You just purchased two coins at a price of $670 each. Because one of the coins is more collectible, you believe that its value will increase at a rate of 7.1 percent per year, while you believe the second coin will only increase at 6.5 percent per year. If you are correct, how much more will the first coin be worth in 15 years?

Answer:

You will requiere 0.34375 oz of the solution for 22 ounces of water.

Step-by-step explanation:

We should start by matching the units: in the example we have 2 gallons of water, while the question refers to 22 ounces of water.

So, by the equivalence we are given, we now know that 2 gallons of water are equal to 256 ounces.

From this point the question can be solved using an arithmetic method known as cross-multiply (sometimes refered as the Rule of Three). This consist of wrinting an equation of the form:

[tex]\frac{a}{b} = \frac{c}{x}\\[/tex]

Where b is the amount of solution required for a ounces of water, and x (unknown variable) is the amount of solution for c ounces of water.

So we have the next equation:

[tex]\frac{256}{4} =\frac{22}{x}[/tex]

We apply the correspondent arithmetic rules so we can calculate the x as follows:

[tex]x=\frac{(4)(22)}{256}[/tex]

[tex]x=\frac{88}{256}[/tex]

[tex]x=0.34375[/tex]

This way we can reply that 0.34375 oz of solution are required for 22 oz of water.

Answer:

A

Step-by-step explanation:

Given

u = <6, -1>

u = 6i-j

and

v=<7,-4>

v=7i-4j

The formula for angle is:

Let x be the angle

[tex]cos\ x = \frac{u.v}{||u||.||v||}[/tex]

where ||u|| is the length and u.v is the dot product or scalar product of both vectors

So,

[tex]||u|| = \sqrt{(6)^2+(-1)^2}\\ = \sqrt{36+1}\\ = \sqrt{37}\\ ||v||=\sqrt{(7)^2+(-4)^2}\\ = \sqrt{49+16}\\ = \sqrt{65}\\[/tex]

[tex]u.v = u_1u_2+v_1v_2\\= (6)(7)+(-1)(-4)\\=42+4\\=46[/tex]

[tex]cos\ x=\frac{46}{\sqrt{37}\sqrt{65}} \\= \frac{46}{\sqrt{2405} }\\Can\ also\ be\ written\ as:\\= \frac{46}{\sqrt{2405} } * \frac{\sqrt{2405} }{\sqrt{2405}} \\=\frac{46\sqrt{2405} }{2405}[/tex]

The calculated angle will be in radians. To find the angle in degrees:

[tex]x = \frac{180}{\pi} cos^{-1} (\frac{46\sqrt{2405} }{2405})\\x = 20.282\\x= 20.3\\[/tex]

Hence Option A is correct ..

Answer:

General Solution is [tex]y=x^{3}+cx^{2}[/tex] and the particular solution is  [tex]y=x^{3}-\frac{1}{2}x^{2}[/tex]

Step-by-step explanation:

[tex]x\frac{\mathrm{dy} }{\mathrm{d} x}=x^{3}+3y\\\\Rearranging \\\\x\frac{\mathrm{dy} }{\mathrm{d} x}-3y=x^{3}\\\\\frac{\mathrm{d} y}{\mathrm{d} x}-\frac{3y}{x}=x^{2}[/tex]

This is a linear diffrential equation of type

[tex]\frac{\mathrm{d} y}{\mathrm{d} x}+p(x)y=q(x)[/tex]..................(i)

here [tex]p(x)=\frac{-2}{x}[/tex]

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

The solution of equation i is given by

[tex]y\times e^{\int p(x)dx}=\int  e^{\int p(x)dx}\times q(x)dx[/tex]

we have [tex]e^{\int p(x)dx}=e^{\int \frac{-2}{x}dx}\\\\e^{\int \frac{-2}{x}dx}=e^{-2ln(x)}\\\\=e^{ln(x^{-2})}\\\\=\frac{1}{x^{2} } \\\\\because e^{ln(f(x))}=f(x)]\\\\Thus\\\\e^{\int p(x)dx}=\frac{1}{x^{2}}[/tex]

Thus the solution becomes

[tex]\tfrac{y}{x^{2}}=\int \frac{1}{x^{2}}\times x^{2}dx\\\\\tfrac{y}{x^{2}}=\int 1dx\\\\\tfrac{y}{x^{2}}=x+c[/tex][tex]y=x^{3}+cx^{2[/tex]

This is the general solution now to find the particular solution we put value of x=2 for which y=6

we have [tex]6=8+4c[/tex]

Thus solving for c we get c = -1/2

Thus particular solution becomes

[tex]y=x^{3}-\frac{1}{2}x^{2}[/tex]

Answer: [tex](0.52,\ 0.68)[/tex]

Step-by-step explanation:

The formula for a [tex]\alpha[/tex]- level confidence interval for the population proportion:-

[tex]p\pm z_{\alpha/2}\times\sqrt{\dfrac{p(1-p)}{n}}[/tex]

Given : n = 225 ; p = 0.60 ; [tex]\alpha= 1-0.99=0.01[/tex]

By using the given information , the confidence interval for the population proportion:-

[tex]0.6\pm z_{0.005}\times\sqrt{\dfrac{0.6(1-0.6)}{225}}\\\\=0.6\pm(2.576)(0.0327)\\\\=0.6\pm0.08\\\\=0.6-0.08,\ 0.6+0.08\\\\=(0.52,\ 0.68)[/tex]

Hence, the resulting confidence interval is [tex](0.52,\ 0.68)[/tex] .

