Ellen said she spent half her money for lunch and half of what was left for a movie. She now has $1.20. How much did she spend for lunch?

Devise a plan

Carry out the plan

Look back (is it reasonable? Did we answer the question?)

Answer:

Since 2 and -1 are eigenvalues of the differential equation,

[tex]x(t) = c_{1}e^{-t} + c_{2}e^{2t}[/tex]

is a solution to the differential equation

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

The solution to the initial value problem is:

[tex]x(t) = \frac{20}{3}e^{-t} +  \frac{10}{3}e^{2t}[/tex]

Step-by-step explanation:

We have the following differential equation:

[tex]x'' - x' - 2x = 0[/tex]

The first step is finding the eigenvalues for this differential equation, that is, finding the roots of the following second order equation:

[tex]r^{2} - r - 2 = 0[/tex]

[tex]\bigtriangleup = (-1)^{2} -4*1*(-2) = 1 + 8 = 9[/tex]

[tex]r_{1} = \frac{-(-1) + \sqrt{\bigtriangleup}}{2*1} = \frac{1 + 3}{2} = 2[/tex]

[tex]r_{2} = \frac{-(-1) - \sqrt{\bigtriangleup}}{2*1} = \frac{1 - 3}{2} = -1[/tex]

Since 2 and -1 are eigenvalues of the differential equation,

[tex]x(t) = c_{1}e^{-t} + c_{2}e^{2t}[/tex]

is a solution to the differential equation.

Solution of the initial value problem:

[tex]x(t) = c_{1}e^{-t} + c_{2}e^{2t}[/tex]

[tex]x(0) = 10[/tex]

[tex]10 = c_{1}e^{-0} + c_{2}e^{2*0}[/tex]

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

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

[tex]x'(t) = -c_{1}e^{-t} + 2c_{2}e^{2t}[/tex]

[tex]x'(0) = 0[/tex]

[tex]0 = -c_{1}e^{-0} + 2c_{2}e^{2*0}[/tex]

[tex]-c_{1} + 2c_{2} = 0[/tex]

[tex]c_{1} = 2c_{2}[/tex]

So, we have to solve the following system:

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

[tex]c_{1} = 2c_{2}[/tex]

[tex]2c_{2} + c_{2} = 10[/tex]

[tex]3c_{2} = 10[/tex]

[tex]c_{2} = \frac{10}{3}[/tex]

[tex]c_{1} = 2c_{2} = \frac{20}{3}[/tex]

The solution to the initial value problem is:

[tex]x(t) = \frac{20}{3}e^{-t} +  \frac{10}{3}e^{2t}[/tex]

Answer:

0.0559

Step-by-step explanation:

Cant prove it but its right

To test the claim that more than 65% homeowners in Omaha possess a lawnmower, a one-tailed test of proportion was performed. After calculating a test statistic (z =1.52), it was determined that the P-value is 0.064, suggesting there is a 6.4% probability that a sample proportion this high could happen by chance given the null hypothesis is true.

In this question, you're asked to calculate the P-value for a test of the claim that the proportion of homeowners with lawn mowers in Omaha is higher than 65%. The proportion from the sample size of Omaha is given as 340/497 = 0.68 or 68%. To test this claim, you would employ a one-tailed test of proportion. The null hypothesis (H0) is that the proportion is equal to 65%, while the alternative hypothesis (Ha) is that the proportion is greater than 65%.

To determine the P-value, you need to first calculate the test statistic (z) using the formula: z = (p - P) / sqrt [(P(1 - P)) / n], where p is the sample proportion, P is the population proportion, and n is the sample size. Substituting into the formula, z = (0.683 - 0.65) / sqrt [(0.65 * 0.35) / 497] = 1.52. The P-value is the probability of getting a z-score that is greater than or equal to 1.52.

Using a standard normal distribution table, or an online z-score calculator, you can find that P(Z > 1.52) = 0.064 (approximately). This is the P-value for the test, which indicates that if the null hypothesis is true, there is a 6.4% probability that a sample proportion this high could occur by chance.

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Answer: 52$

Step-by-step explanation:

The money Dave spent on a t-shirt is obtained multiplying 120$ by the fraction 2/5

120(2/5) = 48$

And the money he spent on mountain bike magazines is obtained multiplying 120$ by the fraction 1/6

120(1/6)= 20$

The money he has left is:

120$ - 48$ -20$ = 52$

Answer:

Equilibrium quantity = 26.92

Equilibrium price is $31.13

Step-by-step explanation:

Given :Demand function : [tex]p^2 + 16q = 1400[/tex]

           Supply function : [tex]700 -p^2 + 10q = 0[/tex]

To Find : find the equilibrium quantity and equilibrium price.

