Which of the following is equivalent to the set {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}?

(you may select more than one)

Question 4 options:

{x | x is one of the seven wonders of the ancient world}

{x | x is one of the six nobel prize categories}

{x | x is the number of humans who can breathe unassisted underwater}

{x | x is a letter of the alphabet

{x | x is one of the two lead singers of Van Halen}

{x | x is a day of the week}

Answer:

The function's graphic is the one that is below the first one in th right.

Step-by-step explanation:

If we want to know how much Chelsea earns for dogsitting her neighbours' dogs, we have to form a linear equation, that must have the following formula:

[tex]y= ax + b[/tex]

In this case, B is the constant value. We know Chelsea charges 12$ for dogsitting, and that doesn't depend on anything (B). But, if you want to Chelsea to walk your dog, then you'd to pay 2.50$ for each walk (X).

[tex]y= 2.50x + 12[/tex]

So, if Chelsea doesn't walk the neighbour's dog (x=0), she would be earning 12 dolars.

[tex]y= 2.50 x 0 + 12 = 12[/tex]

If she walks four times the dog, she would be earning:

[tex]y= 2.50 x 4 + 12 = 10 + 12 = 22 [/tex]

Knowing these two values, we can graph the equation. When x=0, y=12, and when x=4, y=22

The function's graphic is the one that is remarked in the attachment.

Answer:

Minimize C =[tex]13x + 3y[/tex]

[tex]12x + 14y \geq  21[/tex]

[tex]15x + 20y \geq 37[/tex]

and x ≥ 0, y ≥ 0.

Plot the the lines on the graph and find the feasible region

[tex]12x + 14y \geq  21[/tex]  -- Blue

[tex]15x + 20y \geq  37[/tex] --- Green

So, the boundary points of feasible region are (-3.267,4.3) , (0,1.85) and (2.467,0)

Substitute the value in Minimize C

Minimize C =[tex]13x + 3y[/tex]

At (-3.267,4.3)

Minimize C =[tex]13(-3.267) + 3(4.3)[/tex]

Minimize C =[tex]-29.571[/tex]

At (0,1.85)

Minimize C =[tex]13(0) + 3(1.85)[/tex]

Minimize C =[tex]5.55[/tex]

At (2.467,0)

Minimize C =[tex]13(2.467) + 3(0)[/tex]

Minimize C =[tex]32.071[/tex]

So, the optimal value of x is -3.267

So, the optimal value of y is 4.3

So, the minimum value of the objective function is -29.571

Answer:

[tex]\displaystyle\frac{d(y(t))}{dt} =\displaystyle\frac{d(f(u(t)))}{dt} = f'(u(t)).u'(t)[/tex]

Step-by-step explanation:

The chain rule helps us to differentiate functions and a composition of two functions.

Let r(u) and s(u) be two function. Then, composition of these two functions can be be differentiated with the help of chain rule. It states that:

[tex]\displaystyle\frac{d(r(s(u)))}{du} = r'(g(u)).s'(u)[/tex]

Now, we are given

[tex]y(t) = f(u(t))[/tex]

Then, by chain rule, we have:

[tex]\displaystyle\frac{d(y(t))}{dt} =\displaystyle\frac{d(f(u(t)))}{dt} = f'(u(t)).u'(t)[/tex]

In a normally distributed scenario, roughly 99.7% of data falls within three standard deviations from the mean, therefore for this question where we need to determine the blood pressure within 3 standard deviations from the mean, the answer is 99.7%.

The topic under discussion pertains to statistics, particularly, the properties of a normal distribution. In a normally distributed dataset, the rule of three standard deviations states that approximately 99.7% of all data falls within three standard deviations from the mean. This is also known as the empirical rule or the 68-95-99.7 rule. Therefore, in this case, since we are asked about the percentage of 18-year-old women who have a systolic blood pressure that falls within three standard deviations of the mean, the answer is 99.7%, which corresponds to choice (D).

