cos x\right)^2\,\mathrm dx\approx\boxed{0.043}[/tex]

4. [tex]f(x)=\cos x[/tex] is continuous and differentiable everywhere, so the the mean value theorem applies. We have

[tex]f'(x)=-\sin x[/tex]

and by the MVT there is at least one [tex]c\in(0,\pi)[/tex] such that

[tex]-\sin c=\dfrac{\cos\pi-\cos0}{\pi-0}[/tex]

[tex]\implies\sin c=\dfrac2\pi[/tex]

[tex]\implies c=\sin^{-1}\dfrac2\pi+2n\pi[/tex]

for integers [tex]n[/tex], but only one solution falls in the interval [tex][0,\pi][/tex] when [tex]n=0[/tex], giving [tex]c=\sin^{-1}\dfrac2\pi\approx\boxed{0.690}[/tex]

5. Take the derivative of the velocity function:

[tex]v'(t)=2t-9[/tex]

We have [tex]v'(t)=0[/tex] when [tex]t=\dfrac92=4.5[/tex]. For [tex]0\le t<4.5[/tex], we see that [tex]v'(t)<0[/tex], while for [tex]4.5<t\le8[/tex], we see that [tex]v'(t)>0[/tex]. So the particle is speeding up on the interval [tex]\boxed{\dfrac92<t\le8}[/tex] and slowing down on the interval [tex]\boxed{0\le t<\dfrac92}[/tex].

Probability of  randomly picking a yellow marble is [tex]\frac{2}{9}[/tex].

Probability of not picking a yellow marble is [tex]\frac{7}{9}[/tex].

Probability denotes the possibility of the outcome of any random event. The meaning of this term is to check the extent to which any event is likely to happen.

According to the question

A bag contains 4 red marbles, 3 green marbles, and 2 yellow marbles.

Probability of  randomly picking a yellow marble is

Number of favorable outcomes = 2

Total number of favorable outcomes = 4 + 3 + 2 = 9

Probability of  randomly picking a yellow marble is [tex]\frac{2}{9}[/tex].

Probability of not picking a yellow marble

= 1 - Probability of  randomly picking a yellow marble

= 1 - [tex]\frac{2}{9}[/tex]

= [tex]\frac{9-2}{9}[/tex]

= [tex]\frac{7}{9}[/tex]

Probability of not picking a yellow marble is [tex]\frac{7}{9}[/tex].

Hence,

Probability of  randomly picking a yellow marble is [tex]\frac{2}{9}[/tex].

Probability of not picking a yellow marble is [tex]\frac{7}{9}[/tex].

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The probability of not picking a yellow marble from a bag with 9 marbles total, where only 2 are yellow, is 7/9 or approximately 77.78%.

The question is related to calculating the probability of an event not occurring—in this case, not picking a yellow marble from a bag that contains different colored marbles. The total number of marbles is 4 red marbles + 3 green marbles + 2 yellow marbles = 9 marbles. To find the probability of not picking a yellow marble, we consider the total marbles that are not yellow, which are 4 red + 3 green = 7 marbles. The probability is then the number of non-yellow marbles divided by the total number of marbles, so it's 7/9. To calculate this, we divide the 7 non-yellow marbles by the total of 9 marbles, resulting in a probability of 7/9, or about 77.78%.

[tex]9^4\cdot26^2=6561\cdot 676=4435236[/tex]

The total number of personal access codes that can be formed is,

= 4435236 possible ways

To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.

Given that;

Total digits of code = 6

Hence, We get;

Code options for first 4 digits = any of 1 - 9 = 9 options

Code option for last 2 digits = A - Z = 26 options

So,

Code number 1 = 9 possible values

Code number 2 = 9 possible values

Code number 3 = 9 possible values

Code number 4 = 9 possible values

Code number 5 = 26 possible values

Code number 6 = 26 possible values

Hence, total number of possible access codes :

= 9 x 9 x 9 x 9 x 26 x 26

= 9⁴ x 26²

= 4435236 possible ways

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Answer:

  t ≈ 1.118 . . . seconds

Step-by-step explanation:

Set h=0 and solve for t.

  0 = -16t^2 +20

  0 = t^2 -20/16 . . . . . . . . . . . . . . . divide by the coefficient of t^2

  t = √(5/4) = (1/2)√5 ≈ 1.118 . . . . . add 5/4 and take the square root

The pinecone hits the ground about 1.12 seconds after it drops.