Solution:

Demand function : [tex]p^2 + 16q = 1400[/tex]  --A

Supply function : [tex]p^2-10q=700[/tex] ---B

Now to find the equilibrium quantity and equilibrium price.

Solve A and B

Subtract B from A

[tex]p^2-10q -p^2-16q=700-1400[/tex]

[tex]-26q=-700[/tex]

[tex]26q=700[/tex]

[tex]q=\frac{700}{26}[/tex]

[tex]q=26.92[/tex]

So, equilibrium quantity = 26.92

Substitute the value of q in A

[tex]p^2 + 16(26.92) = 1400[/tex]

[tex]p^2 + 430.72 = 1400[/tex]

[tex]p^2 = 1400- 430.72[/tex]

[tex]p^2 = 969.28[/tex]

[tex]p = \sqrt{969.28}[/tex]

[tex]p = 31.13[/tex]

So, equilibrium price is $31.13

To calculate the expected winnings of the spinner game, one needs the probabilities of landing on specific segments. The expected value is found by summing the products of each outcome's probability and its monetary value, subtracting the cost of playing. Without these probabilities, an exact calculation cannot be provided.

To calculate the player's expected winnings in the game with the spinner, we need to understand the concept of expected value, which is essentially the average outcome if the game was played many times. For this game, we are given the following payouts: if the spinner lands on B, the player wins $8; if the spinner lands on C, the player wins $0.10 (a dime); otherwise, the player wins nothing. In addition, each spin costs $3, which will be factored into the expected winnings as a negative value.

Unfortunately, we do not have the probabilities of landing on B or C. Expected value is usually calculated by multiplying the probability of each outcome by its corresponding value and then summing those products. The general formula is Expected value = Σ(Probability of outcome × Value of outcome) - Cost per play.

Without the specific probabilities or the number of segments on the spinner, we cannot calculate the exact expected winnings. However, if hypothetical probabilities were provided, the calculation would follow the structure of: (Probability of landing on B × $8) + (Probability of landing on C × $0.10) - $3.

Answer:

Step-by-step explanation:

the answers are the points in the shaded region so plot the points and see which one is in the blue area so 3,-1

Answer:

A. (3,-1)

Step-by-step explanation:

In order to solve this you just have to search for the point in the graph, if the points are located in theline that the graph shows then they are actually a solution for the inequality shown, since the only point that is actually on the line that is shown in the graph is (3,-1) then that is the correct answer.

Answer:

According to the rule of 72, the doubling time for this interest rate is 8 years.

The exact doubling time of this amount is 8.04 years.

Step-by-step explanation:

Sometimes, the compound interest formula is quite complex to be solved, so the result can be estimated by the rule of 72.

By the rule of 72, we have that the doubling time D is given by:

[tex]D = \frac{72}{Interest Rate}[/tex]

The interest rate is in %.

In our exercise, the interest rate is 9%. So, by the rule of 72:

[tex]D = \frac{72}{9} = 8[/tex].

According to the rule of 72, the doubling time for this interest rate is 8 years.

Exact answer:

The exact answer is going to be found using the compound interest formula.

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

In which A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

So, for this exercise, we have:

We want to find the doubling time, that is, the time in which the amount is double the initial amount, double the principal.

is double the initial amount, double the principal.

[tex]A = 2P[/tex]

[tex]r = 0.09[/tex]

The interest is compounded anually, so [tex]n = 1[/tex]

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

[tex]2P = P(1 + \frac{0.09}{1})^{t}[/tex]

[tex]2 = (1.09)^{t}[/tex]

Now, we apply the following log propriety:

[tex]\log_{a} a^{n} = n[/tex]

So:

[tex]\log_{1.09}(1.09)^{t} = \log_{1.09} 2[/tex]

[tex]t = 8.04[/tex]

The exact doubling time of this amount is 8.04 years.