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Answer:

164.6 mg

Step-by-step explanation:

Given:

Weight of the patient= 120 lbs

Height of patient = 5'8" = 5 × 12 + 8 = 68 inches

Dose of Taxol to be administered= 100 mg/ m²

Now,

the surface area of the body of patient = [tex]\textup{(Weight in kg)}^{0.425}\times\textup{(Height in cms)}^{0.725}\times0.007184[/tex]

Also,

weight of patient in kg = 120 × 0.454 = 54.48 kg

Height of patient in cm = 68" × 2.54 = 172.72 cm

therefore,

Body surface area = [tex]\textup{(54.48)}^{0.425}\times\textup{(172.72)}^{0.725}\times0.007184[/tex]

or

= 5.47 × 41.89 × 0.007184

or

= 1.646 m²

Hence,  

Dose of Taxol to be received by the patient

= 100 mg/m²  × surface area of the patient

= 100 × 1.646

= 164.6 mg

Answer:

450 mL/hr

Step-by-step explanation:

Given:

Order for vasopressin = 18 units/hour

Available vasopressin = 200 units in 5000 mL

Now,

Volume of vasopressin per unit =  [tex]\frac{\textup{Volume of vasopressin}}{\textup{Number of units}}[/tex]

or

Volume of vasopressin per unit =  [tex]\frac{\textup{5000}}{\textup{200}}[/tex]

or

Volume of vasopressin per unit = 25 mL/unit

Thus,

Drip rate in mL/hr  

= volume of vasopressin per unit × order for vassopressin

or

Drip rate in mL/hr  = 25 × 18 = 450 mL/hr

The drip rate for an order of Vasopressin 18 units/hr, given a solution concentration of 200 units in 5000 mL, is calculated to be 450 mL/hr.

To find the drip rate in mL/hr, we start by determining the concentration of the vasopressin solution. It is 200 units in 5000 mL D5W, so the concentration is 0.04 units/mL (200 units/5000 mL).

Next, we know the doctor prescribed 18 units/hr of vasopressin. To find out how many mL this corresponds to, we divide the order of 18 units/hr by the concentration in units/mL,  which gives us 450 mL/hr (18 units/hr / 0.04 units/mL).

Therefore, the drip rate for the Vasopressin order is 450 mL/hr.

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Answer: i guess the problem is with P(x) => "x = [tex]x^{2}[/tex]", then P(x) is true if that equality is true, and is false if the equality is false.

so lets see case for case.

a) x = 0, and [tex]0^{2}[/tex] = 0. So p(0) is true.

b) x = 1 and [tex]1^{2}[/tex] = 1, so P(1) is true.

c) x = 2, and [tex]2^{2}[/tex] = 4, and 2 ≠ 4, then P(2) is false.

d) x= -1 and [tex]1^{2}[/tex] = 1, and 1 ≠ -1, so P(-1) is false.

The truth value of P(0) and P(1) is true while the truth value of P(2) and P(-1) is false

The statement is given as:

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

For P(0), we have:

[tex]0 = 0^2[/tex]

[tex]0 = 0[/tex] --- this is true

For P(1), we have:

[tex]1 = 1^2[/tex]

[tex]1 = 1[/tex] --this is true

For P(2), we have:

[tex]2 = 2^2[/tex]

[tex]2= 4[/tex] -- this is false

For P(-1), we have:

[tex](-1) = (-1)^2[/tex]

[tex](-1) = 1[/tex] --- this is false

Hence, the truth value of P(0) and P(1) is true while the truth value of P(2) and P(-1) is false

Read more about truth values at:

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Notice that

[tex](n+1)^4-n^4=4n^3+6n^2+4n+1[/tex]

so that

[tex]\displaystyle\sum_{i=1}^n((n+1)^4-n^4)=\sum_{i=1}^n(4i^3+6i^2+4i+1)[/tex]

We have

[tex]\displaystyle\sum_{i=1}^n((i+1)^4-i^4)=(2^4-1^4)+(3^4-2^4)+(4^4-3^4)+\cdots+((n+1)^4-n^4)[/tex]

[tex]\implies\displaystyle\sum_{i=1}^n((i+1)^4-i^4)=(n+1)^4-1[/tex]

so that

[tex]\displaystyle(n+1)^4-1=\sum_{i=1}^n(4i^3+6i^2+4i+1)[/tex]

You might already know that

[tex]\displaystyle\sum_{i=1}^n1=n[/tex]

[tex]\displaystyle\sum_{i=1}^ni=\frac{n(n+1)}2[/tex]

[tex]\displaystyle\sum_{i=1}^ni^2=\frac{n(n+1)(2n+1)}6[/tex]

so from these formulas we get

[tex]\displaystyle(n+1)^4-1=4\sum_{i=1}^ni^3+n(n+1)(2n+1)+2n(n+1)+n[/tex]

[tex]\implies\displaystyle\sum_{i=1}^ni^3=\frac{(n+1)^4-1-n(n+1)(2n+1)-2n(n+1)-n}4[/tex]

[tex]\implies\boxed{\displaystyle\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}4}[/tex]

If you don't know the formulas mentioned above:

Answer:

[tex]\frac{6\cdot 8 \cdot 10 \cdot 12}{13 \cdot 15 \cdot 17\cdot 19}\approx 0.091 [/tex]

Step-by-step explanation:

To compute the probability of not including any matching pairs, we can compute the number of ways in which he could pick 8 gloves with no matching pairs, and divide it by the total number of ways in which he could pick 8 gloves.

