5x=k-14 solve for x (literal equation)

Answer: 0.7881446

Step-by-step explanation:

Given : Mean : [tex]\mu = 267\text{ dollars} [/tex]

Standard deviation : [tex]\sigma =20 \text{ dollars}[/tex]

Sample size : [tex]n=64[/tex]

The formula to calculate the z-score :-

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

For x=  267 dollars

[tex]z=\dfrac{267-267}{\dfrac{20}{\sqrt{64}}}=0[/tex]

For x= 269 dollars.

[tex]z=\dfrac{269-267}{\dfrac{20}{\sqrt{64}}}=0.80[/tex]

The P-value : [tex]P(0<z<0.8)=P(z<0.8)-P(z<0)[/tex]

[tex]= 0.7881446-0.5= 0.2881446\approx 0.7881446[/tex]

Hence, the probability that they have a mean cost between 267 dollars and 269 dollars.= 0.7881446

Answer:

There is a 28.81% probability that they have a mean cost between 267 dollars and 269 dollars.

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex]

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 267, \sigma = 20, n = 64, s = \frac{20}{\sqrt{64}} = 2.5[/tex].

Find the probability that they have a mean cost between 267 dollars and 269 dollars.

This probability is the pvalue of Z when X = 269 subtracted by the pvalue of Z when X = 267. So:

X = 269

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{269 - 267}{2.5}[/tex]

[tex]Z = 0.8[/tex]

[tex]Z = 0.8[/tex] has a pvalue of 0.7881.

X = 267

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{267 - 267}{2.5}[/tex]

[tex]Z = 0[/tex]

[tex]Z = 0[/tex] has a pvalue of 0.50.

So there is a 0.7881 - 0.50 = 0.2881 = 28.81% probability that they have a mean cost between 267 dollars and 269 dollars.

The probability that you win exactly one game is:

                               0.37

The probability that you win exactly one game is:

Probability you win first but not second+Probability you win second but not first.

=  0.60×0.25+0.40×0.55

=  0.37

( since probability of losing second game when you win first  is: 1-0.75=0.25

and probability that you lose first game is: 1-0.60=0.40 )

Final answer:

The probability of winning exactly one game, rounded to two decimal places, is 0.37.

Explanation:

Probability of winning the first game = 0.60

Probability of winning the second game if the first game is won = 0.75
Probability of winning the second game if the first game is lost = 0.55

To find the probability of winning exactly one game, we calculate the probability of winning the first game and losing the second game, plus the probability of losing the first game and winning the second game:

Answer:

27, 29, 31

Step-by-step explanation:

We want to find a, b, c so that

a + c = b + 29

and

a = b - 2, c = b + 2

and

b is odd.

This results in

b - 2 + b + 2 = b + 29

with one solution:

b = 29.

[tex]2n-1,2n+1,2n+3[/tex] - 3 consecutive odd integers

[tex]2n-1+2n+3=2n+1+29\\2n=28\\n=14\\\\2n-1=27\\2n+1=29\\2n+3=31[/tex]

27,29,31

Answer:

B

Step-by-step explanation:

(f-g)(x) just means f(x)-g(x)

so let's do that

make sure you distribute the minus in front of g to it's terms

-12x^3+19x^2-5-7x^2-15

-12x^3+12x^2-20

B.

Answer:

owen

Step-by-step explanation:

Answer:

Shaquille

Step-by-step explanation:

To determine the unit rate for each, divide the number of words by the number of minutes typed.

Ella: 640÷16=40

Harper: 450÷15=30

Owen: 560÷14=40

Shaquille=540÷12=45

Since Shaquille's is the most, we can tell he typed fastest.

Hope this helps!