To construct a 99% confidence interval for the population proportion, we can use the sample proportion, critical value, and standard error. The resulting confidence interval is (0.516, 0.684).

To calculate a confidence interval for a population proportion, we can use the formula:

CI = sample proportion ± (critical value)(standard error)

Given that the sample proportion is 60% (0.6) and the confidence level is 99%, we need to find the critical value corresponding to a 99% confidence level. Using a normal distribution, the critical value is approximately 2.576. The standard error can be calculated using the formula:

SE = √((sample proportion)(1 - sample proportion) / sample size)

Let's assume the sample size is 225. Plugging these values into the formula, we can calculate the standard error:

SE = √((0.6)(1 - 0.6) / 225) ≈ √(0.24 / 225) ≈ √0.001067 ≈ 0.0326

Now we can calculate the confidence interval:

CI = 0.6 ± (2.576)(0.0326) ≈ 0.6 ± 0.084

Therefore, the 99% confidence interval for the population proportion is approximately (0.516, 0.684).

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The correct option is to reject the null hypothesis. The mean life spans for whites and non-whites born in 1900 in the certain county are not the same.

To conduct the hypothesis test, we will use a two-sample z-test for the difference in two means. The null hypothesis (H0) states that there is no difference in the mean life spans between whites and non-whites, while the alternative hypothesis (Ha) states that there is a difference.

The formula for the z-test statistic is:

[tex]\[ z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \][/tex]

where:

- [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_2\)[/tex] are the sample means for whites and non-whites, respectively.

- [tex]\(\mu_1\)[/tex] and [tex]\(\mu_2\)[/tex] are the population means for whites and non-whites, respectively.

- [tex]\(\sigma_1\)[/tex] and [tex]\(\sigma_2\)[/tex] are the population standard deviations for whites and non-whites, respectively.

- [tex]\(n_1\)[/tex] and [tex]\(n_2\)[/tex] are the sample sizes for whites and non-whites, respectively.

Given:

- [tex]\(\bar{x}_1 = 45.3\) years, \(s_1 = 12.7\) years, \(n_1 = 124\)[/tex] (for whites)

- [tex]\(\bar{x}_2 = 34.1\)[/tex] years, [tex]\(s_2 = 15.6\) years, \(n_2 = 82\)[/tex] (for non-whites)

- [tex]\(\mu_1 = 47.6\)[/tex] years (for whites)

- [tex]\(\mu_2 = 33.0\)[/tex] years (for non-whites)

Since we do not have the population standard deviations, we will use the sample standard deviations as an estimate. This is appropriate given the sample sizes are large enough (generally [tex]\(n > 30\)[/tex] is considered sufficient).

First, we calculate the standard error (SE) of the difference in means:

[tex]\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{12.7^2}{124} + \frac{15.6^2}{82}} \][/tex]

[tex]\[ SE = \sqrt{\frac{161.29}{124} + \frac{243.36}{82}} \][/tex]

[tex]\[ SE = \sqrt{1.299 + 2.968} \][/tex]

[tex]\[ SE = \sqrt{4.267} \][/tex]

[tex]\[ SE \approx 2.066 \][/tex]

Now, we calculate the z-statistic:

[tex]\[ z = \frac{(45.3 - 34.1) - (47.6 - 33.0)}{2.066} \][/tex]

[tex]\[ z = \frac{11.2 - 14.6}{2.066} \][/tex]

[tex]\[ z = \frac{-3.4}{2.066} \][/tex]

[tex]\[ z \approx -1.646 \][/tex]

Next, we find the p-value for this z-statistic. Since we are conducting a two-tailed test, we will look at the probability of a z-score being less than -1.646 or greater than 1.646. Using a standard normal distribution table or a calculator, we find that the p-value is approximately 0.100.

Finally, we compare the p-value to our significance level (commonly denoted as [tex]\(\alpha\))[/tex]. If the p-value is less than [tex]\(\alpha\)[/tex], we reject the null hypothesis. If we choose [tex]\(\alpha = 0.05\)[/tex], then since [tex]\(0.100 > 0.05\)[/tex], we fail to reject the null hypothesis.

However, if we choose a significance level of [tex]\(\alpha = 0.10\)[/tex], then since [tex]\(0.100 \leq 0.10\)[/tex], we would reject the null hypothesis, indicating that there is a statistically significant difference in the mean life spans between whites and non-whites in the county.