The process of choosing 8 gloves with no matching pair can be seen as follows:

He first picks any random gloves out of the 20, in this 1st step he has then 20 available choices. Then he needs to pick some other glove, BUT it cannot be the other glove from the pair he already picked one from. So at this step there aren't 19 choices, but 18 available choices. Now he has 2 gloves from different pairs, and needs to pick another glove. There are 18 gloves left, BUT he cannot pick the remaining glove from any of the 2 pairs he already has chosen one from. Therefore he only has 16 choices left. The process continues like this, until he chooses 8 gloves in total. Hence the total number of ways to choose 8 gloves with no matching pair is

[tex] 20 \cdot 18 \cdot 16 \cdot 14 \cdot 12 \cdot 10 \cdot 8 \cdot 6[/tex]

Now, the total numer of ways in which he could pick any 8 gloves out of 20 can be seen as follows: At the start he has 20 available gloves, he chooses any of them, so having 20 available choices on that first step. He then needs to choose any other glove, so he has 19 choices. He then picks any other glove, so he has 18 choices. And so on, until he has chosen the 8 gloves. Hence the total number of ways to choose 8 gloves is

[tex] 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 [/tex]

Therefore the probability of choosing 8 gloves with no matching pair is

[tex]\frac{20 \cdot 18 \cdot 16 \cdot 14 \cdot 12 \cdot10 \cdot 8 \cdot 6}{20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13}=\frac{6\cdot 8 \cdot 10 \cdot 12}{13 \cdot 15 \cdot 17\cdot 19}\approx 0.091 [/tex]

Answer:

The solution of this system is x=9/4, y=5/2, and z=-13/8

Step-by-step explanation:

1. Writing the equations in matrix form

The system of linear equations given can be written in matrix form as

[tex]\left[\begin{array}{ccc}1&0&2\\2&-1&0\\0&3&4\end{array}\right]\left[\begin{array}{c}x&y&z\end{array}\right] = \left[\begin{array}{c}-1&2&1\end{array}\right][/tex]

Writing

A = [tex]\left[\begin{array}{ccc}1&0&2\\2&-1&0\\0&3&4\end{array}\right][/tex]

X = [tex]\left[\begin{array}{c}x&y&z\end{array}\right][/tex]

B = [tex]\left[\begin{array}{c}-1&2&1\end{array}\right][/tex]

we have

AX=B

This is the matrix form of the simultaneous equations.

2. Solving the simultaneous equations

Given

AX=B

we can multiply both sides by the inverse of A

[tex]A^{-1}AX=A^{-1}B[/tex]

We know that [tex]A^{-1}A=I[/tex], the identity matrix, so

[tex]X=A^{-1}B[/tex]

All we need to do is calculate the inverse of the matrix of coefficients, and finally perform matrix multiplication.

3. Calculate the inverse of the matrix of coefficients

A = [tex]\left[\begin{array}{ccc}1&0&2\\2&-1&0\\0&3&4\end{array}\right][/tex]

To find the inverse matrix, augment it with the identity matrix and perform row operations trying to make the identity matrix to the left. Then to the right will be inverse matrix.

[tex]\left[\begin{array}{ccc|ccc}1&0&2&1&0&0\\2&-1&0&0&1&0\\0&3&4&0&0&1\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|ccc}1&0&2&1&0&0\\0&-1&-4&-2&1&0\\0&3&4&0&0&1\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|ccc}1&0&2&1&0&0\\0&-1&-4&-2&1&0\\0&0&-8&-6&3&1\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|ccc}1&0&2&1&0&0\\0&1&4&2&-1&0\\0&0&-8&-6&3&1\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|ccc}1&0&2&1&0&0\\0&1&4&2&-1&0\\0&0&1&3/4&-3/8&-1/8\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|ccc}1&0&0&-1/2&3/4&1/4\\0&1&4&2&-1&0\\0&0&1&3/4&-3/8&-1/8\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|ccc}1&0&0&-1/2&3/4&1/4\\0&1&0&-1&1/2&1/2\\0&0&1&3/4&-3/8&-1/8\end{array}\right][/tex]

As can be seen, we have obtained the identity matrix to the left. So, we are done.