[tex]\bf 5y^4+11y^2+2\implies 5(y^2)^2+11y^2+2\implies (5y^2+1)(y^2+2)[/tex]

For this case we must factor the following polynomial:

[tex]5y ^ 4 + 11y ^ 2 + 2[/tex]

We rewrite [tex]y ^ 4[/tex]as [tex](y^ 2) ^ 2[/tex]:

[tex]5 (y ^ 2) ^ 2 + 11y ^ 2 + 2[/tex]

We make a change of variable:

[tex]u = y ^ 2[/tex]

We replace:

[tex]5u ^ + 11u + 2[/tex]

we rewrite the middle term as a sum of two terms whose product of 5 * 2 = 10 and the sum of 11.

So:

[tex]5u ^ 2 + (1 + 10) u + 2[/tex]

We apply distributive property:

[tex]5u ^ 2 + u + 10u + 2[/tex]

We factor the highest common denominator of each group.

[tex](5u ^ 2 + u) + (10u + 2)\\u (5u + 1) +2 (5u + 1)[/tex]

We factor again:

[tex](u + 2) (5u + 1)[/tex]

Returning the change:

[tex](y ^ 2 + 2) (5y ^ 2 + 1)[/tex]

ANswer:

[tex](y ^ 2 + 2) (5y ^ 2 + 1)[/tex]

Answer:

a) False. Because you can live in California AND watch American Idol at the same time

b) True. Because the number of patients is a whole number, like 1, 2 or 3. There is no 1.5 patient

c) False. The actual probability should become closer to the theoretical probablity

d) True

Answer:

  r ≈ 13.01

Step-by-step explanation:

The mnemonic SOH CAH TOA reminds you that ...

  Cos = Adjacent/Hypotenuse

  cos(67.4°) = 5.00/r . . . . . . filling in the given values

Solving for r gives ...

  r = 5.00/cos(67.4°) ≈ 13.01

_____

Check your requirements for rounding. We rounded to 2 decimal places because the x-coordinate, 5.00, was expressed using 2 decimal places.

Positive product:

[tex](\frac{-2}{5})(\frac{-2}{5})\\(\frac{2}{5})(\frac{2}{5})[/tex]

With multiplication if two negative numbers are being multiplied together the answer is positive. If two positive numbers are being multiplied the answer is also positive. Another way to think of it is, that if two numbers with the SAME sign are being multiplied then the product will always be positive

Negative product:

[tex](\frac{-2}{5})(\frac{2}{5})[/tex]

[tex](\frac{2}{5})(\frac{-2}{5})[/tex]

If a negative and positive number are being multiplied then the product is ALWAYS negative

Hope this helped!

~Just a girl in love with Shawn Mendes

For this case we have the following expression:

[tex]\frac {b ^ 7} {b ^ 6}[/tex]

By definition of division of powers of the same base, we have to place the same base and subtract the exponents, that is:

[tex]\frac {a ^ m} {a ^ n} = a ^ {n-m}[/tex]

So:

[tex]\frac {b ^ 7} {b ^ 6} = b ^ {7-6} = b ^ 1 = b[/tex]

Answer:

b

Answer: [tex]b[/tex]

Step-by-step explanation:

 You need to remember a property called "Quotient of powers property". This property states the following:

[tex]\frac{a^m}{a^n}=a^{(m-n)}[/tex]

You can observe that the bases of the expression [tex]\frac{b^7}{b^6}[/tex] are equal, then you can apply the property mentioned before.

Therefore, you can make the division indicated in the exercise.

Then you get this result:

[tex]\frac{b^7}{b^6}=b^{(7-6)}=b[/tex]

Answer:

f(-3)=5x^2+5x*3

-f*3=5x^2+5*3x

-3f=5x^2+15x

f=-5x(x+3)/3

f(-9)=5x^2+5x*3

-f*9=5x^2+5*3x

-9f=5x^2+15x

f=-5x(x+3)/9

Step-by-step explanation:

hope it helps you?