Given the p-value is on the boundary of common significance levels, the conclusion may vary depending on the chosen level of significance. However, the correct option based on the provided conversation is to reject the null hypothesis, suggesting that the mean life spans are not the same for whites and non-whites in the county.

Answer:[tex]\frac{^{10}C_5}{^{12}C_5}[/tex]

Step-by-step explanation:

Given a total of 12 diesel engines

out of which 2 are defective

Company have to choose 5 good engines to ship.

Therefore no of ways in good engines are selected is [tex]^{10}C_5[/tex]

no of ways in which any 5 engines are selected from a total of 12 engines is

[tex]^{12}C_5[/tex]

Therefore the required probability =[tex]\frac{^{10}C_5}{^{12}C_5}[/tex]

=[tex]\frac{7}{22}[/tex]=0.318

Answer:

2500

Step-by-step explanation:

They tell you that only 40% of the total complete the quality test, and you want to know how many represent 100%.

If this 40% is 1000 valid responses use cross multiplication to know how many people are the total.

40%-----1000 valid responses

100%----x=

[tex]\frac{100*1000}{40} =X\\2500=X[/tex]

So, you can tell they need at least 2500 persons to have 1000 valid responses.

Answer:

Number of people = 3473

Step-by-step explanation:

Probability is typically the rate at which an event or occurrence is likely to happen. Whenever we're not so sure about an event outcome, we can then deliberate about the probabilities or possibility of certain outcomes that is how likely they are to happen.

Probability of success, =0.8×0.4×0.9 =0.288

Expected, E=1000

Expected = np

1000=n×0.288

n=​0.288​​1000​​≈34

Answer:

DNE

Step-by-step explanation:

Given that two particles travel along the space curves

[tex]r_1(t) = (t, t^2, t^3)\\ r_2(t) = (1 + 2t, 1 + 6t, 1 + 14t )[/tex]

To find the points of intersection:

At points of intersection both coordinates should be equal.

i.e. r1 =r2

Equate corresponding coordinates

[tex]t=1+2t\\t^2=1+6t\\t^3=1+14t[/tex]

I equation gives t =-1

Substitute in II equation to get [tex]t^2 = -5[/tex]

i.e. t cannot be real

Hence no point of intersection

DNE

Answer:

Step-by-step explanation:

Given lines in parametric form

line [tex]L_1[/tex]

[tex]\frac{x}{3}=\frac{y-1}{-2} =\frac{z-2}{-3}[/tex]

direction vector of [tex]L_1 v_1=<3,-2,-3 >[/tex]

Line [tex]L_2[/tex]

direction vector of [tex]L_2 v_2=<-9,6,9 >[/tex]

therefore

[tex]v_2=-3v_1[/tex]

thus lines are parallel.

(ii)distance between two lines is

[tex]L_2[/tex] is given by

[tex]\frac{x-3}{-9}=\frac{y-1}{6} =\frac{z-4}{9}[/tex]=s

[tex]\frac{x-3}{-3}=\frac{y-1}{2} =\frac{z-4}{3}[/tex]=3s

[tex]\left | \frac{\begin{vmatrix}x_2-x_1 &y_2-y_1 &z_2-z_1 \\ a_1&b_1&c_1 \\ a_2&b_2&c_2\end{vmatrix}}{\sqrt{\left ( a_1b_2-a_2b_1 \right )^2+\left ( b_1c_2-b_2c_1 \right )^2+\left ( c_1a_2-c_2a_1 \right )^2}}\right |[/tex]

where [tex]a_1[/tex]=3

[tex]b_1[/tex]=-2

[tex]c_1[/tex]=-3

[tex]a_2[/tex]=-9

[tex]b_2[/tex]=6

[tex]c_2[/tex]=9

distance(d)=0 units since value of the matrix

[tex]\begin{vmatrix}x_2-x_1 &y_2-y_1 &z_2-z_1\\ a_1&b_1&c_1 \\a_2&b_2 &c_2 \end{vmatrix}[/tex]

is zero

Answer:

1.  [tex]x=3[/tex]; 2. Domain:  [tex]x>1[/tex]  Range: all real numbers

Step-by-step explanation:

Let's find the solutions.

1. Solve the equation [tex]x=\sqrt{2x+3}[/tex] so:

[tex](x)^2=(\sqrt{2x+3})^2[/tex]

[tex]x^2=2x+3[/tex]

[tex]x^2-2x-3=0[/tex]

[tex]x1=\frac{-b+\sqrt{b^{2}-4ac}}{2a}[/tex]

[tex]x1=\frac{2+\sqrt{(-2)^{2}-(4*1*(-3))}}{2*1}[/tex]

[tex]x1=3[/tex]

[tex]x2=\frac{-b-\sqrt{b^{2}-4ac}}{2a}[/tex]

[tex]x2=\frac{2-\sqrt{(-2)^{2}-(4*1*(-3))}}{2*1}[/tex]

[tex]x2=-1[/tex]

Although we have two answers, remember that from the original equation the result of [tex]\sqrt{2x+3} > 0[/tex] is never negative. So -1 do not solve the equation.