[tex]A^{-1} = \left[\begin{array}{ccc}-1/2&3/4&1/4\\-1&1/2&1/2\\3/4&-3/8&-1/8\end{array}\right][/tex]

4. Find the solution [tex]X=A^{-1}B[/tex]

[tex]X= \left[\begin{array}{ccc}-1/2&3/4&1/4\\-1&1/2&1/2\\3/4&-3/8&-1/8\end{array}\right]\cdot \left[\begin{array}{c}-1&2&1\end{array}\right] = \left[\begin{array}{c}9/4&5/2&-13/8\end{array}\right][/tex]

Answer: 686 N

Step-by-step explanation:

Hi!

Second Newton's law is: F=m*a, where F is force, m is mass, and a acceleration

On the Earth's surface, weight is the gravity force W=m*g, where g=9.8 m/s² is the acceleretion of gravity on Earth. So the weight of someone with mass m=70 kg is W=70*9.8 kg*m/s² = 686 N.

The unit N (Newton) is defined as 1 N = 1 kg*m/s²

Answer:

[tex]300\cdot \$1.50+\$125=\$450+\$125=\$575[/tex]

Step-by-step explanation:

The marginal cost is said to be $1.50. Marginal cost is just how much the cost increases per additional unit produced. In this case we're dealing with a shop of slices of pizza, so the marginal cost just represents how much the cost increases per additional slice of pizza produced, or in simpler words, how much it costs to produce a slice of pizza.

We want to compute the total cost to sell 300 slices in one day, so we have to compute how much it costs to produce those 300 slices and add up the fixed costs (which is $125, no matter how many slices we produce). Since the marginal cost is $1.50, that means each slice costs $1.50 to produce. So the 300 slices cost [tex]300\cdot \$1.50=\$450[/tex] to produce. And so the total cost is

[tex]\$450+\$125=\$575[/tex]

Answer:

Yusuf got 19 toffees

Step-by-step explanation:

The problem tells us that if Yusuf counts his toffees in 4's, he has 3 left. Let's call the number of toffees "t", then we know that [tex]t[/tex]≡[tex]3(mod4)[/tex]

We also know that [tex]t[/tex]≡[tex]4(mod 5)[/tex]

Between 10 and 20, there are only 2 numbers t≡[tex]4(mod 5)[/tex], these numbers are 14 and 19.

However, 14≡[tex]2(mod 4)[/tex] so it doesn't add up.

On the other hand, 19 ≡3[tex](mod4)[/tex] and therefore, this is the answer.

Answer:

19 toffees.

Step-by-step explanation:

It is given that Yusuf has between 10 and 20 toffees.

Let n be the number of toffees.

If he counts his toffees in 4s, he has 3 left over.

[tex]n\equiv 3(mod4)[/tex]      ... (1)

So, the possible values of n between 10 and 20 are 11, 15 and 19.

If he counts his toffees in 5s, he has 4 left over.

[tex]n\equiv 4(mod5)[/tex]       ... (2)

So, the possible values of n between 10 and 20 are 14 and 19.

From (1) and (2) the possible value of n between 10 and 20 is 19.

Therefore, Yusuf has 19 toffees.

The initial position has no effect on the velocity, so you can ignore that value (unless there's another part to the question not included, of course).

We have

[tex]v(t)=v(0)+\displaystyle\int_0^t a(u)\,\mathrm du[/tex]

[tex]v(t)=55+\int_0^t\cos(\pi u)\,\mathrm du[/tex]

[tex]\boxed{v(t)=55+\dfrac1\pi\sin(\pi t)}[/tex]

The velocity of an object given the acceleration function a(t) = cos(πt) and an initial velocity of v(0) = 55 is found by integrating the acceleration function. This gives v(t) = (1/π)sin(πt) + 55.

The object's acceleration, velocity, and position can be determined using principles of calculus. The acceleration function is given as a(t) = cos(πt). We find the velocity by integrating the acceleration function. Therefore, v(t) = ∫a(t) dt = ∫cos(πt) dt. Using fundamental calculus principles, the integral of cos(πt) with respect to time (t) is (1/π)sin(πt).

However, the initial velocity is provided as v(0) = 55. To account for this initial condition, we add this known velocity to our integral, giving us v(t) = (1/π)sin(πt) + 55. Thus, the velocity of the object at any time t is given by v(t) = (1/π)sin(πt) + 55.