The particle has position function

[tex]\vec r(t)=\cos t\,\vec\imath+\sin t\,\vec\jmath+3t\,\vec k[/tex]

Taking the derivative gives its velocity at time [tex]t[/tex]:

[tex]\vec v(t)=\dfrac{\mathrm d\vec r(t)}{\mathrm dt}=-\sin t\,\vec\imath+\cos t\,\vec\jmath+3\,\vec k[/tex]

a. The particle never moves downward because its velocity in the [tex]z[/tex] direction is always positive, meaning it is always moving away from the origin in the upward direction. DNE

b. The particle is situated 15 units above the ground when the [tex]z[/tex] component of its posiiton is equal to 15:

[tex]3t=15\implies\boxed{t=5}[/tex]

c. At this time, its velocity is

[tex]\vec v(5)=-\sin 5\,\vec\imath+\cos5\,\vec\jmath+3\,\vec k\approx\boxed{0.959\,\vec\imath+0.284\,\vec\jmath+3\,\vec k}[/tex]

d. The tangent to [tex]\vec r(t)[/tex] at [tex]t=5[/tex] points in the same direction as [tex]\vec v(5)[/tex], so that the parametric equation for this new path is

[tex]\vec r(5)+\vec v(5)t\approx\boxed{(0.284+0.959t)\,\vec\imath+(-0.959+0.284t)\,\vec\jmath+(15+3t)\,\vec k}[/tex]

where [tex]0\le t<\infty[/tex].

Answer:

[tex]\large\boxed{y=\dfrac{-9x-2}{5}}[/tex]

Step-by-step explanation:

[tex]9x+5y=-2\qquad\text{subtract}\ 9x\ \text{from both sides}\\\\5y=-9x-2\qquad\text{divide both sides by 5}\\\\y=\dfrac{-9x-2}{5}[/tex]

Answer:

[tex]\large\boxed{\lim\limits_{x\to7}f(x)=0}[/tex]

Step-by-step explanation:

[tex]f(x)=\left\{\begin{array}{ccc}x^2-8x+7&if&x<7\\-x^2+8x-7&if&x\geq7\end{array}\right\\\\\lim\limits_{x\to7}f(x)=?\\\\\lim\limits_{x\to7^-}(x^2-8x+7)=7^2-(8)(7)+7=49-56+7=0\\\\\lim\limits_{x\to7^+}(-x^2+8x-7)=-7^2+(8)(7)-7=-49+56-7=0\\\\\lim\limits_{x\to7^-}=\lim\limits_{x\to7^+}=0\Rightarrow\lim\limits_{x\to7}f(x)=0[/tex]

In this question, we need to find the limit of the piecewise function f(x) as x approaches 7. After substitution of x=7 in both conditions of the function, we find that the limit of function f(x) exists and equals 0.

To find the limit of the function f(x) as x approaches 7, we need to look at both cases as defined by the function, because f(x) is a piecewise function.

1. For x < 7, we substitute x as 7 in the function x^2 - 8x + 7:

((7)^2 - 8*(7) + 7 = 0)

2. For x >= 7, we substitute x as 7 in the function -x^2 + 8x - 7:

(-(7)^2 + 8*(7) - 7 = 0)

In this case, for both conditional statements the results are the same, so the limit of the function exists and equals:

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Answer:

19) The domain is (12 , ∞) ⇒ answer C

17) The radical notation is [tex]\sqrt[7]{18y^{2} }[/tex]

10) The definition of number i is √(-1) ⇒ answer C

Step-by-step explanation:

19)

* Lets explain the meaning of the domain of the function

- The domain of any function is the values of x which makes the

  function defined

- Examples:

# In the fraction the denominator con not be zero, then if the function

  is a rational fraction then the domain is all the values of x except

  the values whose make the denominator = 0

# In the even roots we can not put negative numbers under the radical

  because there is no even roots for the negative number belonges to

  the real numbers, then the domain is all the values of x except the

  values whose make the quantity under the radical negative

* Now lets solve the question

∵ f(x) = 3 √(x - 12)

