Read the following scenario, which is represented by a polynomial expression. Then answer the questions to interpret the parts of the expression in terms of the given context.

The volunteers at a high school football team’s concession stand are trying to decide on the price of the hot dogs they are selling. When they charge $2 for a hot dog, they sell an average of 70 hot dogs per game. With every $1 increase in the price, the number of hot dogs sold per game decreases by 8.

The volunteers can calculate the revenue earned from selling the hot dogs at each game using the expression -8x2 + 54x + 140, where x is the number of $1 increases in price.

Part A
What is the constant term in the polynomial expression, and what does it represent?

Answer: x = 2

Step-by-step explanation:

P-----------------------T----------------------Q

         (3x-3)                   (5x-7)

Since T is the midpoint we know that PT and TQ are equal

Just solve the equation: 3x-3 = 5x-7

[tex]3x-3 = 5x-7[/tex]

now move the x to one side

[tex]-3 = 2x-7[/tex] (I subtracted the 3x)

then get the 2x by itself

[tex]4=2x[/tex]

lastly, divide by 2 to get x by itself

[tex]x=2\\[/tex]

Answer:

Option C

Step-by-step explanation:

Step 1:  Find the correct method

Option A is incorrect because we don't have m + 6 and 5m - 2

Option B is incorrect because that wouldn't show us the correct value

Option C is correct, once we solve for v, we can plug in v and get the value of m.  For example:  v + 6 = 5v - 2 → v + 8 = 5v → 8 = 4v → 2 = v.  Then we plug it into the other equation m = 2 + 6 → m = 8

Option D is incorrect because that wouldn't show us the correct value.

Answer: Option C

Answer:

C has a zero slope

Step-by-step explanation:

A horizontal line has a slope of zero

A vertical line has an undefined slope

A positive slope goes up from left to right

A negative slope goes down from left to right

Step-by-step explanation:

Recall that [tex]\log 10 = 1[/tex] so we can rewrite the equation above as

[tex]\log 10 + \log 2 = \log (x + 1)[/tex]

Also recall that [tex]\log(ab) = \log a + \log b[/tex] so the left-hand side becomes

[tex]\log [(10)(2)] = \log (x + 1)[/tex]

or

[tex]20 = x + 1 \Rightarrow x = 19[/tex]

Answer:

a. 0.042 = 4.2% probability that the chosen applicants are either all managers or all engineers.

b. 0.2398 = 23.98% probability that the number of managers is the same as the number of engineers on the task force.

c. The expected number of engineers chosen is 4.

d. 0.958 = 95.8% probability that at least one manager is chosen for the task force.

Step-by-step explanation:

The positions are chosen without replacement, which means that the hypergeometric distribution is used to solve this question.

Hypergeometric distribution:

The probability of x successes is given by the following formula:

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]

In which:

x is the number of successes.

N is the size of the population.

n is the size of the sample.

k is the total number of desired outcomes.

Combinations formula:

[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

In this question:

5 + 10 = 15 applicants, which means that [tex]N = 15[/tex]

10 are engineers, which means that [tex]k = 10[/tex]

Six members are chosen, which means that [tex]k = 6[/tex]

a. What is the probability that the chosen applicants are either all managers or all engineers?

Not possible having all managers(five applicants are manager, while there are 6 open positions), so this is P(X = 6), that is, all engineers.

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]

[tex]P(X = 6) = h(6,15,6,10) = \frac{C_{10,6}*C_{5,0}}{C_{15,6}} = 0.042[/tex]

0.042 = 4.2% probability that the chosen applicants are either all managers or all engineers.

b. What is the probability that the number of managers is the same as the number of engineers on the task force?

3 engineers, so this is P(X = 3).

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]

[tex]P(X = 3) = h(3,15,6,10) = \frac{C_{10,3}*C_{5,3}}{C_{15,6}} = 0.2398[/tex]

0.2398 = 23.98% probability that the number of managers is the same as the number of engineers on the task force.

c. What is the expected number of engineers chosen?