For the mathematical model h = -16t² + 20, corresponding to a pinecone dropping from a tree, the pinecone would hit the ground after approximately 1.118 seconds.

In order to know when a pinecone hits the ground, we would need to solve the equation provided for the variable t when h equals zero, as that would represent the pinecone being on the ground. The equation given is quadratic in nature: h = -16t² + 20. In this equation, h represents the height of the pinecone, and t represents time in seconds.

To find when the pinecone hits the ground (h=0), we set h to zero and solve for t:

0 = -16t² + 20
Therefore, 16t² = 20
So, t² = 20/16 = 1.25
Then, t = sqrt(1.25) = 1.118 (remember we exclude negative root as it doesn't go with time).

The pinecone hits the ground approximately at t = 1.118 seconds.

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Answer:

x-2y = -12

Step-by-step explanation:

Standard form of a line is in the form Ax + By = C   where A is a positive integer

y − 3 = 1/2(x + 6)

Multiply each side by 2 to eliminate the fractions

2(y-3)= 1/2*2 (x+6)

Distribute

2y -6 = x+6

Subtract x from each side

-x +2y -6 = x-x +6

-x+2y -6 = 6

Add 6 to each side

-x+ 2y -6+6 = 6+6

-x +2y = 12

Multiply each side by -1 to make A a positive integer

x-2y = -12

Answer:

889

Step-by-step explanation:

The required proof that the measure of the angle A is  60° is below in the solution part.

Triangle is defined as a basic polygonal shape of a triangle that has three sides and three interior angles.

By the triangle sum theorem, the sum of angles in a triangle is equal to 180°.

Therefore,  m∠A + m∠B + m∠C = 180  using the Substitution property (3x)° + 90° + (x+10)° = 180°

To solve for x, first combine like terms to get 4x + 100 = 180,

using the subtraction property of equality, 4x = 80

using the Division property of equality,  x = 20.

To find the measure of the angle A, use the substitution property to get m∠A = 3(20)°.

Finally, simplify the expression gets m∠A = 60°

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The missing figure has been attached below.

The tangent half angle formula, one of several, is

[tex]\tan \dfrac a 2 = \dfrac{1 - \cos a}{\sin a}[/tex]

We have θ is opposite 4 in the 3/4/5 right triangle so

[tex]\cos \theta = \dfrac{3}{5}[/tex]

[tex]\sin \theta = \dfrac{4}{5}[/tex]

[tex]\tan \dfrac{\theta}{2} = \dfrac{1 - 3/5}{4/5} = \dfrac{5-3}{4}=\dfrac{1}{2}[/tex]

Answer: 1/2

This is actually pretty deep.  It says half the big acute angle in the 3/4/5 triangle is the small diagonal angle of the 1x2 rectangle.   Similarly, the small acute angle in 3/4/5 triangle is twice the small diagonal angle of the 1x3 rectangle.

To find the value of tan(1/2)θ, we calculate θ using the fact that tan(θ) = opposite/adjacent = 4/3, then apply the half-angle formula from trigonometry. We cannot complete the calculation as we don't have the exact cosine value of θ.

The question asks to find the value of tan(1/2)θ where θ is an angle in standard position, and its terminal side passes through the point (3, 4). In this case, first, we need to find the value of θ. This can be found using the formula tan(θ)=opposite/adjacent. Given the point (3, 4), let's consider the coordinates as (x,y). Here, 3 is the x-coordinate, which acts as the adjacent side, and 4 is the y-coordinate, which acts as the opposite side. Therefore, θ=tan^-1(4/3).

To find the value of tan(1/2)θ, we can use the half-angle formula from trigonometry: tan(1/2)θ = √((1-cos(θ))/(1+cos(θ)))

However, this question does not provide enough information to determine which option is the solution, as the cosine of θ is needed for completing the calculation.

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Answer:

  B)  T = 2πr/v

Step-by-step explanation:

To solve the given equation for T, multiply it by T/v.

[tex]v=\dfrac{2\pi r}{T}\\\\v\dfrac{T}{v}=\dfrac{2\pi r}{T}\cdot\dfrac{T}{v}\\\\T=\dfrac{2\pi r}{v} \qquad\text{simplify}[/tex]