In conlcusion, the equation is solved by x=3.

2A. Domains and ranges of f1(x) and f2(x)

[tex]f1(x)=ln(x+1)+ln(x-1)[/tex]

Using logarithmic property [tex]ln(a)+ln(b)=ln(a*b)[/tex] we have:

[tex]f1(x)=ln(x^2-1)[/tex] because:

[tex]ln(x)[/tex] is defined by  [tex]x>0[/tex] then:

[tex]x^2-1>0[/tex]

[tex]x>\sqrt{1}[/tex] so the domain of f1(x) is [tex]x>1[/tex]

Now for the range:

[tex]f1(x)=ln(x+1)+ln(x-1)[/tex]

[tex]y=ln(x^2-1)[/tex]

[tex]e^y=x^2-1[/tex]

[tex]\sqrt{e^y+1}=x^2-1[/tex] notice that [tex]e^y+1[/tex] is always positive, so the range of f1(x) is all real numbers.

Be aware that although point number two of the problem mentioned two equations, f1(x)=f2(x) by logarithmic properties, so their domains and ranges are the same.

2B. Graph of f1(x) is attached. Because f1(x)=f2(x) both functions plot equal.

Answer:

  $21012.50

Step-by-step explanation:

$20,000 today will be inflated to $20,000·(1.025)^2 ≈ $21012.50 in 2 years. We presume this is the nominal terminal value you want.

Answer:

By 2086

Step-by-step explanation:

The provided equation is:

[tex]A=A0*e^{k*t}[/tex] , where:

A=total of population after t years

A0=initial population

k= rate of growth

t= time in years

Given information:

The final population will be 15 million, then A=15.

We start in 2000 with a 5.82 million population, then A0=5.82.

Missing information:

Although k is not given, we can calculate k by using the following statement, from 2000 to 2040 (within 40 years) population is proyected to grow to 9 million, which means a passage from 5.8 to 9 million (3.2 million increament).

Then we can use the same expression to calculate k:

[tex]A=A0*e^{k*t}[/tex]

[tex]9=5.8*e^{40*k}[/tex]

[tex]ln(9/5.8)/40=k[/tex]

[tex]0.010984166494596147=k[/tex]

[tex]0.011=k[/tex]

Now that we have k=0.011, we can find the time (t) by which population will be 15 million:

[tex]A=A0*e^{k*t}[/tex]

[tex]15=5.8*e^{0.011t}[/tex]

[tex]ln(15/5.8)/0.011=t[/tex]

[tex]86.38111668634878=t[/tex]

[tex]86.38=t[/tex]

Because the starting year is 2000, and we need 86.38 years for increasing the population from 5.8 to 15 million, then by 2086 the population will be 15 million.

Answer with  explanation:

Given L: V\rightarrow W is a linear transformation of a vector space  V into a vector space W.

Let Dim V= DimW=n

a.If L is one-one

Then nullity=0 .It means dimension of null space is zero.

By rank- nullity theorem we have

Rank+nullity= Dim V=n

Rank+0=n

Rank=n

Hence, the linear transformation is onto. Because dimension of range is equal to dimension of codomain.

b.If linear transformation is onto.

It means  dimension of range space is equal to dimension of codomain

Rank=n

By rank nullity theorem  we have

Rank + nullity=dimV

n+nullity=n

Nullity=n-n=0

Dimension of null space is zero.Hence, the linear transformation is one-one.

To prove Corollary 6.2, we show that a one-to-one linear transformation between vector spaces of equal dimensions is onto, and conversely, an onto transformation is one-to-one. This conclusion is based on the properties of linear transformations and the significance of preserving vector space dimensions.

To prove Corollary 6.2 regarding a linear transformation L: V → W, where both vector spaces V and W have the same dimension, we must show two parts:

If L is one-to-one, then it is onto. Assuming L is one-to-one, for every vector v in V, there is a unique image in W, guaranteeing that all vectors in W can be reached since dim V = dim W. Thus, L covers all of W, making it onto.

If L is onto, then it is one-to-one. When assuming L is onto, every vector in W is the image of some vector in V. Because dim V = dim W, there can't be more vectors in V than in W, preventing multiple vectors in V from mapping to the same vector in W, thus L is one-to-one.

These parts rely heavily on the concept that linear transformations maintain vector operations, such as the addition and scalar multiplication, and the relationship between dimensions of the domain and codomain for linear transformations.

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Answer:

80,00[tex]ft^{2}[/tex]

Step-by-step explanation:

According to my research, the formula for the Area of a rectangle is the following,

[tex]A = L*W[/tex]

Where

Since the building wall is acting as one side length of the rectangle. We are left with 1 length and 2 width sides. To maximize the Area of the parking lot we will need to equally divide the 800 ft of fencing between the Length and Width.