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Here are some possible answers

Answer:

The correct option is A) [tex]3d+C[/tex].

Step-by-step explanation:

Consider the provided information.

Paco bought 3 CDs that cost d dollars each.

Let d is the cost of each CDs.

The cost of 3 CDs will be 3 times of d.

This can be written as:

[tex]3d[/tex]

He bought a pack of gum for C cent. Thus, we can say that the cost of pack of gum is C.

Now add the cost of gum in above expression.

[tex]3d+C[/tex]

Hence, the required expression is [tex]3d+C[/tex].

Thus, the correct option is A) [tex]3d+C[/tex].

Answer:

D = 170.6Km

Step-by-step explanation:

First of all, we set the reference (origin) at Grand Bahama.

Nw, from the first displacement of 3h we calculate the distance:

D1 = V1*t = 43.5 * 3 = 130.5 Km

The coordinates of this new location is given by:

r1 = ( -D1*cos(60°); D1*sin(60°)) = (-62.5; 158.835) Km

For the second displacement, the duration was of 1.95 hours (4.95 -3), so the distance traveled was:

D2 = V2*t = 23.5 * 1.95 = 45.825 Km

The coordinates of this new location is given by:

r2 = r1 + ( 0; D2) = (-62.25; 158.835) Km

Now we just need to calculate the magnitude of that vector to know the distance to Grand Bahama:

[tex]Dt = \sqrt{D_{2x}^{2}+D_{2y}^{2}}=170.6 Km[/tex]

Answer: The dimensions of rectangle are 32 feet and 16 feet.

Step-by-step explanation:

Let the width of rectangle be 'x'.

Let the length of rectangle be '2x'.

Perimeter of fence = 96 feet

As we know the formula for "Perimeter":

[tex]Perimeter=2(l+b)\\\\96=2(2x+x)\\\\\dfrac{96}{2}=3x\\\\48=3x\\\\x=\dfrac{48}{3}\\\\x=16\ ft[/tex]

Hence, the length of rectangle is 2x=2×16 = 32 feet and width is 16 feet.

Therefore, the dimensions of rectangle are 32 feet and 16 feet.

Answer:

The value of given expression is [tex]\frac{21}{128}[/tex].

Step-by-step explanation:

Given information: n=7, x=2, p=1/2

[tex]q=1-p=1-\frac{1}{2}=\frac{1}{2}[/tex]

The given expression is

[tex]C(n,x)p^xq^{n-x}[/tex]

It can be written as

[tex]^nC_xp^xq^{n-x}[/tex]

Substitute n=7, x=2, p=1/2 and q=1/2 in the above formula.

[tex]^7C_2(\frac{1}{2})^2(\frac{1}{2})^{7-2}[/tex]

[tex]\frac{7!}{2!(7-2)!}(\frac{1}{2})^2(\frac{1}{2})^{5}[/tex]

[tex]\frac{7!}{2!5!}(\frac{1}{2})^{2+5}[/tex]

[tex]\frac{7\times 6\times 5!}{2\times 5!}(\frac{1}{2})^{2+5}[/tex]

[tex]21(\frac{1}{2})^{7}[/tex]

[tex]\frac{21}{128}[/tex]

Therefore the value of given expression is [tex]\frac{21}{128}[/tex].

Answer:

  It is a multiple of 8

Step-by-step explanation:

The product may or may not be a factor of 8. We usually think of the factors of an integer as being positive integers, so the factors of 8 would be 1, 2, 4, or 8. If 8n is to be a factor of 8, then n must be 1/8, 1/4, 1/2, or 1. This will not be the case in general.

__

The product of 8 and any number is a multiple of 8. (Again, we usually think of a multiple of 8 as being an integer, which would require the number n to be an integer.)

__

No product of two (integer) numbers is a prime number. If 8n is to be a prime, then the value of n must be (some prime number)/8. Again, this will not be the case in general.

__

n is a factor of the product; not the other way around.

In the equation n x 8, the product is always a multiple of 8. The product is not necessarily a factor of 8, a prime number, or a factor of n.

In the given equation, n x 8, the product of n and 8 is always a multiple of 8. This is because when we multiply any number by 8, the resulting product is included in the sequence of multiples of 8 (i.e., 8, 16, 24, 32, and so forth). Hence, regardless of the value of n, the product is always a multiple of 8. Note that the product is not necessarily a factor of 8, a prime number, or a factor of n, as these properties depend on the specific value of n.