Answer:

The minimum approximate size to reach a maximum estimation error of 0.03 and a 99% confidence is 752 units

Step-by-step explanation:

In calculating the sample size to estimate a population proportion in which there is no information on an initial sample proportion, the principle of maximum uncertainty is assumed and a ratio [tex]P = 1/2[/tex] is assumed. The expression to calculate the size is:

[tex] n=\frac{z_{\alpha /2}^2}{4 \epsilon^2} [/tex]

With

Z value (for 0.005) [tex] Z _ {\alpha / 2} = 1.64485 [/tex]

Significance level [tex] \alpha = 0.01 [/tex]

Estimation error [tex] \epsilon = 0.03 [/tex]

[tex] n=\frac{(1.64485)^2}{(4)(0.03)^2} = 751.5398[/tex]

The minimum approximate size to reach a maximum estimation error of 0.03 and a 99% confidence is 752 units

To determine the percentage of male customers with a margin of error of 3% at a 99% confidence level, the manager would need to survey approximately 1847 customers.

The question is related to the concept of statistics, and more specifically to the idea of a confidence interval

for a population proportion. When you want to be very sure about your estimates, you use a high level of confidence. The standard formula for calculating the sample size needed in order to get a certain margin of error at a certain confidence level is n = [Z^2 * P * (1-P)] / E^2. In this formula, n is the sample size, Z is the z-score associated with your desired level of confidence, P is the preliminary estimate of the population proportion, and E is the desired margin of error. If the manager doesn't have a precursory idea of what the proportion of male customers is, it's standard to use P = 0.5. The Z score for a 99% confidence interval is approximately 2.576. Substituting these values, the manager would need a sample size of approximately 1846 customers. For more accuracy, it's better to round up to the next nearest whole number, so the minimum sample size required would be 1847.

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Answer:

Ans. the present value of $1,300/month, at 6.4% compounded monthly for 360 months (30 years) is $207,831.77

Step-by-step explanation:

Hi, first, we have to turn that 6.4% compound monthly rate into an effective rate, one that meets the units of the payment, in our case, effective monthly, that is:

[tex]r(EffectiveMonthly)=\frac{r(CompMonthly)}{12} =\frac{0.063}{12} =0.005333[/tex]

Therefore, our effective monthly rate is 0.5333%, and clearly the time of the investment is 30 years*12months=360 months.

Now, we need to use the following formula.

[tex]Present Value=\frac{A((1+r)^{n}-1) }{r(1+r)^{n} }[/tex]

Everything should look like this.

[tex]Present Value=\frac{1,300((1+0.005333)^{360}-1) }{0.005333(1+0.005333)^{360} }[/tex]

Therefore

[tex]PresentValue=207,831.77[/tex]

Best of luck.

Answer:

$9142.46

Step-by-step explanation:

Use the compounded interest formula: [tex]A=P(1+\frac{r}{m} )^{m*t}[/tex]

Where

A is the accumulated amount after compounding (our unknown)

P is the principal ($7500 in our case)

r is the interest rate in decimal form (0.05 in our case)

m is the number of compositions per year (2 in our semi-annually case)

and t is the number of years (5 in our case)

[tex]A=P(1+\frac{r}{m} )^{m*t}= 7500 (1+\frac{0.05}{2} )^{2*5} =9142.4581996....[/tex]

We round the answer to $9142.46

Answer:

See explanation below.

Step-by-step explanation:

The prime numbers are bold:

1  2  3 4  5 6  7  8  9  10  11 12  13  14  15  16  17

18  19  20  21  22  23  24  25  26  27  28  29  30  31

a) We can see that as we go higher, twin primes seem less frequent but even considering that, there is an infinite number of twin primes. If you go high enough you will still eventually find a prime that is separated from the next prime number by just one composite number.

b) I think it's interesting the amount of time that has been devoted to prove this conjecture and the amount of mathematicians who have been involved in this. One of the most interesting facts was that in 2004 a purported proof (by R. F. Arenstorf) of the conjecture was published but a serious error was found on it so the conjecture remains open.

Answer:

r=1 and t= -21/2.

Step-by-step explanation:

Two vectors are parallel if both are multiples. That is, for a vector (x,y), the parallel vector to (x,y) will be of the form k(x,y) with k a real number. Then,

a) (8r, -2) = 2(4r,-1). Then, we need to have that r=1, in other case the first component wouldn't be 4 or the second component wouldn't be -1 and the vector (8r,-2) wouldn't be parallel to (4, -1).

b) for the case of (8t, 21) we need -1 in the second component and 4 in the first component, then let t= -21/2 to factorize the -21 and get 4 in the fisrt component and -1 in the second component.

