A mother who is 40 years old has a daughter and a son. The son is twice as old as the daugther. In 15 years the sum of all their ages will be 100 years. How old are the siblings now?

Answer:

Step-by-step explanation:

The total sold is the sum of the individual numbers sold, hence c+h.

We assume "3 times more" means "3 times as many", so the number of hotdogs sold (h) is 3 times the number of cheeseburgers sold (c), hence 3c.

  c + h = 220

  h = 3c

_____

55 cheeseburgers and 165 hotdogs were sold.

Answer:

x = 52/9

Step-by-step explanation:

The exterior angle is half the difference of the intercepted arcs, so we have ...

9x -5 = (158 -64)/2

9x = 52 . . . . . . . . . . . add 5

x = 52/9 = 5 7/9

Answer:

165

Step-by-step explanation:

[tex]5c^{2} -3d+15[/tex]

c = 6 and d = 10

[tex]5c^{2}[/tex] = 5 × 6² = 5 × 36 = 180

[tex]5c^{2}[/tex] - ( 3 d ) = 180 - ( 3 × 10 ) = 180 - 30 = 150

[tex]5c^{2}[/tex] -  3 d  ( + 15 ) = 150 + 15 = 165

Answer:

165

Step-by-step explanation:

Substitutet 6 for c and 10 for d in  5c^2 - 3d + 15​ .

Note that " ^ " is used here to denote exponentiation; c2 is meaningless.

Then we have 5(6)^2 - 3(10) + 15, or   180 - 30 + 15, or 165.

Answer:

C. there is a common difference of 3 for f(x) when x increases by 1

Step-by-step explanation:

As 3 is the slope of this function, there will be a common difference of 3 when x increases by 1.

f(x) = 3x  - 7

Let's think, whenever we add 1 to x it i'll increase 3 in the result

f(0) = 3.0 - 7 = 0 - 7 = -7

f(1) = 3.1 - 7 = 3 - 7 = -4

f(2) = 3.2 - 7 = 6 - 7 = -1

So we can know that there's a common difference of 3 for f(x) when x increase by 1.

To convert 88 square yards to square meters, we use the conversion factor 1 square yard = 1.196 square meters. By multiplying 88 by 1.196, we find that 88 square yards is approximately 105.3 square meters.

The question is part of the mathematics subject, specifically in the area of unit conversion. We have a conversion factor to use, which is 1 square yard = 1.196 square meters, based on the provided reference information.

So to convert 88 square yards to square meters, you multiply 88 by 1.196.

88 yards2 * 1.196 m2/yard2 = 105.3 m2.

so 88 square yards is approximately = 105.3 square meters.

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Answer:

P(cream and sugar) = 0.4

Step-by-step explanation:

* Lets study the meaning of or , and on probability

- The use of the word or means that you are calculating the probability

 that either event A or event B happened

-  Both events do not have to happen

- The use the word and, means that both event A and B have to happen

* The addition rules are:

# P(A or B) = P(A) + P(B) ⇒ mutually exclusive (events cannot happen

 at the same time)

# P(A or B) = P(A) + P(B) - P(A and B) ⇒ non-mutually exclusive (if they

 have at least one outcome in common)

- The union is written as A∪B or “A or B”.

- The intersection is written as A∩B or “A and B”

* Lets solve the question

∵ P(cream) = 0.5

∵ P(sugar) = 0.6

∵ P(cream or sugar) = 0.7

- To find P(cream and sugar) lets use the rule of non-mutually exclusive

∵ P(A or B) = P(A) + P(B) - P(A and B)

∴ P(cream or sugar) = P(cream) + P(sugar) - P(cream and sugar)

- Lets substitute the values of P(cream) , P(sugar) , P(cream or sugar)

 in the rule

∵ 0.7 = 0.5 + 0.6 - P(cream and sugar) ⇒ add the like terms

∴ 0.7 = 1.1 - P(cream and sugar) ⇒ subtract 1.1 from both sides

∴ 0.7 - 1.1 = - P(cream and sugar)

∴ - 0.4 = - P(cream and sugar) ⇒ multiply both sides by -1

∴ 0.4 = P(cream and sugar)

* P(cream and sugar) = 0.4

Answer:

0.4

Step-by-step explanation:

The answer is:

The number of teeth of Gear 2 is 90 teeth.

[tex]N_{2}=90teeth[/tex]

To calculate the number of teeth for the Gear 2, we need to use the following formula that establishes a relation between the number of RPM and the number of teeth of two or more gears.

[tex]N_{1}Z_{1}=N_{2}Z_{2}[/tex]

Where,

N, are the rpm of the gears

Z, are the teeth of the gears.