800 / 2 = 400ft

So We have 400 ft for the length and 400 ft for the width. Since the width has 2 sides we need to divide 60 by 2.

400/2 = 200 ft

Now we can calculate the maximum Area using the values above.

[tex]A = 400ft*200ft[/tex]

[tex]A = 80,000ft^{2}[/tex]

So the Maximum area we are able to create with 800 ft of fencing is 80,00[tex]ft^{2}[/tex]

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

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The largest possible parking lot, given 800 feet of fencing with one side not requiring a fence, would be a rectangular lot with dimensions of 400 feet by 200 feet.

We're dealing with a rectangular parking lot here where one of its sides is bordered by a building, so it doesn't need a fence. We have 800 feet of fence available for the three remaining sides. Let's denote the length of the rectangular parking lot by 'x' and the width by 'y'.

Because we only need to fence three sides, we can establish the following equation based on the total amount of fence available:

x + 2y = 800

To calculate the maximum possible area of a rectangle, we need to use the formula for the area of a rectangle, which is length times width (Area = x * y). However, we want to express the area in terms of a single variable. To do this, we can rearrange our fence equation to solve for y:

y = (800 - x) / 2

Now replace y in the area equation:

Area = x * (800 - x) / 2

For the area to be maximum, the derivative of the area with respect to x must be equal to zero. Differentiating and solving for x, we get the dimensions as x = 400 and y = 200.

So the largest possible parking lot would have dimensions of 400 feet by 200 feet.

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Answer:

2 ties of 4 feet and 1 tie of 5 feet

2 ties of 6 feet and 1 tie of 9 feet

Step-by-step explanation:

Given data

2 ties = 4 feet

2 ties = 6 feet

1 tie = 9 feet

1 tie = 5 feet

to find out

what are the total possible dimensions for each set of flower beds

solution

there are many combination but

the best possible combination are for 1st triangular

when we use 2 ties of 4 feet and 1 tie of 5 feet

and

for 2nd triangular

when we use 2 ties of 6 feet and 1 tie of 9 feet

only these are best combination

Answer:

Alfred has 250 socks in his collection.

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.

Based on the information given to us we can see that out of All his Brown socks he has 20% in the wash and the rest are in his drawer. Meaning 80% of the Brown socks are in his drawer. So we first need to find how many Brown socks are in the wash. We can solve this using the Rule of Three property as shown in the picture below.

120 drawer   ⇒   80%

      x wash    ⇒  20%

[tex]\frac{120*20}{80} = 30[/tex]

Now that we have the amount of Brown socks in the washer we can add that to the amount in the drawer to find the total amount of Brown socks.

[tex]Br = 120+30\\Br = 150[/tex]

So we now know that there are a total of 150 Brown socks. Since the question states that the Brown socks are 60% of the total we can use the Rule of Three to find the total.

150 Brown   ⇒   60%

    T  Total   ⇒   100%

[tex]\frac{150*100}{60} = T[/tex]

[tex]250 socks = T[/tex]

Finally, we can see that Alfred has 250 socks in his collection.

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

Alfred owns a total of 250 socks. This was derived by first finding out that Alfred has 150 brown socks, which represent 60% of his total socks' collection. Thus, the total number of socks Alfred owns is 250.

The question deals with the calculation of a total number of socks owned by Alfred.

If we know that 120 brown socks represent 80% of all of Alfred's brown socks (because 20% of them are in the wash), we can calculate the total number of brown socks. To do this, divide 120 by 0.8, which equals 150. Hence, Alfred has 150 brown socks.

We know from the problem that the brown socks account for 60% of all his socks. Hence the total number of socks (brown and black) is calculated by dividing the total number of brown socks (150) by 0.6. After performing this percentage calculation, we find that Alfred owns 250 socks.

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[tex]f(x,y,z)=2z^2x+y^3[/tex]

[tex]f[/tex] has gradient

[tex]\nabla f(x,y,z)=2z^2\,\vec\imath+3y^2\,\vec\jmath+4xz\,\vec k[/tex]

which at the point (-1, 4, 3) has a value of

[tex]\nabla f(-1,4,3)=18\,\vec\imath+48\,\vec\jmath-12\,\vec k[/tex]

I'm not sure what the given direction vector is supposed to be, but my best guess is that it's intended to say [tex]\vec u=15\,\vec\imath+25\,\vec\jmath[/tex], in which case we have

[tex]\|\vec u\|=\sqrt{15^2+25^2}=5\sqrt{34}[/tex]

Then the derivative of [tex]f[/tex] at (-1, 4, 3) in the direction of [tex]\vec u[/tex] is

[tex]D_{\vec u}f(-1,4,3)=\nabla f(-1,4,3)\cdot\dfrac{\vec u}{\|\vec u\|}=\boxed{\dfrac{294}{\sqrt{34}}}[/tex]

To find the directional derivative, we need to find the gradient of the function and dot product it with the given direction vector.