[tex](8\frac{-21}{2}, 21) = -21(\frac{8}{2}, -1) = -21(4,-1)[/tex]. In other case,  the vector (8t, 21) wouldn't be parallel to (4,-1).

Vector (8r,-2) is parallel to (4,-1) when r = 1, whereas (8t, 21) cannot be made parallel to it. To determine this, we look for a scalar multiple relation between the vectors.

The question asks for what value(s) of, if any, the given vector is parallel to (4,-1). To determine if two vectors are parallel, we need to see if one is a scalar multiple of the other, which means their components in each dimension multiply by the same scalar. Let's examine the given options:

(a) (8r,-2) is parallel to (4,-1) if there exists a scalar 'k' such that 4k = 8r and -1k = -2. By solving these equations, we find that k = 2 satisfies both, meaning if r = 1, the vector is parallel to (4,-1).

(b) (8t, 21) cannot be made parallel to (4,-1) through any scalar multiplication, as there's no single scalar that would simultaneously satisfy the required equations for both components.

Therefore, vector (8r,-2) is parallel to (4,-1) for r = 1, while vector (8t, 21) cannot be parallel to it under any circumstances.

Answer:

[tex]volume = 1536 m^3[/tex]

Step-by-step explanation:

given data;

B = 16m

b =8 m

height   H = 4 m

length  L = 32 m

volume of any right cylinder = (Area of bottom) \times (length)

Volume = A* L

The area of a trapezoid is

[tex]A=\frac{1}{2} H*(b+B)[/tex]

[tex]A =\frac{1}{2} 4*(8+16)[/tex]

[tex]A = 48 m^2[/tex]

therefore volume is given as  

volume = 48*32

[tex]volume = 1536 m^3[/tex]

Final answer:

To find the volume of the right trapezoidal cylinder, calculate the area of the trapezoid base and multiply by the cylinder's length. With B = 16m, b = 8m, and height = 4m, the trapezoid's area is 48m². The volume of the cylinder is 1536m³.

Explanation:

The volume of a right trapezoidal cylinder can be calculated using the area of the trapezoid as the base area and then multiplying by the height of the cylinder. Firstly, to calculate the area of the trapezoid (the base of the cylinder), we use the formula for the area of a trapezoid, which is ½ × (sum of the parallel sides) × height of the trapezoid. In this case, the parallel sides are B = 16m and b = 8m, and the height is 4m.

The area A of the trapezoid is then ½ × (16m + 8m) × 4m = ½ × 24m × 4m = 48m2. To find the volume of the trapezoidal cylinder, we multiply this area by the length of the cylinder, which is 32m. So, the volume V = 48m2 × 32m = 1536m3.

Answer:

4 parameters are necessary to specify all solutions and correspond to the number of free variables of the system.

Step-by-step explanation:

Remember that the number of free variables of a system is equal to m-rank(A) where m is the number of unknowns variables and A is the matrix of the system.

Since the system is consistent and the rank of the matrix is 3 then echelon form of the augmented matrix has two rows of zeros.

Then m-rank(A)=7-3=4.

Answer:

[tex][0,\infty)[/tex]

Step-by-step explanation:

We have been given that the height, h, of a ball that is tossed into the air is a function of the time, t, it is in the air. The height in feet fort seconds is given by the function [tex]h(t)=-16t^2+96t[/tex].

We are told that the height of the ball is function of time, which means time is independent variable.

We know that the domain of a function is all real values of independent variable for which function is defined.

We know that time cannot be negative, therefore, the domain of our given function would be all values of t greater than or equal to 0 that is [tex][0,\infty)[/tex].

Hey!

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

Steps To Solve:

~Subtract n to both sides

3 - n = n + 4 - n

~Simplify

3 - n = 4

~Subtract 3 to both sides

3 - n - 3 = 3 - 4

~Simplify

n = -1

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

Answer:

[tex]\large\boxed{n~=~-1}[/tex]

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

Hope This Helped! Good Luck!

Answer:

The correct option is: (D) The psychology test score is relatively better because its z score is greater than the z score for the economics test score.

Step-by-step explanation:

Consider the provided information.

For psychology test:

Scores on the psychology test have a mean of 86 and a standard deviation of 15.

Use the Z score test as shown:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

Substitute x=73, μ=86 and σ=15 in the above formula.