We are given the following information:

[tex]Z_{1}=30teeth\\N_{1}=150RPM\\N_{2}=50RPM[/tex]

Then, substituting and calculating we have:

[tex]N_{1}Z_{1}=N_{2}Z_{2}[/tex]

[tex]150RPM*30teeth=N_{2}50RPM[/tex]

[tex]N_{2}=\frac{150RPM*30teeth}{50RPM}=90teeth[/tex]

[tex]N_{2}=90teeth[/tex]

Hence, we have that the number of teeth of Gear 2 is 90 teeth.

Have a nice day!

Answer:

3.5 seconds, B

Step-by-step explanation:

This is an upside down parabola, a function that is extremetly useful in helping us to understand position and velocity and time and how they are all related.  Her upwards velocity is 16 ft/sec and she starts from a height of 140 feet, according to the problem.  The h is the height she ends up at after a certain amount of time has gone by.  You want to know how long it will take her to hit the water.  When she hits the water, she has no more height.  Therefore, her height above the water when she hits the water is 0.  Plug in a 0 for h and factor the quadratic to get t = -2.5 seconds and t = 3.5 seconds.  The only two things in math that will never ever be negative is a distance measure and time, so we can disregard the -2.5 and go with 3.5 seconds as our answer.

Answer:

B

Step-by-step explanation:

Answer:

  1557/1892

Step-by-step explanation:

Your calculator can do this:

[tex]\dfrac{18M}{K}=\dfrac{18\cdot 3\cdot 11\cdot 173}{2\cdot 33\cdot 11\cdot 172}=\dfrac{18\cdot 173}{2\cdot 11\cdot 172}\\\\=\dfrac{1557}{1892}[/tex]

1. C. -512√3+512i

2. B. 16(cos240°+i sin240°)

3. D. 3√2+3√6i, -3√2-3√6i

4. A. cos60°+i sin60°, cos180°+i sin180°, cos300°+i sin300°

5. D. 2√3(cos π/6+i sin π/6), 2√3(cos 7π/6+i sin 7π/6)

We will see that the equivalent expression is:

[tex]8*(cos(240\°) + i*sin(240\°))[/tex]

So the correct option is the first one.

We have the expression:

[2*(cos(240°) + i*sin(240°))]^4

Remember that Euler's formula says that:

[tex]e^{ix} = cos(x) + i*sin(x)[/tex]

Then we can rewrite our expression as:

[tex][2*(cos(240\°) + i*sin(240\°)]^4 = [2*e^{i*240\°}]^4[/tex]

Now we distribute the exponent:

[tex]2^4*e^{4*i*240\°} = 8*e^{i*960\°}[/tex]

Now, we need to find an angle equivalent to 960°.

Remember that the period of the trigonometric functions is 360°, then we can rewrite:

960° - 2*360° = 240°

This means that 960° is equivalent to 240°. Then we can write:

[tex]8*e^{i*960\°} = 8*e^{i*240\°} = 8*(cos(240\°) + i*sin(240\°))[/tex]

So the correct option is the first one.

If you want to learn more about complex numbers, you can read:

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Answer:

B) 34.1 cm

Step-by-step explanation:

The longer diagonal is longer than either side, but shorter than their sum. The only answer choice in the range of 24–39 cm is choice B.

_____

You are given sufficient information to use the Law of Cosines to find the diagonal length. If we call it "c", then the angle opposite that diagonal is the larger of the angles in the parallelogram: 120°. The law of cosines tells you ...

c^2 = a^2 +b^2 -2ab·cos(C)

Here, we have a=24, b=15, C=120°, so ...

c^2 = 24^2 +15^2 -2·24·15·cos(120°) = 576 +225 +360 = 1161

c = √1161 ≈ 34.073 . . . . cm

Rounded to tenths, the diagonal length is 34.1 cm.

[tex]\bf \textit{Pythagorean Identities} \\\\ sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{csc(\theta )-sin(\theta )}{cos(\theta )}\implies \cfrac{~~\frac{1}{sin(\theta )}-sin(\theta )~~}{cos(\theta )}\implies \cfrac{~~\frac{1-sin^2(\theta )}{sin(\theta )}~~}{cos(\theta )}[/tex]

[tex]\bf \cfrac{1-sin^2(\theta )}{sin(\theta )}\cdot \cfrac{1}{cos(\theta )}\implies \cfrac{\stackrel{cos(\theta )}{\begin{matrix} cos^2(\theta ) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}} }{sin(\theta )}\cdot \cfrac{1}{\begin{matrix} cos(\theta ) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix} }\implies \cfrac{cos(\theta )}{sin(\theta )}\implies cot(\theta )[/tex]

Answer:

cot Ф

Step-by-step explanation:

Recall that sin²Ф + cos²Ф = 1, (which also says that cos²Ф - 1 = sin²Ф).

Also recall the definitions of the csc, sin and cos functions.