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By considering the given information, we have

Null hypothesis : [tex]H_0: \mu=35.0[/tex]

Alternative hypothesis : [tex]H_1: \mu\neq35.0[/tex]

Since the alternative hypothesis is two-tailed , so the test is a two-tailed test.

Given : Sample size : n= 20, since sample size is less than 30 so the test applied is a t-test.

[tex]\overline{x}=41.0[/tex]    ;    [tex]\sigma= 3.7[/tex]

Test statistic : [tex]t=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

i.e. [tex]t=\dfrac{41.0-35.0}{\dfrac{3.7}{\sqrt{20}}}=7.252112359\approx7.25[/tex]

Degree of freedom : n-1 = 20-1=19

Significance level = 0.01

For two tailed,  Significance level [tex]=\dfrac{0.01}{2}=0.005[/tex]

By using the t-distribution table, the critical value of t =[tex]t_{19, 0.005}=2.861[/tex]

Since , the observed t-value (7.25) is greater than the critical value (2.861) .

So we reject the null hypothesis, it means we have enough evidence to support the alternative hypothesis.

We conclude that there is some significance difference between the mean score for sober women and 35.0.

Answer:

1.176 grams

Step-by-step explanation:

Given:

Recommended dose

21 mg per day for 6 weeks

Now,

1 week = 7 days

Thus,

number of days in 6 weeks = 6 × 7 = 42 days

Therefore, the total dose = dose per days × number of days

= 21 × 42 = 882 mg

further,

14 mg per day for 2 weeks

Now,

1 week = 7 days

Thus,

number of days in 2 weeks = 2 × 7 = 14 days

Therefore, the total dose = dose per days × number of days

= 14 × 14 = 196 mg

further,

7 mg per day for 2 weeks

Now,

1 week = 7 days

Thus,

number of days in 6 weeks = 2 × 7 = 14 days

Therefore, the total dose = dose per days × number of days

= 7 × 14 = 98 mg

Hence, the total dose = 882 + 196 + 98 = 1176 mg

also,

1 g = 1000 mg

thus,

1176 mg = 1.176 grams

total quantity received during this course is 1.176 grams

Answer:

0.84

Step-by-step explanation:

Given that Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows:

P(A) = 0.86, P(B) = 0.79, P(A') = 0.14, P(AUB) = 0.95

We are to find out P(A'UB)

We have

[tex]P(AUB) =P(A)+P(B)-P(A\bigcap B)\\0.95=0.86+0.79-P(A\bigcap B)\\P(A\bigcap B)=0.70[/tex]

[tex]P(A'UB) = P(A')+P(B)-P(A' \bigcap B)\\= 1-P(A) +P(B)-[P(B)-P(A \bigcap B)]\\= 1-0.86+0.79-P(B)+[tex]P(A'UB)=0.14+0.79-0.79+0.70\\=0.84[/tex]P(A \bigcap B)[/tex]

Answer:

1.[tex]\frac{dy}{dt}=ky[/tex]

2.543.6

Step-by-step explanation:

We are given that

y(0)=200

Let y be the number of bacteria at any time

[tex]\frac{dy}{dt}[/tex]=Number of bacteria per unit time

[tex]\frac{dy}{dt}\proportional y[/tex]

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

Where k=Proportionality constant

2.[tex]\frac{dy}{y}=kdt[/tex],y'(0)=100

Integrating on both sides then, we get

[tex]lny=kt+C[/tex]

We have y(0)=200

Substitute the values then , we get

[tex]ln 200=k(0)+C[/tex]

[tex]C=ln 200[/tex]

Substitute the value of C then we get

[tex]ln y=kt+ln 200[/tex]

[tex]ln y-ln200=kt[/tex]

[tex]ln\frac{y}{200}=kt[/tex]

[tex]\frac{y}{200}=e^{kt}[/tex]

[tex]y=200e^{kt}[/tex]

Differentiate w.r.t

[tex]y'=200ke^{kt}[/tex]

Substitute the given condition then, we get

[tex]100=200ke^{0}=200 \;because \;e^0=1[/tex]

[tex]k=\frac{100}{200}=\frac{1}{2}[/tex]

[tex]y=200e^{\frac{t}{2}}[/tex]

Substitute t=2

Then, we get [tex]y=200e^{\frac{2}{2}}=200e[/tex]

[tex]y=200(2.718)=543.6=543.6[/tex]

e=2.718

Hence, the number of bacteria after 2 hours=543.6

Answer:

The initial population was approximatedly 3535 inhabitants.