[tex]z=\frac{73-86}{15}[/tex]

[tex]z=-0.866[/tex]

For economics test:

Scores on the economics test have a mean of 48 and a standard deviation of 7.

Use the Z score test as shown:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

Substitute x=41, μ=48 and σ=7 in the above formula.

[tex]z=\frac{41-48}{7}[/tex]

[tex]z=-1[/tex]

The Z score of psychology test is greater than the Z score of economic test.

Thus, the correct option is: (D) The psychology test score is relatively better because its z score is greater than the z score for the economics test score.

Answer: Our required probability is [tex]\dfrac{11}{20}[/tex]

Step-by-step explanation:

Since we have given that

Number of red marbles = 8

Number of blue marbles = 2

Number of yellow marbles = 7

Number of green marbles = 3

So, Total number of marbles = 8 + 2 + 7 + 3 = 20

So, Probability for selecting a red marble and then selecting a green marbles is given by

P(red) + P(green) is equal to

[tex]\dfrac{8}{20}+\dfrac{3}{20}\\\\=\dfrac{8+3}{20}\\\\=\dfrac{11}{20}[/tex]

Hence, our required probability is [tex]\dfrac{11}{20}[/tex]

Answer:

4/12 or 1/3

Step-by-step explanation:

If you have 12 equal sections of a clock face, and 1/12 or 1 section is done Monday, then 4/12 or 1/3 is done on Tuesday if you exclude Monday's section. It's 4/12 or simplified to be 1/3 because it is only asking you what Miki has painted on Tuesday not Monday and Tuesday combined. How you get 4/12 to be 1/3 is that you take both the top number, (numerator), and the bottom number, (demoninator), and you divide them by the greatest common factor for both. Which is 4, so 4 divided by 4 is 1, and 4 divided into 12 is 3, (3 x 4 = 12), and that's how you get 1/3 for a fraction.

Hope this helps! :)

Miki paints 4 out of 12 sections of the clock face on Tuesday which is 1/3rd of the clock face.

Miki divides the clock face into 12 equal sections and paints 4 sections on Tuesday.

To find the fraction of the clock face painted on Tuesday, we look at the number of sections painted on that day compared to the total number of sections.

Given that Miki paints 4 out of 12 sections, we write this as a fraction:

4 (sections painted on Tuesday) / 12 (total sections)

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

(4 / 4) / (12 / 4) = 1 / 3

Answer:

$ 293.40

Step-by-step explanation:

Given,

The allowance given by the store = $ 10.50,

The original cost of the dress = $ 285.99,

So, the cost of the dress after allowance = 285.99 - 10.50

= $ 275.49,

Now, the sales tax = 6.5 %,

Hence, the final amount of the dress after tax = (100+6.5)% of 275.49

= 106.5% of 275.49

[tex]=\frac{106.5\times 275.49}{100}[/tex]

[tex]=\frac{29339.685}{100}[/tex]

= $ 293.39685

≈ $ 293.40

Final answer:

The customer paid $293.40 for the dress after subtracting the allowance and adding the sales tax, which was calculated and rounded to the nearest penny.

Explanation:

To calculate the final price the customer paid for the dress after receiving an allowance and including sales tax, we follow these steps:

Therefore, the customer paid $293.40 for the dress after the allowance and including the 6.5% sales tax.

Answer:

There is not enough statistical evidence in the sample taken by the veterinarian to support his skepticism

Step-by-step explanation:

To solve this problem, we run a hypothesis test about the population proportion.

Proportion in the null hypothesis [tex]\pi_0 = 0.37[/tex]

Sample size [tex]n = 150[/tex]

Sample proportion [tex]p = 54/150 = 0.36[/tex]

Significance level [tex]\alpha = 0.05[/tex]

[tex]H_0: \pi_0 = 0.36\\H_a: \pi_0<0.36[/tex]

Test statistic [tex] = \frac{(p - \pi_0)\sqrt{n}}{\sqrt{\pi_0(1-\pi_0)}}[/tex]

Left critical Z value (for 0.01) [tex]Z_{\alpha/2}= -1.64485[/tex]

Calculated statistic = [tex]= \frac{(0.36 - 0.37)\sqrt{150}}{\sqrt{0.37(0.63)}} = -0.254[/tex]

[tex]p-value = 0.6003[/tex]

Since, test statistic is greater than critical Z, the null hypothesis cannot be rejected. There is not enough statistical evidence to state that the true proportion of pet owners who talk on the phone with their pets is less than 37%. The p - value is 0.79860.