To find the directional derivative, we need to find the gradient of the function and dot product it with the given direction vector. The gradient of the function f(x, y, z) = 2z^2x + y^3 is (∂f/∂x, ∂f/∂y, ∂f/∂z) = (4zx, 3y^2, 4xz). The directional derivative in the direction of the vector (15-√i + 25-√j) is given by the dot product of the gradient and the direction vector: (4(-1)(15-√) + 3(4^2)(25-√) + 4(3)(-1)(15-√))/√((15-√)^2 + (25-√)^2). Simplifying this expression gives the directional derivative.

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Answer with explanation:

It is given that, a, b and c belong to the set of integers.

→ gcd(a,c)=d

→GCD=Greatest Common Divisor

→The greatest number which divides both a and c is d.

It means d divides a, and d divides c.

 a=d k, for some integer k.-------(1)

  c= d m, for some integer m.-------(2)

Now, it is given that, a divides bc.

So,→ 'a' will divide "bdm".--------[using 2, as c=d m]

It shows that, a divides bd, that is a| bd.

Hence proved

Answer: 38

Step-by-step explanation:

                 i.e. [tex]P-Q=n(P)-n(P\cap Q)[/tex]

Let A be the number of students who correctly answered the first question and B be the number of students who correctly answered the second question .

Given : [tex]n(A)=76[/tex]

[tex]n(B)=60[/tex]

[tex]n(A\cap B)=38[/tex]

Then the number of students who answered the first question correctly, but not the second is given by :-

[tex]A-B=n(A)-n(A\cap B)\\\\=76-38=38[/tex]

Hence, the number of students who answered the first question correctly, but not the second is 38.

Answer:

(x² -7)/(2x + 1)

Step-by-step explanation:

f(x) = 2x+1 and g(x) = x² -7

thus: (g/f)(x) = g(x)/f(x) = x² -7/2x + 1

The answer is:

[tex](g \circ f)(x)=4x^{2}-4x-1[/tex]

To solve the problem, we need to remember that composing functions means evaluate a function into another different function.

Also, we need to remember how to solve the following notable product:

[tex](a-b)^{2}=a^{2}-2ab+b^{2}[/tex]

We have that:

[tex](g \circ f)(x)=g(f(x))[/tex]

Now, we are given the equations:

[tex]f(x)=2x-1\\g(x)=x^{2}-2[/tex]

So, composing we have:

[tex](g \circ f)(x)=g(f(x))[/tex]

[tex](g \circ f)(x)=(2x-1)^{2}-2[/tex]

Now, we have to solve the notable product:

[tex](g \circ f)(x)=((2x)^{2}-2(2x*1)+1^{2})-2[/tex]

[tex](g \circ f)(x)=4x^{2}-4x+1-2[/tex]

Hence, we have that:

[tex](g \circ f)(x)=4x^{2}-4x-1[/tex]

Have a nice day!

Answer:

0.48

Step-by-step explanation:

Probability that the first person chooses a hotel

⁵C₁

[tex]^5C_1=\frac{5!}{(5-1)!1!}\\=\frac{120}{24}=5[/tex]

Probability that the second person chooses a different hotel

⁴C₁

[tex]^4C_1=\frac{4!}{(4-1)!1!}\\=\frac{24}{6}=4[/tex]

because the choice of hotels has reduced by 1 as one hotel is occupied by the first person

Probability that the second person chooses a different hotel

³C₁

[tex]^3C_1=\frac{3!}{(3-1)!1!}\\=\frac{6}{2}=3[/tex]

because the choice of hotels has reduced by 2 as two different hotels are occupied by the first person and second person

∴ The favorable outcomes are =⁵C₁×⁴C₁׳C₁=5×4×3=60

The total number of outcomes=5³=125

∴Probability that they each check into a different hotel=60/125=0.48

Answer:

$933.12

Step-by-step explanation:

This is a composite figure:  an upper rectangle and a lower one.  We need to find the volumes of each one individually, add the volumes together, then multiply the total volume by .02

The upper rectangle has a volume of:

V = 72×12×24

V = 20,736 cubic inches

The lower rectangle has a volume of:

V = 72×36×10

V = 25,920 cubic inches

The sum of the two volumes is

V = 46,656 inches cubed

Multiply that by .02:

46,656(.02) = $933.12

Answer:

2/9

Step-by-step explanation:

We are given Susan (S) is 3 times as old as Paul (P) so S=3P.

And Paul (P) is 3/2 times as old as Mary (M) so P=(3/2)M.

What fraction of Susan's age is Mary?

So if S=3P and P=(3/2)M, then by substitution we have

[tex]S=3(\frac{3}{2})M[/tex]

[tex]S=\frac{9}{2}M[/tex]

Multiply both sides by the reciprocal of 9/2 resulting in:

[tex]\frac{2}{9}S=M[/tex]

This says Mary is two-ninths the age of Susan.