The expected value of the hypergeometric distribution is:

[tex]E(X) = \frac{nk}{N}[/tex]

So

[tex]E(X) = \frac{6(10)}{15} = 4[/tex]

The expected number of engineers chosen is 4.

d. What is the probability that at least one manager is chosen for the task force?

At most five engineers, which is:

[tex]P(X \leq 5) = 1 - P(X = 6)[/tex]

Since in item a. we already have P(X = 6).

[tex]P(X \leq 5) = 1 - 0.042 = 0.958[/tex]

0.958 = 95.8% probability that at least one manager is chosen for the task force.

Answer:

Step-by-step explanation:

[tex]R(x)=\frac{x+3}{x-5} \geq 0\\R(x)=0,gives~x+3=0,x=-3\\R(x)>0 ,if~both~numerator ~and~denominator~are~of~same~sign.\\let~x+3>0,x>-3\\and~x-5>0,x>5\\combining \\x>5\\\\again~let~x+3<0,x<-3\\x-5<0,x<5\\combining\\x<-3\\Hence~R(x)\geq 0\\if ~x \in ~[- \infty,-3]U(5,\infty)[/tex]

Answer:

The point at the bottom has to move over 2 to the left to be aligned with the point at the top however they will have a 3 space in between the 2 same for the point at the top, the top point moves over 2 to the right to be aligned with the bottom point, then they will have a 3 square space between each other.

Answer:Around 3 spaces between?

Step-by-step explanation:

Answer:

Step-by-step explanation:

For a function given as,

f(x)= 2x + 2

Domain of the given function is → {-5, -1, 2, 3}

For the Range of the given function,

f(-5) = 2(-5) + 2

      = -8

f(-1) = 2(-1) + 2

      = -2 + 2

      = 0

f(2) = 2(2) + 2

     = 6

f(3) = 2(3) + 2

     = 8

Therefore, set for the range will be → {-8, 0, 6, 8}

Now plot the ordered pairs on the graph,

(-5, -8), (-1, 0), (2, 6), (3, 8)

Answer:

The 90% confidence interval for the mean waste recycled per person per day for the population of Montana is between 2.68 and 2.92 pounds.

Step-by-step explanation:

We have the standard deviation for the sample, which means that the t-distribution is used to solve this question.

The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So

df = 7 - 1 = 6

90% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 6 degrees of freedom(y-axis) and a confidence level of [tex]1 - \frac{1 - 0.9}{2} = 0.95[/tex]. So we have T = 1.9432.

The margin of error is:

[tex]M = T\frac{s}{\sqrt{n}} = 1.9432\frac{0.16}{\sqrt{7}} = 0.12[/tex]

In which s is the standard deviation of the sample and n is the size of the sample.

The lower end of the interval is the sample mean subtracted by M. So it is 2.8 - 0.12 = 2.68 pounds.

The upper end of the interval is the sample mean added to M. So it is 2.8 + 0.12 = 2.92 pounds.

The 90% confidence interval for the mean waste recycled per person per day for the population of Montana is between 2.68 and 2.92 pounds.

Answer:

The second car traveled 280 miles farther than the first car.

Step-by-step explanation:

Car A: 50 mi/1 hr for 4 hrs

Car B: 60 mi/1 hr for 8 hrs

Solve for Car A:

50 mi/1 hr × 4 hrs

50 × 4

200 miles

Solve for Car B:

60 mi/1 hr × 8 hrs

60 × 8

480 miles

Find the difference between Car A and Car B:

480 miles - 200 miles

280 miles

Answer:

Option B

Step-by-step explanation:

Option A

a² - b² = (a+ b)(a - b)

It's a polynomial identity.

Option B

a³ + b³ = (a - b)(a² - ab + b²)

It's not a polynomial identity.

Because the identity is,

a³ + b³ = (a + b)(a² - ab + b²)

Option C

a³ - b³ = (a - b)(a² + ab + b²)

It's a polynomial identity.

Option D

(a²+ b²)(c² + d²) = (ac - bd)² + (ad + bc)²

                         = a²c² - 2abcd + b²d² + a²d² + b²c² + 2abcd

                         = a²c² + b²c² + b²d² + a²d²

                         = c²(a² + b²) + d²(a² + b²)

                         = (a²+ b²)(c² + d²)

Therefore, it's a polynomial identity.