Answer:

2400 laborers

Step-by-step explanation:

Let N be the amount of laborers employed at the plant. The amount of heaters produced by all workers in an hour is:

heaters = 0.15heaters/hour * N

In a month the total amunt of heaters will be:

heaters = 0.15 * N * 160

Since this has to be 57600 to meet the expected demand:

57600 = 0.15 * N * 160   Solving for N we get:

N = 57600 / (0.15 * 160) = 2400 laborers

The volume of the ark in cubic feet is  697500 feet³

Calculation:-

1 cubit= 6 palm

1 palm=3.10 inches (given in the question)

⇒the volume of a cuboid is =(lenght*breath*height)

lenght=300 cubit

wide=50 cubit

height=30 cubit

volume=300*50*30

            450000 cubit³

since 1 cubit = 6 palm

        450000 cubit = 6*450000 palm

                                  2700000 palm

Again 1 palm = 3.10 inches (given in question)

       ∴ 2700000 palm= 2700000*3.10 inches

                                    =    8370000 inches³

   12 inch = 1 feet

  8370000 inch = 1/12*8370000 feet³

                             = 697500 feet³ (answer)

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The estimated volume of Noah's Ark, assuming a shoe-box shape and given dimensions in cubits with one cubit equaling 1.55 feet, is approximately 1,583,737.5 cubic feet.

To estimate the volume of Noah's Ark using the measurements provided in cubits and converting them to a modern unit like feet, we first need to determine the length of a cubit in inches. As we are given that one cubit is equal to six palms and one palm is 3.10 inches, we can calculate the length of one cubit as follows:

Now, to convert inches to feet, we know that:

Therefore, one cubit in feet is:

Using this conversion, we can calculate the dimensions of the ark in feet:

To find the volume of the ark, we will multiply these dimensions together:

Therefore, the estimated volume of Noah's Ark is approximately 1,583,737.5 cubic feet.

Answer:

[tex]v_1=(\frac{1}{10},-\frac{3}{10})[/tex]

[tex]v_2=(-\frac{1}{10},\frac{3}{10})[/tex]

Step-by-step explanation:

First we define two generic vectors in our [tex]\mathbb{R}^2[/tex] space:

By definition we know that Euclidean norm on an 2-dimensional Euclidean space [tex]\mathbb{R}^2[/tex] is:

[tex]\left \| v \right \|= \sqrt{x^2+y^2}[/tex]

Also we know that the inner product in [tex]\mathbb{R}^2[/tex] space is defined as:

[tex]v_1 \bullet v_2 = (x_1,y_1) \bullet(x_2,y_2)= x_1x_2+y_1y_2[/tex]

So as first condition we have that both two vectors have Euclidian Norm 1, that is:

[tex]\left \| v_1 \right \|= \sqrt{x^2+y^2}=1[/tex]

and

[tex]\left \| v_2 \right \|= \sqrt{x^2+y^2}=1[/tex]

As second condition we have that:

[tex]v_1 \bullet (3,1) = (x_1,y_1) \bullet(3,1)= 3x_1+y_1=0[/tex]

[tex]v_2 \bullet (3,1) = (x_2,y_2) \bullet(3,1)= 3x_2+y_2=0[/tex]

Which is the same:

[tex]y_1=-3x_1\\y_2=-3x_2[/tex]

Replacing the second condition on the first condition we have:

[tex]\sqrt{x_1^2+y_1^2}=1 \\\left | x_1^2+y_1^2 \right |=1 \\\left | x_1^2+(-3x_1)^2 \right |=1 \\\left | x_1^2+9x_1^2 \right |=1 \\\left | 10x_1^2 \right |=1 \\x_1^2= \frac{1}{10}[/tex]

Since [tex]x_1^2= \frac{1}{10}[/tex] we have two posible solutions, [tex]x_1=\frac{1}{10}[/tex] or [tex]x_1=-\frac{1}{10}[/tex]. If we choose [tex]x_1=\frac{1}{10}[/tex], we can choose next the other solution for [tex]x_2[/tex].

Remembering,

[tex]y_1=-3x_1\\y_2=-3x_2[/tex]

The two vectors we are looking for are:

[tex]v_1=(\frac{1}{10},-\frac{3}{10})\\v_2=(-\frac{1}{10},\frac{3}{10})[/tex]

The two vectors in R2 with Euclidean Norm 1 that are orthogonal to (3,1) are (1/√10, -3/√10) and (-1/√10, 3/√10).