Your expression is equivalent to:

   1               sin Ф

----------  -  -------------

sin Ф              1

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

           cos Ф

There are three terms in your expression:  csc, sin and cos.  Multiply all of them by sin Ф.  The result should be:

 1 - sin²Ф

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

sin Ф · cos Ф

Using the Pythagorean identity (see above), this simplifies to

  cos²Ф

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

sin Ф·cos Ф

and this whole fraction reduces to

   cos Ф

--------------   and this ratio is the definition of the cot function.

   sin Ф

Thus, the original expression is equivalent to cot Ф

Answer:

116.75 oceanic miles

Step-by-step explanation:

A graph of the data shows the distance to be between 110 and 120 miles (closer to 120). There is only one answer choice in that range.

In 2 hours, the ship travels 42.5 miles, so in 1 hour will travel 21.25 miles. Adding that distance to the distance at 4 hours gives the distance at 5 hours, ...

  95.5 +21.25 = 116.75 . . . . "oceanic" miles

_____

In order for the distance from the lighthouse to be uniformly increasing, the ship must be traveling directly away from the lighthouse. Traveling at any other angle, the distances will not fall on a straight line. (That is one reason I wanted to graph the data.)

Answer:

Associative property of multiplication

Step-by-step explanation:

To show that 9(3x) = 27(x), we need to show that 9(3x) = (9 * 3)x.

The APM does just that. By this property, in multiplication, the order of which numbers are multiplied do not matter.

So, 9(3x) = (9 * 3)x.

And by multiplication, (9 * 3)x = 27x.

So 9(3x) = 27x

0.3 or 30%. The probability that Ace pick a banana from a basket that content others fruits is 0.3.

The key to solve this problem is using the equation of probability [tex]P(A)=\frac{n(A)}{n}[/tex] where n(A) the numbers of favorables outcomes and n the numbers of possible outcomes.

There are in the basket 10 fruits in total (3 bananas + 4 apples + 3 oranges = 10fruits). Then, extract a fruit can occur in 10 ways, this is n. There is only 3 bananas in the basket, so the fruit that ACE will pick be a banana can occur in 3 ways out of 10,  so 3 is n(A).

Solving the equation:

[tex]P(A)=\frac{3}{10}=0.3[/tex]

The probability that Ace will pick a banana from the basket is 3/10, as there are 3 bananas out of a total of 10 pieces of fruit.

The question asks for the probability that Ace will pick a banana from a basket containing 3 bananas, 4 apples, and 3 oranges. To calculate this, you sum up the total number of pieces of fruit, which is 3 bananas + 4 apples + 3 oranges = 10 pieces of fruit. The probability is then the number of desired outcomes (bananas) over the total number of possible outcomes (all pieces of fruit), which is 3 bananas / 10 pieces of fruit = 3/10 or 30%.

Answer:

x=2

Step-by-step explanation:

Answer:

Eric’s share in the profit is $30,000 ⇒ answer B

Step-by-step explanation:

* We will use the ratio to solve this problem

- At first lets find the ratio between their invested

∵ Eric has invested $400,000

∵ George has invested $300,000

∵ Denzel has invested $300.000

- To find the ratio divide each number by 100,000

∴ Eric : George : Denzel = 4 : 3 : 3

- They will divide the profits among themselves in the ratio of their

  respective investments

- The total profit will divided by the total of their ratios

∵ The total of the ratios = 4 + 3 + 3 = 10

∴  Eric : George : Denzel : Sum = 4 : 3 : 3 : 10

- That means the profit will divided into 10 equal parts

- Eric will take 4 parts, George will take 3 parts and Denzel will take

 3 parts

∵ The profit = $75,000

- Divide the profit by the sum of the ratio

∴ Each part of the profit = 75,000 ÷ 10 = $7,500

- Now lets find the share of each one

∴ The share of Eric = 4 × 7,500 = $30,000

∴ The share of George = 3 × 7,500 = $22,500

∴ The share of Denzel = 3 × 7,500 = $22,500

* Eric’s share in the profit is $30,000

# If you want to check your answer add the shares of them, the answer

  will be the total profit (30,000 + 22,500 + 22,500 = $75,000), and if

  you find the ratio between their shares it will be equal the ratio

  between their investments (divide each share by 7,500 to simplify

  them the answer will be 4 : 3 : 3)

Answer:

B

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

The volume of a sphere is given by

[tex]V=\frac{4}{3}\pi r^3[/tex]

where r is the radius

For $2.75, we can get 1 large  OR  2 small scoops.