Answer:

b) 2/9.

Step-by-step explanation:

Let Mary's age be x , then:

Paul's age is (3/2)x.

Susan's age =  3(3/2)x = 9/2 x.

So Mary is  2/9 the age of Susan.

We are asked to prove by the method of mathematical induction that:

      3n(n+1) is divisible by 6 for all positive integers.

[tex]3n(n+1)=3\times 1(1+1)\\\\i.e.\\\\3n(n+1)=3\times 2\\\\i.e.\\\\3n(n+1)=6[/tex]

which is divisible by 6.

Hence, the result is true for n=1

i.e. 3k(k+1) is divisible by 6.

Let n=k+1

then

[tex]3n(n+1)=3(k+1)\times (k+1+1)\\\\i.e.\\\\3n(n+1)=3(k+1)(k+2)\\\\i.e.\\\\3n(n+1)=(3k+3)(k+2)\\\\i.e.\\\\3n(n+1)=3k(k+2)+3(k+2)\\\\i.e.\\\\3n(n+1)=3k^2+6k+3k+6\\\\i.e.\\\\3n(n+1)=3k^2+3k+6k+6\\\\i.e.\\\\3n(n+1)=3k(k+1)+6(k+1)[/tex]

Since, the first term:

[tex]3k(k+1)[/tex] is divisible by 6.

( As the result is true for n=k)

and the second term [tex]6(k+1)[/tex] is also divisible by 6.

Hence, the sum:

[tex]3k(k+1)+6(k+1)[/tex]  is divisible by 6.

Hence, the result is true for n=k+1

Hence, we may say that the result is true for all n where n belongs to positive integers.

To prove by induction that the expression 3n(n+1) is divisible by 6, we start with the base case of n=1, which is divisible by 6, and then show that if it holds for an integer k, it also holds for k+1. By factoring and using the induction hypothesis, we demonstrate the expression's divisibility by 6 for all positive integers.

Proof by Induction of Divisibility by 6

To prove by induction that 3n(n+1) is divisible by 6 for all positive integers, we follow two steps: the base case and the inductive step.

Base Case

Let's check for n=1:

Inductive Step

Assume the statement holds for a positive integer k, so 3k(k+1) is divisible by 6.

Now, we must show that 3(k+1)((k+1)+1) = 3(k+1)(k+2) is also divisible by 6.

Factoring out the common term we get:

Notice that (k+1) is an integer, hence 3(k+1) is divisible by 3. If k is even, then k+1 is odd, so 3(k+1) is still divisible by 3 but not necessarily by 6. However, if k is odd, k+1 is even and 3(k+1) is divisible by 6. Since the divisible by 3 part is always true, and the divisible by an additional factor of 2 part is true every other time, the sum 6m + 3(k+1) is divisible by 6 regardless of whether k is odd or even. This is because adding a multiple of 6 to either another multiple of 6 or a multiple of 3 always results in a multiple of 6.

Therefore, 3n(n + 1) is divisible by 6 for all positive integers n.

Final answer:

To calculate the monetary value of Bill Mason's job offers, you subtract the taxes from the salary and add the nontaxable benefits. Job 1 results in a total monetary value of $36,500, while Job 2 is higher, with a total monetary value of $37,425. Therefore, Job 2 offers a higher total monetary value.

Explanation:

Bill Mason is considering two job offers, each with a different combination of salary and nontaxable benefits. To determine the monetary value for both jobs, considering a 25 percent tax rate, we must calculate the Net Annual Income for each job separately. Net Annual Income is the salary after taxes have been subtracted. Nonetheless, nontaxable benefits do not affect the Net Annual Income as they do not get taxed.

Calculations for Job 1:

Gross Salary: $41,300

Taxable Income (Tax Rate 25%): 0.25 x $41,300 = $10,325

Net Salary after Tax: $41,300 - $10,325 = $30,975

Nontaxable Benefits: $5,525

Total Monetary Value: Net Salary + Benefits = $30,975 + $5,525 = $36,500

Calculations for Job 2:

Gross Salary: $40,400

Taxable Income (Tax Rate 25%): 0.25 x $40,400 = $10,100

Net Salary after Tax: $40,400 - $10,100 = $30,300

Nontaxable Benefits: $7,125

Total Monetary Value: Net Salary + Benefits = $30,300 + $7,125 = $37,425

The final answer for the total monetary value of Job 1 is $36,500 and for Job 2 is $37,425. As per the given calculations, Job 2 offers a higher total monetary value compared to Job 1 when taking into account the salary after tax plus nontaxable benefits. This explanation should provide clarity on how to assess job offers based on their financial merits.

Answer:

We conclude that population mean is equal to 100 at the α=0.01 level of​ significance.

Step-by-step explanation:

Given information:

Sample size, n=40

Sample mean=104.3

sample standard deviation, s=18.2

We need to check whether the population mean greater than 100 at the α=0.01 level of​ significance.