Option B will be the answer.

Answer:

B. 11.5

Step-by-step explanation:

That missing length can be solved via the equation a^2+b^2=c^2, where the hypotenuse is c. We know A and B, which is 7 and 9.

7^2+9^2=130

sqrt 130 is 11.4017543

Here, you kinda have to break the rules of rounding to say that it is B.

There could be a more efficient route for resolving this answer, but this is the method that I was taught.

AS IN THE PICTURE.............

Answer:

The endpoints of the latus rectum are [tex](12, -7)[/tex] and [tex](-8, -7)[/tex].

Step-by-step explanation:

A parabola with vertex at point [tex]C(x, y) = (h,k)[/tex] and whose axis of symmetry is parallel to the y-axis is defined by the following formula:

[tex](x-h)^{2} = 4\cdot p \cdot (y-k)[/tex] (1)

Where:

[tex]y[/tex] - Independent variable.

[tex]x[/tex] - Dependent variable.

[tex]p[/tex] - Distance from vertex to the focus.

[tex]h[/tex], [tex]k[/tex] - Coordinates of the vertex.

The coordinates of the focus are represented by:

[tex]F(x,y) = (h, k+p)[/tex] (2)

The latus rectum is a line segment parallel to the x-axis which contains the focus. If we know that [tex]h = 2[/tex], [tex]k = -2[/tex] and [tex]p = -5[/tex], then the latus rectum is between the following endpoints:

By (2):

[tex]F(x,y) = (2, -2-5)[/tex]

[tex]F(x,y) = (2,-7)[/tex]

By (1):

[tex](x-2)^{2} = -20\cdot (-7+2)[/tex]

[tex](x-2)^{2} = 100[/tex]

[tex]x - 2 = \pm 10[/tex]

There are two solutions:

[tex]x_{1} = 2 + 10[/tex]

[tex]x_{1} = 12[/tex]

[tex]x_{2} = 2-10[/tex]

[tex]x_{2} = -8[/tex]

Hence, the endpoints of the latus rectum are [tex](12, -7)[/tex] and [tex](-8, -7)[/tex].

2 Answers:

z = 1 + i   and   z = -1 - i

========================================================

Explanation:

We want z to be a complex number in the form z = a+bi, where a,b are real numbers and [tex]i = \sqrt{-1}[/tex] is imaginary.

Let's plug that into the equation your teacher gave you

[tex]z^2 = 2i\\\\(a+bi)^2 = 2i\\\\(a+bi)(a+bi) = 2i\\\\a(a+bi)+bi(a+bi) = 2i\\\\a^2+abi+abi+b^2*i^2 = 2i\\\\a^2+2abi+b^2*(-1) = 2i\\\\a^2+2abi-b^2 = 2i\\\\(a^2-b^2)+2abi = 0+2i\\\\[/tex]

You could use the FOIL rule to take a shortcut. I'm deciding to be a bit more wordy to show a further breakdown how everything is multiplying out.

Notice that the real part a^2-b^2 must be 0 so that it matches the real part on the right hand side.

a^2-b^2 = 0

(a-b)(a+b) = 0 .... difference of squares rule

a-b = 0 or a+b = 0

a = b or a = -b

So whatever solution z = a+bi is, it must have either a = b or a = -b.

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

If a = b, then the 2abi portion on the left side turns into 2a^2*i

Set this equal to 2i on the right hand side and isolate 'a'

[tex]2a^2*i = 2i\\\\2a^2 = 2\\\\a^2 = 1\\\\a = 1 \text{ or } a = -1\\\\[/tex]

So a = 1 leads to b = 1

Or a = -1 leads to b = -1

Two complex solutions so far are:   z = 1 + i   and   z = -1 - i  based on those two cases above.

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

Now consider the case that a = -b

We'll effectively have the same steps as the previous section, but the equation to solve now is [tex]-2a^2*i = 2i\\\\[/tex]

The only difference is that negative is out front. You should find that it leads to a^2 = -1, but this has no solutions because we stated earlier that a,b were real numbers.