Giant scoop has diameter 6, so radius is half of that, which is 3, hence the volume is:

[tex]V=\frac{4}{3}\pi r^3\\V=\frac{4}{3}\pi (3)^3\\V=113.1[/tex]

Regular scoop's diameter is 4, hence radius is 2. So volume of 1 regular scoop is:

[tex]V=\frac{4}{3}\pi r^3\\V=\frac{4}{3}\pi (2)^3\\V=33.51[/tex]

We can get 2 of those, so total volume is 33.51 + 33.51 = 67.02

Hence, the max volume for $2.75 is around 113, answer choice B.

Explanation:

To find the value, put 320 where x is and do the arithmetic.

f(320) = 0.11·320 +43 = 35.20 +43 = 78.20

The meaning is described by the problem statement:

"how much Derek pays for phone service" for "the number of minutes, [320], used for international calls in a month."

Derek pays 78.20 for 320 minutes of international calls in a month.

__

The units (dollars, rupees, euros, pounds, ...) are not specified.

Answer:

Given the function f(x) = 0.11x + 43, this shows the relationship between how much Derek has to pay for phone service for the amount of minutes he uses on international calls a month.  f(320) can be solved by substituting x = 320, and this is shown below: f(x) = 0.11x + 43 f(320) = 0.11(320) + 43 f(320) = 78.2 This means that Derek has to pay $78.20 for the 320 minutes of calls. Among the choices, the correct answer is B.

By Pythagoras' Theorem:

Sum of the squares of the two side  = Square of longest side

a² + b² = c²

a)

So let's check 7, 24, 25

Is 7² + 24² = 25²  ?

7*7 + 24*24

49 + 576  

=625.

Let us perform the other side 25²

25² = 25 * 25 = 625

Therefore the left hand side = Right hand side.

Therefore 7, 24, 25 is a Pythagorean Triple

b)

Let's check 9, 40, 41

Is 9² + 40² = 41²  ?

9² + 40²

9*9+ 40*40  

81 + 1600

=1681

Let us perform the other side 41²

41² = 41 * 41 = 1681

Therefore the left hand side = Right hand side.

Therefore 9, 40, 41 is a Pythagorean Triple.

Answer:

a. X is the number of adults in America that need to be surveyed until finding the first one that will watch the Super Bowl.

b. X can take any integer that is greater than or equal to 1. [tex]\rm X\in \mathbb{Z}^{+}[/tex].

c. [tex]\rm X \sim NB(1, 0.40)[/tex].

d. [tex]E(\rm X) = 2.5[/tex].

e. [tex]P(\rm X = 7) = 0.0187[/tex].

f. [tex]P(\text{X} = 3) +P(\text{X} = 4) = 0.230[/tex].

Step-by-step explanation:

In this setting, finding an adult in America that will watch the Super Bowl is a success. The question assumes that the chance of success is constant for each trial. The question is interested in the number of trials before the first success. Let X be the number of adults in America that needs to be surveyed until finding the first one who will watch the Super Bowl.

It takes at least one trial to find the first success. However, there's rare opportunity that it might take infinitely many trials. Thus, X may take any integer value that is greater than or equal to one. In other words, X can be any positive integer: [tex]\rm X\in \mathbb{Z}^{+}[/tex].

There are two discrete distributions that may model X:

[tex]\rm NB(1, p)[/tex] (note that [tex]r=1[/tex]) is equivalent to [tex]\sim Geo(p)[/tex]. However, in this question the distribution of [tex]\rm X[/tex] takes two parameters, which implies that [tex]\rm X[/tex] shall follow the negative binomial distribution rather than the geometric distribution. The probability of success on each trial is [tex]40\% = 0.40[/tex].

[tex]\rm X\sim NB(1, 0.40)[/tex].

The expected value of a negative binomial random variable is equal to the number of required successes over the chance of success on each trial. In other words,

[tex]\displaystyle E(\text{X}) = \frac{r}{p} = \frac{1}{0.40} = 2.5[/tex].

[tex]P(\rm X = 7) = 0.0187[/tex].

Some calculators do not come with support for the negative binomial distribution. There's a walkaround for that as long as the calculator supports the binomial distribution. The r-th success occurs on the n-th trial translates to (r-1) successes on the first (n-1) trials, plus another success on the n-th trial. Find the chance of (r-1) successes in the first (n-1) trials and multiply that with the chance of success on the n-th trial.

[tex]P(\text{X} = 3)+P(\text{X} = 4) = 0.230 [/tex].

Answer:

D.

Step-by-step explanation:

I also haven't learned this yet but i could tell that in the second image if A.F = 1/2AC and DE = A.F, therefore DE = 1/2AC. The problem is that i don't know if it is B or D.

Sorry .-.

The ODE

[tex]M(x,y)\,\mathrm dx+N(x,y)\,\mathrm dy=0[/tex]

is exact if

[tex]\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}[/tex]

We have

[tex]M=-xy\sin x+2y\cos x\implies M_y=-x\sin x+2\cos x[/tex]

[tex]N=2x\cos x\implies N_x=2\cos x-2x\sin x[/tex]

so the ODE is indeed not exact.