Null hypothesis:

[tex]H_0:\mu=100[/tex]

Alternative hypothesis:

[tex]H_1:\mu>100[/tex]

Let as assume that the data follow the normal distribution. It is a right tailed test.

The formula for z score is

[tex]z=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}[/tex]

[tex]z=\frac{104.3-100}{\frac{18.2}{\sqrt{40}}}[/tex]

[tex]z=1.494263[/tex]

[tex]z\approx 1.49[/tex]

Using the standard normal table the p-value at z=1.49 and 0.01 level of significance is 0.068112.

(0.068112 > 0.01) p-value is greater than α, so we accept the null hypothesis.

Therefore, we conclude that population mean is equal to 100 at the α=0.01 level of​ significance.

Answer:

There are three transformation i.e. vertically stretch, downward and towards left.            

Step-by-step explanation:

Given : Parent function [tex]f(x)=|x|[/tex] and transformed function [tex]f(x)=2|x+3|-5[/tex]

To find : Name the type of function and describe each of the three transformations?

Solution :

Parent function [tex]f(x)=|x|[/tex] get transformed into [tex]f(x)=2|x+3|-5[/tex]

The transformations are as follow :

1) The function is multiplied by 2 which means there is a transformation vertically by a stretch factor '2'.

2) The function is subtracted by 5 which means there is a transformation downward by a factor '5'.

3) The function inside x value get added by 3 which means there is a transformation left by a factor '3'.

Therefore, There are three transformation i.e. vertically stretch, downward and towards left.

Answer:

Step-by-step explanation:

Put hypotheses as:

[tex]H_0: mu =40 mg\\H_a: mu \neq 40 mg.[/tex]

(Two tailed test at 4%)

Since population std deviation is  given we can do Z test.

[tex]Sample size n =30\\Sample mean = 42.3 mg\\Mean diff = 2.3 mg\\Std error of sample = \frac{7.1}{\sqrt{30} } =1.296[/tex]

Test statistic t = mean diff/se = [tex]\frac{2.3}{1.296} =1.775[/tex]

p value=0.075

Since p >0.04 our alpha, we accept null hypothesis.

At alphaequals0.04​, we cannot reject the​ company's claim

Step-by-step explanation:

Put hypotheses as:

\begin{gathered}H_0: mu =40 mg\\H_a: mu \neq 40 mg.\end{gathered}

H

0

:mu=40mg

H

a

:mu



=40mg.

(Two tailed test at 4%)

Since population std deviation is given we can do Z test.

\begin{gathered}Sample size n =30\\Sample mean = 42.3 mg\\Mean diff = 2.3 mg\\Std error of sample = \frac{7.1}{\sqrt{30} } =1.296\end{gathered}

Samplesizen=30

Samplemean=42.3mg

Meandiff=2.3mg

Stderrorofsample=

30

7.1

=1.296

Test statistic t = mean diff/se = \frac{2.3}{1.296} =1.775

1.296

2.3

=1.775

p value=0.075

Since p >0.04 our alpha, we accept null hypothesis.

At alphaequals0.04, we cannot reject the company's claim

Answer:

System A has 2 real solutions.

System B has 0 real solutions.

System C has 1 real solutions.

Step-by-step explanation:

If the graph of system of equation intersect each other at n points then the system of equation has n real solutions.

System A:

[tex]x^2+y^2=17[/tex]               .... (1)

[tex]y=-\frac{1}{2}x[/tex]             .... (2)

Plot the graph of these equations.

From graph (1) it is clear that the graph of equation (1) and (2), intersect each other at two points, (-3.688,1.844) and (3.688,-1.844).

Therefore, System A has two real solutions.

System B:

[tex]y=x^2-7x+10[/tex]               .... (3)

[tex]y=-6x+5[/tex]             .... (4)

Plot the graph of these equations.

From graph (2) it is clear that the graph of equation (3) and (4), never intersect each other.

Therefore, System B has 0 real solutions.

System C:

[tex]y=-2x^2+9[/tex]               .... (5)

[tex]8x-y=-17[/tex]             .... (6)

Plot the graph of these equations.

From graph (3) it is clear that the graph of equation (5) and (6), intersect each other at one points, (-2,1).

Therefore, System C has 1 real solutions.

Answer:

Step-by-step explanation:

The pigeonhole principal states that if n items are put into m containers, with n > m then at least one container must contain more than one item.

In other words, to find the case where maximum attempts are required, we eliminate all the cases where our criterion is not met and we will be left with the desired result.

For this case, the box has 10 black socks and 12 blue socks.

10 black socks = 5 left foot ones and  5 right foot ones

12 blue socks = 6 left foot ones and 6 right foot ones

If we draw all left foot or right foot ones then we will not have a pair till 11 draws have been made.

The next socks drawn will be a right foot one of blue or black color and a pair will be made.

Therefore, the maximum number of socks needed to be drawn from the box are 12.