Step-by-step explanation:

The population of the city can be given by the following differential equation.

[tex]\frac{dP}{dt} = Pr[/tex],

In which r is the rate of growth of the population.

We can solve this diffential equation by the variable separation method.

[tex]\frac{dP}{dt} = Pr[/tex]

[tex]\frac{dP}{P} = r dt[/tex]

Integrating both sides:

[tex]ln P = rt + c[/tex]

Since ln and the exponential are inverse operations, to write P in function of t, we apply ln to both sides.

[tex]e^{ln P} = e^{rt + C}[/tex]

[tex]P(t) = Ce^{rt}[/tex]

C is the initial population, so:

[tex]P(t) = P(0)e^{rt}[/tex]

Now, we apply the problem's statements to first find the growth rate and then the initial population.

The problem states that:

In two years the population has doubled:

[tex]P(2) = 2P(0)[/tex]

[tex]P(t) = P(0)e^{rt}[/tex]

[tex]2P(0) = P(0)e^{2r}[/tex]

[tex]2 = e^{2r}[/tex]

To isolate r, we apply ln both sides

[tex]e^{2r} = 2[/tex]

[tex]ln e^{2r} = ln 2[/tex]

[tex]2r = 0.69[/tex]

[tex]r = \frac{0.69}{2}[/tex]

[tex]r = 0.3466[/tex]

So

[tex]P(t) = P(0)e^{0.3466t}[/tex]

In two years the population has doubled and a year later there were 10,000 inhabitants.

[tex]P(3) = 10,000[/tex]

[tex]P(t) = P(0)e^{0.3466t}[/tex]

[tex]10,000= P(0)e^{0.3466*3}[/tex]

[tex]P(0) = \frac{10,000}{e^{1.04}}[/tex]

[tex]P(0) = 3534.55[/tex]

The initial population was approximatedly 3535 inhabitants.

Answer:

Actual length = 883 feet

Actual height = 175 feet

A)  If a model is built to have a scale ratio of 1 in : 36ft , how long will the model be?

36 feet = 1 inch

Actual length = 883 feet

So, 883 feet = [tex]\frac{883}{36} inch[/tex]

883 feet = [tex]24.527 inch[/tex]

So, If a model is built to have a scale ratio of 1in : 36ft ,the model will be 24.527 inch long.

B)  If a model is built to have a scale ratio of 1 in : 22 ft , how tall will the model be?

22 feet = 1 inch

Actual height = 175 feet

So, 175 feet = [tex]\frac{175}{22} inch[/tex]

883 feet = [tex]7.9545 inch[/tex]

So,  If a model is built to have a scale ratio of 1 in : 22 ft , the model will be 7.9545 inch tall.

C) If a model is built to have a ratio of 1: 30 , how tall will the model be?

Let the height be x

Actual length = 883 feet

Ratio of 1: 30

So, [tex]\frac{1}{30}=\frac{883}{x}[/tex]

[tex]x=\frac{883}{30}[/tex]

[tex]x=29.433[/tex]

So, If a model is built to have a ratio of 1: 30 ,the model will be 29.433 inches tall .

Answer:

The gcd(474,147) = 3 and the linear combination is [tex]3=9\cdot 474 - 29\cdot 147[/tex] and the proof is below.

Step-by-step explanation:

The greatest common divisor (GCD) of two whole numbers is the largest natural number that divides evenly into both without a remainder.

To find the GCD you can use the Euclidean algorithm which is an efficient method for computing the greatest common divisor (GCD) of two integers, without explicitly factoring the two integers. Here is an outline of the steps:

To compute gcd(474,147), divide the larger number by the smaller number, using the division algorithm we have

[tex]\frac{474}{147} \\= 474-147=327\\327-147=180\\180-147=33\\[/tex]

At this point, we cannot subtract 147 again. Hence 3 is the quotient ( we subtract 147 from 474 3 times) and 33 is the remainder. We can express this as a linear combination [tex]474 = 147*3+33[/tex]

Using the same reasoning and the steps of the Euclidean algorithm we have

[tex]gcd(474,147) = \\474 =147\cdot 3+33\\147=33\cdot 4 +15\\33=15\cdot 2+3\\15=3\cdot 5+0[/tex]

To find the linear combination you need to use the Bezout's identity that says that the equation [tex]ax+by=gcd(a,b)[/tex] has solutions x, y. So we need to find the solution of the equation [tex]474x+147y=3[/tex].