Multiplying both sides of the ODE by [tex]\mu(x,y)=xy[/tex] gives

[tex]\mu M=-x^2y^2\sin x+2xy^2\cos x\implies(\mu M)_y=-2x^2y\sin x+4xy\cos x[/tex]

[tex]\mu N=2x^2y\cos x\implies(\mu N)_x=4xy\cos x-2x^2y\sin x[/tex]

so that [tex](\mu M)_y=(\mu N)_x[/tex], and the modified ODE is exact.

We're looking for a solution of the form

[tex]\Psi(x,y)=C[/tex]

so that by differentiation, we should have

[tex]\Psi_x\,\mathrm dx+\Psi_y\,\mathrm dy=0[/tex]

[tex]\implies\begin{cases}\Psi_x=\mu M\\\Psi_y=\mu N\end{cases}[/tex]

Integrating both sides of the second equation with respect to [tex]y[/tex] gives

[tex]\Psi_y=2x^2y\cos x\implies\Psi=x^2y^2\cos x+f(x)[/tex]

Differentiating both sides with respect to [tex]x[/tex] gives

[tex]\Psi_x=-x^2y^2\sin x+2xy^2\cos x=2xy^2\cos x-x^2y^2\sin x+\dfrac{\mathrm df}{\mathrm dx}[/tex]

[tex]\implies\dfrac{\mathrm df}{\mathrm dx}=0\implies f(x)=c[/tex]

for some constant [tex]c[/tex].

So the general solution to this ODE is

[tex]x^2y^2\cos x+c=C[/tex]

or simply

[tex]x^2y^2\cos x=C[/tex]

We are to verify and confirm if the given differential equations are exact or not. Then solve for the exact equation.

The first differential equation says:

[tex]\mathbf{(-xy \ sin x + 2y \ cos x) dx + 2(x \ cos x) dy = 0 }[/tex]

Recall that:

A differential equation that takes the form [tex]\mathbf{M(x,y)dt + N(x, y)dy = 0 }[/tex] will be exact if and only if:

From equation (1), we can represent M and N as follows:

Thus, taking the differential of M and N, we have:

[tex]\mathbf{ \dfrac{\partial M}{\partial y }= M_y = -x sin x + 2cos x}[/tex]

[tex]\mathbf{ \dfrac{\partial N}{\partial x }= N_x = 2 cos x + 2x sin x}[/tex]

From above, it is clear that:

[tex]\mathbf{\dfrac{\partial M }{\partial y} \neq \dfrac{\partial N }{\partial x}}[/tex]

∴

We can conclude that the equation is not exact.

Now, after multiplying the given differential equation in (1) by the integrating factor μ(x, y) = xy, we have:

Representing the equation into form M and N, then:

[tex]\mathbf{M = -x^22y^2 sin x +2xy^2 cos x}[/tex]

[tex]\mathbf{N = 2x^2y cos x}[/tex]

Taking the differential, we have:

[tex]\mathbf{\dfrac{\partial M}{\partial y }= M_y = -2x^2y sin x + 4xy cos x }[/tex]

[tex]\mathbf{\dfrac{\partial N}{\partial x} =N_x= 4xycos \ x -2x^2 y sin x}[/tex]

Here;

[tex]\mathbf{\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x} }[/tex]

Therefore, we can conclude that the second equation is exact.

Now, the solution of the second equation is as follows:

[tex]\int_{y } M dx + \int (not \ containing \ 'x') dy = C[/tex]

[tex]\rightarrow \int_{y } (-x^2y^2 sin(x) +2xy^2 cos (x) ) dx + \int(0)dy = C[/tex]

[tex]\rightarrow-y^2 \int x^2 sin(x) dx +2y ^2 \int x cos (x) dx = C[/tex]    ---- (3)

Taking integrations by parts:

[tex]\int u v dx = u \int v dx - \int (\dfrac{du}{dx} \int v dx) dx[/tex]

∴

[tex]\int x^2 sin (x) dx = x^2 \int sin(x) dx - \int (\dfrac{d}{dx}(x^2) \int (sin \ (x)) dx) dx[/tex]

[tex]\to x^2 (-cos (x)) \ - \int 2x (-cos \ (x)) \ dx[/tex]

[tex]\to -x^2 (cos (x)) \ + \int 2x \ cos \ (x) \ dx[/tex]   ----- replace this equation into (3)

∴

[tex]\rightarrow-y^2( -x^2 cos (x) \ + \int 2x \ cos \ (x) \ dx) +2y ^2 \int x cos (x) dx = C[/tex]

[tex]\mathbf{\rightarrow -x^2 y^2 cos (x) \ -2y ^2 \int x \ cos \ (x) \ dx +2y ^2 \int x cos (x) dx = C}[/tex]

[tex]\mathbf{x^2y^2 cos (x) = C\ \text{ where C is constant}}[/tex]

Therefore, from the explanation, we've can conclude that the first equation is not exact and the second equation is exact.