- To find the domain let (x - 12) greater than zero because there is

 no square root for negative value

∵ x - 12 > 0 ⇒ add 12 to both sides

∴ x ≥ 12

∴ The domain is all values of x greater than 12

* The domain is (12 , ∞)

17)

* Lets talk about the radical notation

- The radical notation for the fraction power is:

  the denominator of the power will be the radical and the numerator

  of the power will be the power of the base

- Ex: [tex]x^{\frac{a}{b}}=\sqrt[b]{x^{a}}[/tex]

* Lets solve the problem

∵ (18 y²)^(1/7)

- The power 1/7 will be the radical over (18 y²)

∴ [tex](18y^{2})^{\frac{1}{7}}=\sqrt[7]{18y^{2}}[/tex]

* The radical notation is [tex]\sqrt[7]{18y^{2} }[/tex]

10)

* Lets talk about the imaginary number

- Because there is no even root for negative number, the imaginary

 numbers founded to solve this problem

- It is a complex number that can be written as a real number multiplied

  by the imaginary unit i, which is defined by i = √(-1) or i² = -1

- Ex: √(-5) = √[-1 × 5] = i√5

* The definition of number i is √(-1)

Answer:

True

Step-by-step explanation:

If two events X and Y are mutually exclusive,

Then,

P(X∪Y) = P(X) + P(Y)

Let A represents the event of a diamond card and B represent the event of a heart card,

We know that,

In a deck of 52 cards there are 4 suit ( 13 Club cards, 13 heart cards, 13 diamond cards and 13 Spade cards )

That is, those cards which are heart can not be diamond card,

⇒ A ∩ B = ∅

⇒ P(A∩B) = 0

Since, P(A∪B) = P(A) + P(B) - P(A∩B)

⇒ P(A∪B) = P(A) + P(B)

By the above statement,

Events A and B are mutually exclusive,

Hence, the probability of selecting a 4 of diamonds or a 4 of hearts is an example of a mutually exclusive event is a true statement.

Answer:

(a) 19.

Step-by-step explanation:

The even numbers in the given range are -14, -12, -10, -8, - 6 , -4 and -2. and  7 more of their positive values. Total 14.

The numbers divisible by 3  and not 2 are -9, -3,  3 and 9.

Also 0 is a multiple of any number

Thus,  number of thrift numbers are  14 + 4 + 1 = 19 (answer).

Answer:

The difference in volume of the two models is [tex]\frac{128}{3}\ in^{3}[/tex]

Step-by-step explanation:

we know that

The volume of a square pyramid is equal to

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

where

b is the length of the side of the square base

h is the height of the pyramid

step 1

Find the volume of Amy's model

we have

[tex]b=8\ in[/tex]

[tex]h=5\ in[/tex]

substitute

[tex]V=\frac{1}{3}(8)^{2}(5)[/tex]

[tex]V=\frac{320}{3}\ in^{3}[/tex]

step 2

Find the volume of Alex's model

we have

[tex]b=8\ in[/tex]

[tex]h=3\ in[/tex]

substitute

[tex]V=\frac{1}{3}(8)^{2}(3)[/tex]

[tex]V=\frac{192}{3}\ in^{3}[/tex]

step 3

Find the difference in volume of the two models

[tex]\frac{320}{3}\ in^{3}-\frac{192}{3}\ in^{3}=\frac{128}{3}\ in^{3}[/tex]

Answer:

[tex]N(L)=\frac{k}{L^2}[/tex]

Step-by-step explanation:

Here, N represents the number of animal species and L represents a certain body length,

According to the question,

[tex]N\propto \frac{1}{L^2}[/tex]

[tex]\implies N=\frac{k}{L^2}[/tex]

Where, k is the constant of proportionality,

Since, with increasing the value of L the value of N is decreasing,

So, we can say that, N is dependent on L, or we can write N(L) in the place of N,