To find two vectors in R2 with Euclidean Norm 1 whose Euclidean inner product with (3,1) is zero, we need to look for vectors that are orthogonal to (3,1). The Euclidean inner product of two vectors (x, y) and (3,1) is calculated by (3x + y). To have an inner product of zero, we need 3x + y = 0. Also, we want the vectors to have a Euclidean Norm (or length) of 1, so we need to satisfy the equation x2 + y2 = 1.

Solving these two equations together, we get that y=-3x for orthogonality, and substituting this into the norm equation gives x2 + 9x2 = 1, or 10x2 = 1. This gives two solutions for x, which are x = 1/√10 or x = -1/√10. For y we get correspondingly y = -3/√10 or y = 3/√10.

The two vectors in R2 with Euclidean Norm 1 that are orthogonal to (3,1) are therefore (1/√10, -3/√10) and (-1/√10, 3/√10).

Answer:  A ∪ B = {a, b, c, d}

               (A ∪ B) ∪ C =  = {a, b, c, d, e}

               B ∪ C =  = {a, b, d, e}

               (B ∪ C) ∪ A  =  = {a, b, c, d, e}

Step-by-step explanation:

A = {a, b, c}       B = {a, b, d}         C = {b, d, e}

Union means "to join" so combine the sets to form a union.

A ∪ B = {A & B}

         = {a, b, c & a, b, d}

         = {a, b, c, d}          because we do not need to list a & b twice

(A ∪ B) ∪ C = {(A ∪ B) & C)

                  = {a, b, c, d & b, d, e}

                  = {a, b, c, d, e}     because we do not need to list b & d twice

B ∪ C = {A & B}

         = {a, b, d & b, d, e}

         = {a, b, d, e}           because we do not need to list b & d twice

(B ∪ C) ∪ A = {(B ∪ C) & A)

                  = {a, b, d, e & a, b, c}

                  = {a, b, c, d, e}     because we do not need to list a & b twice

Final answer:

We utilized set theory to find the union of sets A, B, and C in different orders and observed that the associative property of union holds true, meaning the order of union operations does not change the outcome which is {a, b, c, d, e}.

Explanation:

To answer this student's question, we can apply set theory concepts, specifically the concepts of the union and associativity.

Given sets A = {a, b, c}, B = {a, b, d}, and C = {b, d, e}, the union of these sets can be found as follows:

From these operations, we observe that regardless of the order in which we take the union of the three sets, the result is the same. This demonstrates the associative property of union in set theory, where the order in which unions are performed does not affect the final outcome. We can generalize this as A ∪ (B ∪ C) = (A ∪ B) ∪ C = A ∪ B ∪ C.

Answer:

The vertical height, h = 34.64 feets

Step-by-step explanation:

Given that,

Length of the ladder, l = 40 ft

The ladder makes an angle of  60 degrees with the ground, [tex]\theta=60^{\circ}[/tex]

We need to find the vertical height of of which the ladder will reach. Let it iss equal to h. Using trigonometric equation,      

[tex]sin\theta=\dfrac{perpendicular}{hypotenuse}[/tex]

Here, perpendicular is h and hypotenuse is l. So,

[tex]sin(60)=\dfrac{h}{40}[/tex]

[tex]h=sin(60)\times 40[/tex]

h = 34.64 feets

So, the vertical height of which the ladder will reach is 34.64 feets. Hence, this is the required solution.

Answer:

The T-score is 2.49216

Step-by-step explanation:

A 98% confidence interval should be estimated for the end times of cyclists. Since the sample is small, a T-student distribution should be used, in such an estimate. The confidence interval is given by the expression:

[tex][\bar x -T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}, \bar x +T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}][/tex]

[tex]n = 25\\\alpha = 0.02\\T_{(n-1;\frac{\alpha}{2})}= T_{(24;0.01)} = 2.49216[/tex]

Then the T-score is 2.49216

Answer:

2.485

Step-by-step explanation:

Answer:

x = 2100 - 20p

Step-by-step explanation:

Let the quantity demanded be 'x'

unit price be 'p'

thus, from the given relation in the question, we have

p (100) = $100

and,

p (1100) = $50

now, from the standard equation for the line

[tex]\frac{\textup{p - p(100)}}{\textup{x - 100}}[/tex]  = [tex]\frac{\textup{p(1100) - p(100)}}{\textup{50 - 100}}[/tex]

or

[tex]\frac{\textup{p - 100}}{\textup{x - 100}}[/tex]  = [tex]\frac{\textup{50 - 100}}{\textup{1100 - 100}}[/tex]

or

1000 × (p - 100) = - 50 × ( x - 100 )

or

20p - 2000 = - x + 100

or

x = 2100 - 20p

The demand equation for the MP3 players can be determined using the given data points. By setting up a system of linear equations and solving for the values of a and b, we can find the demand equation Qd = 1500 - 10P.