So if a = -b, then it concludes with a,b being nonreal numbers. Ultimately we rule out the case that a = -b is possible.

Put another way, note how -2a^2 is always negative which clashes with the idea that the right hand side is positive (ignore the 'i' portions). This contradiction means that no real values of 'a' will make the equation [tex]-2a^2*i = 2i\\\\[/tex] to be true.

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

To wrap things up, we only have two solutions and they are  

z = 1 + i   and   z = -1 - i

You can use a tool like WolframAlpha to confirm this.

That's a question about exponentiation.

Answer:

Kenji is wrong because he does not aply the porperty correctly.

Step-by-step explanation:

A exponetiation has this form:

[tex]\boxed{a^b}[/tex]

a is the base

b is the power or exponent

To understand that situation it's important to know a property about exponentiation. When we have a multiplication with the same exponent and diferent bases, the result is the multiplication of the bases with the same exponent. Let's see this above, in mathematical language:

[tex]\boxed{a^b \cdot c^b = (a\cdot c) ^b}[/tex]

Examples:

Now, we can say why Kenji is wrong. It's easy simplify [tex]3^5 \cdot 4^5[/tex]! We know that the result is [tex](3 \cdot 4) ^5 = 12^5[/tex], but Kenji multiplied the bases and added the exponents, that's why he is wrong.

I hope I've helped. ^^

Enjoy your studies! \o/

Answer:

u still doing school :)

Step-by-step explanation:

9514 1404 393

Answer:

  (b)  (negative 2, StartFraction 5 over 3 EndFraction)

Step-by-step explanation:

The value of x is given, so you only need to substitute that into the first equation to find y.

  y = 2/3(-2) +3 = -4/3 +9/3

  y = 5/3

The solution is (x, y) = (-2, 5/3).

Answer:

negative 2, StartFraction 5 over 3 EndFraction)

Step-by-step explanation:

9514 1404 393

Answer:

  f(x) = 3

Step-by-step explanation:

Replace sin(x) with 0 and you will have it.

  f(x) = 12·0 +3

  f(x) = 3

Answer:

y= 3

Step-by-step explanation:

I took the test

Answer:

See answers below

Step-by-step explanation:

From the given functions, the equivalent function for when x = 0 is -(x-1)²

h(x) = -(x-1)²

h(0) = -(0-1)²

h(0)= -(-1)²

h(0) = -1

when x = 2, the equivalent function is -1/2x - 1

h(x) =  -1/2x - 1

h(2) =  -1/2(2) - 1

h(2) = -1-1

h(2) = -2

when x = 5, the equivalent function is -1/2x - 1

h(x) =  -1/2x - 1

h(5) =  -1/2(5) - 1

h(5) = -5/2-1

h(5) = -7/2

Answer: I dont know if this is right

Step-by-step explanation:

EC/CA= ED/DB

EC/EA = DE/BA

EB/ED= AE/DE

DB/EB = CD/AE

Answer:

a) 0.0797 = 7.97% probability that the next home run will be on his fifth at-bat.

b) The expected number of at-bats until the next home run is 6.25.

Step-by-step explanation:

For each at bat, there are two possible outcomes. Either it is a home run, or it is not. The probability of an at bat resulting in a home run is independent of any other at-bat, which means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

The probability that Barry Bonds hits a home run on any given at-bat is 0.16

This means that [tex]p = 0.16[/tex]

Part A: What is the probability that the next home run will be on his fifth at-bat?

0 on his next 4(P(X = 0) when n = 4)

Home run on his 5th at-bat, with 0.16 probability. So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 0) = C_{4,0}.(0.16)^{0}.(0.84)^{4} = 0.49787136 [/tex]

0.49787136 *0.16 = 0.0797.

0.0797 = 7.97% probability that the next home run will be on his fifth at-bat.

Part B: What is the expected number of at-bats until the next home run?