Hence, the required function formula is,

[tex]N(L)=\frac{k}{L^2}[/tex]

Certainly! When we say that the number of animal species \( N \) is inversely proportional to the square of the body length \( L \), what we mean mathematically is that as the body length increases, the number of species decreases at a rate that is the square of the increase in length. This can be represented by the following formula:

\[ N = \frac{k}{L^2} \]

Here \( N \) is the number of species, \( L \) is the body length, and \( k \) is the constant of proportionality. This constant \( k \) represents the number of species at the unit body length (when \( L = 1 \)). The constant of proportionality is determined by the specific biological context, based on empirical data or theoretical considerations.

In this formula, \( L^2 \) denotes the body length squared, and the fraction represents the inverse relationship.

In summary, to find the number of species \( N \) for a given body length \( L \), we use the inverse square relationship with the constant of proportionality \( k \).

The probability of an event can be computed by the probability formula by simply dividing the favorable number of outcomes by the total number of possible outcomes.

The probability of an event can be computed by the probability formula by simply dividing the favorable number of outcomes by the total number of possible outcomes.

Utilizing the TI-83, 83+, 84, 84+ Calculator to estimate these probabilities

Go to 2nd DISTR, and select item 2: normalcdf

The syntax is: normalcdf (lower bound, upper bound, mean, standard deviation)

a) P(1100 <= X <= 1500)

= normalcdf(1100, 1500, 1252, 129)

= 0.8534

b) P(X < 1125)

= normalcdf(-1E99, 1125, 1252, 129)

= 0.1624

c) P(X > 1200)

= normalcdf(1200, 1E99, 1252, 129)

= 0.6566 = 65.66%

d) P(X < 1000)

= normalcdf(-1E99, 1000, 1252, 129)

= 0.0254 = approx. 3rd percentile

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Answer:

x =  3 ±sqrt(5)

Step-by-step explanation:

(x - 3)^2 = 5

Take the square root of each side

Sqrt( (x - 3)^)2 =±sqrt( 5)

x-3 = ±sqrt(5)

Add 3 to each side

x-3+3 = 3 ±sqrt(5)

x =  3 ±sqrt(5)

Answer:

The probability is 0.20

Step-by-step explanation:

a) Lets revise how to find the z-score

- The rule the z-score is z = (x - μ)/σ , where

# x is the score

# μ is the mean

# σ is the standard deviation

* Lets solve the problem

- Bob's golf score at his local course follows the normal distribution

- The mean is 92.1

- The standard deviation is 3.8

- The score on his next round of golf will be between 82 and​ 89

- Lets find the z-score for each case

# First case

∵ z = (x - μ)/σ

∵ x = 82

∵ μ = 92.1

∵ σ = 3.8

∴ [tex]z=\frac{82-92.1}{3.8}=\frac{-10.1}{3.8}=-2.66[/tex]

# Second case

∵ z = (x - μ)/σ

∵ x = 89

∵ μ = 92.1

∵ σ = 3.8

∴ [tex]z=\frac{89-92.1}{3.8}=\frac{-3.1}{3.8}=-0.82[/tex]

- To find the probability that the score on his next round of golf will

  be between 82 and​ 89 use the table of the normal distribution

∵ P(82 < X < 89) = P(-2.66 < z < -0.82)

∵ A z-score of -2.66 the value is 0.00391

∵ A z-score of -0.82 the value is 0.20611

∴ P(-2.66 < z < -0.82) = 0.20611 - 0.00391 = 0.2022

* The probability is 0.20

Normal distribution and z-scores are applied in this context. The z-scores for the given range are calculated, followed by finding the correlating probabilities from the z-table, resulting in the probability of Bob's next score falling within the range of 82-89.

To answer this question, we will utilize the concept of z-scores in a normal distribution. A z-score basically explains how many standard deviations a data point (in this case, a golf score) is from the mean.