Learn more about differential equations here:

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Answer:

Step-by-step explanation:

The absolute value function prevents the expression from being a polynomial. The degree of 3 in y^3 is an odd number so that polynomial function will not be even.

Answer:

The equation [tex]y=x^3[/tex] is not an equation of a simple , even polynomial function.

Step-by-step explanation:

Even  function : A function  is even when its graph is symmetric with respect to y-axis.

Algebrically , the function f is even if and only if

f(-x)=f(x) for all x in the domain of f.

When the function does not satisfied the above condition then the function is called non even function.

f(x)[tex]\neq[/tex] f(-x)

Now , we check given function is even or not

A. y= [tex]\mid x\mid[/tex]

If x is replaced by -x

Then we get the function

f(-x)=[tex]\mid -x \mid[/tex]

f(-x)=[tex]\mid x \mid[/tex]

Hence, f(-x)=f(x)

Therefore , it is even  polynomial function.

B. [tex]y=x^2[/tex]

If x is replace by -x

Then we get

f(-x)=[tex](-x)^2[/tex]

f(-x)=[tex]x^2[/tex]

Hence, f(-x)=f(x)

Therefore, it is even polynomial function.

C. [tex]y=x^3[/tex]

If x is replace by -x

Then we get

f(-x)=[tex](-x)^3[/tex]

f(-x)=[tex]-x^3[/tex]

Hence, f(-x)[tex]\neq[/tex] f(x)

Therefore, it is not even polynomial function.

D.[tex]y= -x^2[/tex]

If x is replace by -x

Then we get

f(-x)= - [tex](-x)^2[/tex]

f(-x)=-[tex]x^2[/tex]

Hence, f(-x)=f(x)

Therefore, it is even polynomial function.

Answer: C. [tex]y=x^3[/tex] is not simple , even polynomial function.

The events “cheering for Team A” and “being a female” are not mutually exclusive. The probability that a randomly selected attendee is female or cheers for Team A is approximately 0.741.

(a) No, the events “cheering for Team A” and “being a female” are not mutually exclusive. This is because there are females who are cheering for Team A. Mutually exclusive events cannot happen at the same time.

(b) To find the probability that a randomly selected attendee is female or cheers for Team A, we need to add the probabilities of each event happening and subtract the probability of both events happening at the same time. We can use the formula:

P(A or B) = P(A) + P(B) - P(A and B)

In this case, P(A) is the probability of cheering for Team A, P(B) is the probability of being female, and P(A and B) is the probability of being a female who cheers for Team A.

Given the numbers provided, the probability of cheering for Team A is 2118/4997 and the probability of being female is 2568/4997. The probability of being a female who cheers for Team A is 982/4997. Plugging these values into the formula, we get:

P(Female or Team A) = P(Team A) + P(Female) - P(Female and Team A) = 2118/4997 + 2568/4997 - 982/4997 = 3704/4997 ≈ 0.741

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ANSWER

[tex]x = \frac{ 5 - \sqrt{ 5} }{4} \: or \: \: x = \frac{ 5 + \sqrt{ 5} }{4} [/tex]

EXPLANATION

The given quadratic equation is

[tex]4 {x}^{2} - 10x + 5 = 0[/tex]

We compare this to

[tex]a {x}^{2} + bx + c = 0[/tex]

to get a=4, b=-10, and c=5.

The quadratic formula is given by

[tex]x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} [/tex]

We substitute these values into the formula to get:

[tex]x = \frac{ - - 10 \pm \sqrt{ {( - 10)}^{2} - 4(4)(5)} }{2(4)} [/tex]

This implies that

[tex]x = \frac{ 10 \pm \sqrt{ 100 - 80} }{8} [/tex]

[tex]x = \frac{ 10 \pm \sqrt{ 20} }{8} [/tex]

[tex]x = \frac{ 10 \pm2 \sqrt{ 5} }{8} [/tex]

[tex]x = \frac{ 5 \pm \sqrt{ 5} }{4} [/tex]

The solutions are:

[tex]x = \frac{ 5 - \sqrt{ 5} }{4} \: or \: \: x = \frac{ 5 + \sqrt{ 5} }{4} [/tex]

Answer:

[tex]\large\boxed{x=\dfrac{5-\sqrt5}{4},\ x=\dfrac{5+\sqrt5}{4}}[/tex]

Step-by-step explanation:

[tex]\text{The quadratic formula for}\ ax^2+bx+c=0\\\\\text{if}\ b^2-4ac<0,\ \text{then the equation has no real solution}\\\\\text{if}\ b^2-4ac=0,\ \text{then the equation has one solution:}\ x=\dfrac{-b}{2a}\\\\\text{if}\ b^2-4ac,\ ,\ \text{then the equation has two solutions:}\ x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\==========================================[/tex]

[tex]\text{We have the equation:}\ 4x^2-10x+5=0\\\\a=4,\ b=-10,\ c=5\\\\b^2-4ac=(-10)^2-4(4)(5)=100-80=20>0\\\\x=\dfrac{-(-10)\pm\sqrt{20}}{2(4)}=\dfrac{10\pm\sqrt{4\cdot5}}{8}=\dfrac{10\pm\sqrt4\cdot\sqrt5}{8}=\dfrac{10\pm2\sqrt5}{8}\\\\=\dfrac{2(5\pm\sqrt5)}{8}=\dfrac{5\pm\sqrt5}{4}[/tex]

Answer:

  80 cm²

Step-by-step explanation:

Trapezoid LPKB has area ...

  A = (1/2)(b1 +b2)h = (1/2)(4 +20)(20) = 240 . . . . cm²

Triangle BPN has area ...

  A = (1/2)bh = (1/2)(20)(20) = 200 . . . . cm²

Triangle BKN has a height that is 4/5 the height of triangle BPN, so will have 4/5 the area:

  ΔBKN = (4/5)(200 cm²) = 160 cm²

The area of quadrilateral LPKB is that of trapezoid LPNB less the area of triangle BKN, so is ...

  240 cm² - 160 cm² = 80 cm²

Answer:

6433.98 ft

Step-by-step explanation:

In order to find what 25% of the tank's capacity is, we know to know the full capacity of the tank then take 25% of that.  The volume formula for a right circular cone is

[tex]V=\frac{1}{3}\pi r^2h[/tex]

We have all the values we need for that:

[tex]V=\frac{1}{3}\pi (16)^2(96)[/tex]

This gives us a volume of 25735.93 cubic feet total.

25% of that:

.25 × 25735.93 = 6433.98 ft

Answer:

The height of the water is [tex]60.5\ ft[/tex]

Step-by-step explanation:

step 1

Find the volume of the tank

The volume of the inverted right circular cone is equal to

[tex]V=\frac{1}{3}\pi r^{2} h[/tex]

we have

[tex]r=16\ ft[/tex]

[tex]h=96\ ft[/tex]

substitute

[tex]V=\frac{1}{3}\pi (16)^{2} (96)[/tex]

[tex]V=8,192\pi\ ft^{3}[/tex]

step 2

Find the 25% of the tank’s capacity

[tex]V=(0.25)*8,192\pi=2,048\pi\ ft^{3}[/tex]

step 3

Find the height, of the water in the tank  

Let

h ----> the height of the water  

we know that

If two figures are similar, then the ratio of its corresponding sides is proportional

[tex]\frac{R}{H}=\frac{r}{h}[/tex]

substitute

[tex]\frac{16}{96}=\frac{r}{h}\\ \\r= \frac{h}{6}[/tex]

where

r is the radius of the smaller cone of the figure

h is the height of the smaller cone of the figure

R is the radius of the circular base of tank

H is the height of the tank

we  have

[tex]V=2,048\pi\ ft^{3}[/tex] -----> volume of the smaller cone

substitute

[tex]2,048\pi=\frac{1}{3}\pi (\frac{h}{6})^{2}h[/tex]

Simplify

[tex]221,184=h^{3}[/tex]

[tex]h=60.5\ ft[/tex]

ANSWER

[tex]|^{ \to} _{YZ}| = \sqrt{13} [/tex]

EXPLANATION

Given the points, y(-2,5),z(1,3)

[tex] ^{ \to} _{YZ} = \binom{1}{3} - \binom{ - 2}{5} = \binom{3}{ - 2} [/tex]

Therefore the ordered pair is <3,-2>

The magnitude is

[tex] |^{ \to} _{YZ}| = \sqrt{ {3}^{2} + ( - 2)^{2} } [/tex]

[tex] |^{ \to} _{YZ}| = \sqrt{ 9 +4} [/tex]

[tex]|^{ \to} _{YZ}| = \sqrt{13} [/tex]

Answer: AAAAAAAAAAAAAAAAAAAAAAAAAAa

ANSWER

[tex]y = 2x -4[/tex]

EXPLANATION

Part a)

Eliminating the parameter:

The parametric equation is

[tex]x = 5 + ln(t) [/tex]

[tex]y = {t}^{2} + 5[/tex]

From the first equation we make t the subject to get;

[tex]x - 5 = ln(t) [/tex]

[tex]t = {e}^{x - 5} [/tex]

We put it into the second equation.