The demand equation can be determined using the given information. We know that when the price is set at $100, the quantity demanded is 100 units, and when the price is $50, the quantity demanded is 1100 units.

We can set up a linear demand equation in the form Qd = a + bP, where Qd is the quantity demanded and P is the unit price. Using the two data points, we can solve for the values of a and b.

Therefore, the demand equation is Qd = 1500 - 10P.

Answer:

In kJ/kg.K - 1.005  kJ/kg degrees Kalvin.

In  J/g.°C  -  1.005 J/g °C

In kcal/ kg °C  0.240 kcal/kg °C

In Btu/lbm-°F   0.240 Btu/lbm degree F

Step-by-step explanation:

given data:

specific heat of air = 1.005 kJ/kg °C

In kJ/kg.K

1.005 kJ./kg °C = 1.005 kJ/kg degrees Kalvin.

In  J/g.°C

[tex]1.005 kJ/kg C \times (1000 J/1 kJ) \times (1kg / 1000 g) = 1.005 J/g °C[/tex]

In kcal/ kg °C

[tex]1.005 kJ/kg C \times (\frac{1 kcal}{4.190 kJ}) = 0.240 kcal/kg C[/tex]  

For   kJ/kg. °C to Btu/lbm-°F  

Need to convert by taking following conversion ,From kJ to Btu, from kg to lbm and from degrees C to F.

[tex]1.005 kJ/kg C \frac{1 Btu}{1.055 kJ} \times \frac{0.453 kg}{1 lbm} \times \frac{(5/9)\ degree C}{ 1\ degree F}  = 0.240 Btu/lbm degree F[/tex]

1.005 kJ/kg C =  0.240 Btu/lbm degree F

Answer: 0.144

Step-by-step explanation:

Formula to find standard deviation: [tex]\sigma=\sqrt{\dfrac{\sum_{i=1}^n(x_i-\overline{x})^2}{n-1}}[/tex]

Given : The growth rate of a random sample of knockout lines:-

0.8, 0.98, 0.72, 1, 0.82, 0.63, 0.63, 0.75, 1.02, 0.97, 0.86

Here ,

[tex]\overline{x}=\dfrac{\sum_{i=1}^{10}x_i}{n}\\\\=\dfrac{0.8+0.98+0.72+1+0.82+0.63+0.63+ 0.75+1.02+ 0.97+ 0.86}{10}\\\\=\dfrac{9.18}{10}\approx0.83[/tex]

[tex]\sum_{i=1}^n(x_i-\overline{x})^2=(-0.03)^2+(0.15)^2+(-0.11)^2+(0.17)^2+(-0.01)^2+(-0.2)^2+(-0.2)^2+(-0.08)^2+(0.19)^2+(0.14)^2+(0.03)^2\\\\=0.2075[/tex]

Now, the standard deviation:

[tex]\sigma=\sqrt{\dfrac{0.2075}{10}}=0.144048602909\approx0.144[/tex]

Hence, the standard deviation of growth rate this sample of yeast lines =0.144

The standard deviation of the growth rates for the given sample of yeast knockout lines is 0.148 when rounded to three decimal places.

To calculate the standard deviation of the growth rates of the yeast knockout lines, we first need to compute the mean (average) of the given data. Then, following the steps, we find the variance by calculating the difference between each value and the mean, squaring those differences, and finding their average. Finally, we take the square root of the variance to find the standard deviation.

Here are the calculations using the given growth rates:

Mean (average) = (0.8 + 0.98 + 0.72 + 1 + 0.82 + 0.63 + 0.63 + 0.75 + 1.02 + 0.97 + 0.86) / 11 = 0.836

Variance = 0.021918

Standard Deviation = sqrt(variance) = sqrt(0.021918) ≈ 0.148

Therefore, the standard deviation of the sample growth rates, rounded to three decimal places, is 0.148.