Firstly, we calculate the z-scores for the limits given. Here, 82 and 89. The formula we use is Z = (X - μ) / σ, where X is the golf score, μ is the mean, and σ is the standard deviation.

The calculated z-scores are then used as references, and we refer a z-table (also known as a standard normal table) to find the probabilities which correspond to these z-scores, and the result is a probability of Bob's next golf score being between 82 and 89.

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The total number of ways are:

                       462

When we are asked to select r items from a set of n items that the rule that is used to solve the problem is:

Method of combination.

Here the total number of bills of different values are: 7

i.e. n=7

(  $1, $2, $5, $10, $20, $50, and $100 )

and there are atleast five of each type of bill.

Also, we have to choose 5 bills i.e. r=5

The repetition is  allowed while choosing bills.

Hence, the formula is given by:

[tex]C(n+r-1,r)[/tex]

Hence, we get:

[tex]C(7+5-1,5)\\\\i.e.\\\\C(11,5)=\dfrac{11!}{5!\times (11-5)!}\\\\C(11,5)=\dfrac{11!}{5!\times 6!}\\\\\\C(11,5)=\dfrac{11\times 10\times 9\times 8\times 7\times 6!}{5!\times 6!}\\\\\\C(11,5)=\dfrac{11\times 10\times 9\times 8\times 7}{5!}\\\\\\C(11,5)=\dfrac{11\times 10\times 9\times 8\times 7}{5\times 4\times 3\times 2}\\\\\\C(11,5)=462[/tex]

           Hence, the answer is:

                  462

There are 462 different possible ways to choose five bills from a set of seven types with repetition allowed.

To determine the number of different ways to choose five bills from a set of bills with values $1, $2, $5, $10, $20, $50, and $100, we can use the combinatorial concept known as "combinations with repetition."

Given:

- Bill values:  [tex]\( \{1, 2, 5, 10, 20, 50, 100\} \)[/tex]

- Total number of types of bills: ( n = 7 )

- Number of bills to choose: ( r = 5 )

The formula for combinations with repetition (also known as "stars and bars") is:

\[

\binom{n + r - 1}{r}

\]

Substituting the values ( n = 7 ) and ( r = 5 ):

[tex]\[\binom{7 + 5 - 1}{5} = \binom{11}{5}\][/tex]

Now, calculate  [tex]\( \binom{11}{5} \):[/tex]

[tex]\[\binom{11}{5} = \frac{11!}{5!(11-5)!} = \frac{11!}{5! \cdot 6!}\][/tex]

First, compute the factorials:

[tex]\[11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\]\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]\[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\][/tex]

Next, compute [tex]\( \frac{11!}{6!} \):[/tex]

[tex]\[\frac{11!}{6!} = 11 \times 10 \times 9 \times 8 \times 7\]\[= 11 \times 10 = 110\]\[110 \times 9 = 990\]\[990 \times 8 = 7920\]\[7920 \times 7 = 55440\][/tex]

Now, compute:

[tex]\[\frac{55440}{120} = 462\][/tex]

Therefore, the number of different possible ways to choose the five bills is:

[tex]\[\boxed{462}\][/tex]

Answer:

  [2, 15]

Step-by-step explanation:

The graph shows the n/4 algorithm to be better (smaller run time) for n in the range 2 to 15.

In this Physics problem, we calculate the horizontal distance a stone travels when thrown horizontally from a cliff to determine the ratio of heights between two buildings.

The horizontal distance the stone will travel can be calculated using the formula:

d = v*t

Where d is the distance, v is the initial velocity, and t is the time of flight. Using the information given, we can calculate the distance the stone from block A will travel, and then find the ratio of building A's height to building B's height.

Answer:   [tex]\bold{\dfrac{h^9}{g^9}}[/tex]

Step-by-step explanation:

The power rule is "multiply the exponents".

You must understand that the exponent of both h and g is 1.

Multiply 1 times 9 for both variables.