The expected number of trials for n successes is given by:

[tex]E = \frac{n}{p}[/tex]

In this question, [tex]n = 1, p = 0.16[/tex]. So

[tex]E = \frac{1}{0.16} = 6.25[/tex]

The expected number of at-bats until the next home run is 6.25.

Answer:

I AM VERY SORRY AARUSH KAPUSH

Step-by-step explanation:

Answer:

Option B.

Step-by-step explanation:

3 points are collinear if we can draw a line that connects the points.

And we know that any 2 or 3 points are always coplanar because we can find a plane such that the 2 or 3 points belong to it.

In the image we can see that B, A, and E are at the same y-value, thus these points are collinear, then the points define a line, and these are 3 points, thus we know that are coplanar.

Then points A, B, and E are collinear and coplanar.

The correct option is B.

Answer:

horizontal expansion factor of 2

2^x =2(2)=4

2^4=2×2×2×2= 16

Answer:

i would switch

Step-by-step explanation:

your chance of picking the right door that has the grand price is 33.33% because 1 out of 3 doors has the prize and you have a 66.66% chance of being wrong.

*just know that the host will never reveal the grand prize first because then the tension of the game show is gone, etc *

after knowing the door that has a goat, are my chances of winning 50/50 (1 out of 2 doors) ?

no.

Always switch!

your chances are still the same as before (33.33%). like i mentioned the host won't reveal the grand prize first.

I will refer to 3 doors: X, Y, Z

So if your 1 out of 3 pick wasn't the money(you chose x), and the money is in, let's say door Y, then the host will reveal Z. If the money is in Z, the host reveals Y. IF you chose the money the first time (X), then the host can reveal either Y or Z.  no matter what, you are still stuck in that initial 33.33% chance that you chose right the very first time. But if you switch, regardless of the prize, you are now in the 66.66% zone. you have actually doubled your chances of winning.

to think about it in another way, when you are being asked to switch, you are given a dilemma : do you want to keep one envelope of do you want both of the others? you already know what is inside one of them (the goat). But since the one that will be revealed won't have the money in it, the chances that the other envelope has the prize are twice as high.

[tex] \frac{dw}{dt} = \frac{dx}{dt} \times \frac{dw}{dx} + \frac{dy}{dt} \times \frac{dw}{dy} + \frac{dz}{dt} \times \frac{dw}{dz} \\ \frac{dw}{dt} = 3t ^{2} \times {e}^{ \frac{y}{z} } + - t \times \frac{x {e}^{ \frac{y}{z} } }{z} + 4 \times - \frac{xy {e}^{ \frac{y}{z} } }{ {z}^{2} } \\ \frac{dw}{dt} = \frac{ {e}^{ \frac{y}{z} } (3 {t}^{2} {z}^{2} - tzx - 4xy)}{ {z}^{2} } [/tex]

========================================================

Explanation:

Since f(x) is linear, this means f(x) = mx+b

Let's plug in x = 6

[tex]f(x) = mx+b\\f(6) = m*6+b\\f(6) = 6m+b[/tex]

Repeat for x = 2

[tex]f(x) = mx+b\\f(2) = m*2+b\\f(2) = 2m+b[/tex]

Now subtract the two function outputs

[tex]f(6)-f(2) = (6m+b)-(2m+b)\\f(6)-f(2) = 6m+b-2m-b\\f(6)-f(2) = 4m\\[/tex]

The b terms cancel out which is very handy.

Set this equal to 12, since f(6)-f(2) = 12, and solve for m

[tex]f(6)-f(2) = 12\\4m = 12\\m = 12/4\\m = 3\\[/tex]

So the slope of f(x) is m = 3

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

Next, plug in x = 12

[tex]f(x) = mx+b\\f(12) = m*12+b\\f(12) = 12m+b[/tex]

We can then say:

[tex]f(12)-f(2) = (12m+b)-(2m+b)\\f(12)-f(2) = 12m+b-2m-b\\f(12)-f(2) = 10m\\[/tex]

Lastly, we plug in m = 3

[tex]f(12)-f(2) = 10m\\f(12)-f(2) = 10*3\\f(12)-f(2) = 30\\[/tex]