[tex]y = { ({e}^{x - 5}) }^{2} + 5[/tex]

[tex]y = { ({e}^{2(x - 5)}) } + 5[/tex]

We differentiate to get;

[tex] \frac{dy}{dx} = 2 {e}^{2(x - 5)} [/tex]

At x=5,

[tex] \frac{dy}{dx} = 2 {e}^{2(5 - 5)} [/tex]

[tex]\frac{dy}{dx} = 2 {e}^{0} = 2[/tex]

The slope of the tangent is 2.

The equation of the tangent through

(5,6) is given by

[tex]y-y_1=m(x-x_1)[/tex]

[tex]y - 6 = 2(x - 5)[/tex]

[tex]y = 2x - 10 + 6[/tex]

[tex]y = 2x -4[/tex]

Without eliminating the parameter,

[tex] \frac{dy}{dx} = \frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } [/tex]

[tex]\frac{dy}{dx} = \frac{ 2t}{ \frac{1}{t} } [/tex]

[tex]\frac{dy}{dx} = 2 {t}^{2} [/tex]

At x=5,

[tex]5 = 5 + ln(t) [/tex]

[tex] ln(t) = 0[/tex]

[tex]t = {e}^{0} = 1[/tex]

This implies that,

[tex]\frac{dy}{dx} = 2 {(1)}^{2} = 2[/tex]

The slope of the tangent is 2.

The equation of the tangent through

(5,6) is given by

[tex]y-y_1=m(x-x_1)[/tex]

[tex]y - 6 = 2(x - 5) =[/tex]

[tex]y = 2x -4[/tex]

The equation of the tangent to the curve at the point (5,6) is [tex]\(y = 2x - 4\)[/tex].

To find the equation of the tangent to the curve given by the parametric equations [tex]\(x = 5 + \ln(t)\)[/tex] and [tex]\(y = t^2 + 5\)[/tex] at the point (5,6), we can approach this problem in two ways: by eliminating the parameter \(t\) and without eliminating the parameter.

Method 1: Eliminating the Parameter

Step 1: Express (t) in terms of (x)

[tex]\[ x = 5 + \ln(t) \implies \ln(t) = x - 5 \implies t = e^{x-5} \][/tex]

Step 2: Substitute (t) into (y)

[tex]\[ y = t^2 + 5 \implies y = (e^{x-5})^2 + 5 \implies y = e^{2(x-5)} + 5 \][/tex]

Step 3: Find [tex]\(\frac{dy}{dx}\)[/tex]

[tex]\[ y = e^{2(x-5)} + 5 \][/tex]

[tex]\[ \frac{dy}{dx} = 2e^{2(x-5)} \][/tex]

Step 4: Evaluate [tex]\(\frac{dy}{dx}\)[/tex] at (x = 5)

[tex]\[ \frac{dy}{dx}\bigg|_{x=5} = 2e^{2(5-5)} = 2e^0 = 2 \][/tex]

Step 5: Equation of the tangent line

The slope (m = 2). The tangent line at (5,6) is:

[tex]\[ y - 6 = 2(x - 5) \][/tex]

[tex]\[ y = 2x - 10 + 6 \][/tex]

[tex]\[ y = 2x - 4 \][/tex]

Method 2: Without Eliminating the Parameter

Step 1: Find [tex]\(\frac{dx}{dt}\)[/tex] and [tex]\(\frac{dy}{dt}\)[/tex]

[tex]\[ x = 5 + \ln(t) \implies \frac{dx}{dt} = \frac{1}{t} \][/tex]

[tex]\[ y = t^2 + 5 \implies \frac{dy}{dt} = 2t \][/tex]

Step 2: Find [tex]\(\frac{dy}{dx}\)[/tex]

[tex]\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{2t}{\frac{1}{t}} = 2t^2 \][/tex]

Step 3: Find (t) at the point (5,6)

From [tex]\(x = 5 + \ln(t)\)[/tex]:

[tex]\[ 5 = 5 + \ln(t) \implies \ln(t) = 0 \implies t = e^0 = 1 \][/tex]

Step 4: Evaluate [tex]\(\frac{dy}{dx}\)[/tex] at (t = 1)

[tex]\[ \frac{dy}{dx}\bigg|_{t=1} = 2(1)^2 = 2 \][/tex]

Step 5: Equation of the tangent line

The slope (m = 2). The tangent line at (5,6) is:

[tex]\[ y - 6 = 2(x - 5) \][/tex]

[tex]\[ y = 2x - 10 + 6 \][/tex]

[tex]\[ y = 2x - 4 \][/tex]

Thus, using both methods, the equation of the tangent to the curve at the point (5,6) is [tex]\(y = 2x - 4\)[/tex].