[tex]\bigg(\dfrac{h}{g}\bigg)^9=\bigg(\dfrac{h^1}{g^1}\bigg)^9=\dfrac{h^{1\times 9}}{g^{1\times 9}}=\large\boxed{\dfrac{h^9}{g^9}}[/tex]

[tex]\bf ~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^{-m}\implies a^{n-m} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{125^2}{125^{\frac{4}{3}}}\implies \cfrac{(5^3)^2}{(5^3)^{\frac{4}{3}}}\implies \cfrac{5^{3\cdot 2}}{5^{3\cdot \frac{4}{3}}}\implies \cfrac{5^6}{5^4}\\\\\\ 5^6\cdot 5^{-4}\implies 5^{6-4}\implies 5^2\implies 25[/tex]

Answer:

  f(x) = f(-x) = x^2 -1 is an even function

Step-by-step explanation:

When f(x) = f(-x), the function is symmetrical about the y-axis. That is the definition of an even function.

___

An odd function is symmetrical about the origin: f(x) = -f(-x).

Check the picture below.

The given vertices define a polygon consisting of a rectangle and a triangle. The areas of the shapes are calculated separately and added together to find the total area of the polygon, which is 20 square units.

The given coordinates define two distinct shapes, a rectangle, and a triangle. Considering point A(0,1), B(0,5), C(4,5), and D(6,1), we can see that the rectangle is A, B, C and a point E(4,1) and the triangle is E, C, D. The area of any rectangle is calculated as

width multiplied by height

. In the case of our rectangle, the width is the distance between A and E - 4 units - and the height is from A to B, or 5-1 = 4 units. Thus, the

area of the rectangle

is 4 * 4 = 16 units

2

. The area of a triangle is calculated as 1/2 * base * height. For triangle ECD, the base is the distance from C to D, or 6-4=2 units, and the height (equal to EC) is 4 units. Thus the

area of the triangle

is 1/2 * 2 * 4 = 4 units

2

. The total

area of the polygon

is the area of the rectangle + the area of the triangle = 16 units

2

+ 4 units

2

= 20 units

2

.

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Answer:

(a) shortest ladder length ≈ 35.7 ft (rounded to tenth)

(b) L = (d/x +1)√(16+x²) . . . . where 16 is fence height squared

Step-by-step explanation:

It works well to solve the second part of the problem first, then put in the specific numbers.

We have not been asked anything about "b", so we can basically ignore it. Using the Pythagorean theorem, we find the length GH in the attached drawing to be ...

  GH = √(4²+x²) = √(16+x²)

Then using similar triangles, we can find the ladder length L to be that which satisfies ...

  L/(d+x) = GH/x

  L = (d +x)/x·√(16 +x²)

The derivative with respect to x, L', is ...

  L' = (d+x)/√(16+x²) +√(16+x²)/x - (d+x)√(16+x²)/x²

Simplifying gives ...

  L' = (x³ -16d)/(x²√(16+x²))

Our objective is to minimize L by making L' zero. (Of course, only the numerator needs to be considered.)

___

(a) For d=24, we want ...

  0 = x³ -24·16

  x = 4·cuberoot(6) ≈ 7.268 . . . . . feet

Then L is

  L = (24 +7.268)/7.268·√(16 +7.268²) ≈ 35.691 . . . feet

__

(b) The objective function is the length of the ladder, L. We want to minimize it.

  L = (d/x +1)√(16+x²)

The objective function for the length of the ladder in terms of x is L = sqrt((x+b)^2 + d^2).

To find the length of the shortest ladder that extends from the ground to the house without touching the fence, we can use the concepts of similar triangles. Let L be the length of the ladder, x be the distance from the base of the ladder to the fence, d be the distance from the fence to the house, and b be the distance from the ground to the point the ladder touches the house. In terms of x, the objective function for the length of the ladder is:

L = sqrt((x+b)^2 + d^2)

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