To find the values of x and y you can run the Euclidean Algorithm backward.

We know that

[tex]33=15\cdot 2+3[/tex]

We can express 3 as linear combination

[tex]3=33- 2\cdot15\\3=33-2\cdot(147-33*4)=9\cdot 33 -2\cdot147\\3=9\cdot 33 -2\cdot147=9\cdot (474-147\cdot 3)-2 \cdot 147\\3= 9\cdot 474-27 \cdot 147-2 \cdot 147\\3=9\cdot 474 - 29\cdot 147[/tex]

The gcd(474,147) = 3 and the linear combination is [tex]3=9\cdot 474 - 29\cdot 147[/tex]

The principle of mathematical induction is stated as follows:

Let n be a natural number and let P(n) be an statement that depends on n. If

Then P(n) is true for all natural numbers n.

Proposition: For all positive integers n, 2+4+6+...+2n = n(n+1).

Proof. Let's let P(n) be the statement "2+4+6+...+2n = n(n+1)" .The proof will now proceed in two steps: the initial step and the inductive step.

Initial step. We must verify that P(1) is true

[tex]n=1\\2\cdot 1=1\cdot (1+1)[/tex]

which is clearly true. So we are done with the initial step.

Inductive step. We must prove the following assertion: "If there is a k such that P(k) is true, then (for this same k) P(k+1) is true." Thus, we assume there is a k such that 2+4+6+...+2k = k(k+1), this is called the inductive assumption. We must prove, for this same k, the formula P(k+1): 2+4+6+...+2k+2(k+1) = (k+1)(k+2)

To prove that P(k+1) holds, we will start  with the expression on the left-hand side of P(k+1) and show that it is equal to the expression on the right-hand side.

[tex]2+4+6+...+2k+2(k+1)[/tex]

we know that [tex]2+4+6+...+2k+2(k+1)=k(k+1)[/tex] for the inductive assumption

[tex]k(k+1)+2(k+1)\\k^{2}+k+2k+2\\k^2+3k+2\\(k+1)(k+2)[/tex]

we see that the result [tex](k+1)(k+2)[/tex], is the expression on the right-hand side of P(k+1). Thus by mathematical induction P(n) is true for all natural numbers n.

Answer:

1 member took literature but not mathematics.

Step-by-step explanation:

We can draw a Venn diagram for the given question.

In a fraternity total number of members = 32

Number of members who took mathematics M = 18

Number of members who took both mathematics and literature (M∩L) = 5

And number of members who took neither mathematics nor literature = 8

Therefore, number of members who took literature but not mathematics

= 32 - [(18 + 5) + 8]

= 32 - [23 + 8]

= 32 - 31

= 1

Therefore, 1 member took literature but not mathematics.

Answer:

1 member took literature but not mathematics.

Step-by-step explanation:

The number of letters in word "ALGORITHM" = 9

The number of combinations to select r things from n things is given by :-

[tex]C(n,r)=\dfrac{n!}{r!(n-r)!}[/tex]

Now, the number of combinations to select 6 letters from 9 letters will be :-

[tex]C(9,8)=\dfrac{9!}{6!(9-6)!}=\dfrac{9\times8\times7\times6!}{6!\times3!}=84[/tex]

Thus , the number of ways can six of the letters of the word ALGORITHM=84

The number of ways to arrange n things in a row :[tex]n![/tex]

So, the number of ways can the letters of the word ALGORITHM be arranged in a be seated together in the row :-

[tex]9!=362880[/tex]

If GOR comes together, then we consider it as one letter,  then the total number of letters will be = 1+6=7

Number of ways to arrange 9 letters if "GOR" comes together :-

[tex]7!=5040[/tex]

Thus, the number of ways to arrange 9 letters if "GOR" comes together=5040

Answer:  30 people

Step-by-step explanation: So you start with 12 people

12+6= 18 because 6 more people moved in then an additional 12 move in because it doubles every month

The total number of people occupying the development on 31 March 2017 is 466.

To determine how many people occupy the development on 31 March 2017, we need to consider the pattern of people moving in each month since January 2016.

Initial Occupants: There were 12 people in the development as of January 2016.

Additional Occupants in January 2016: During January 2016, 6 more people moved in.

Monthly Pattern: Each subsequent month, 4 more people move in than the previous month. So, we need to figure out this sequence from February 2016 to March 2017.

Calculating Monthly Increase:

Sum of Monthly Increases: We need to determine the total number of new occupants from February 2016 to March 2017. This period includes 14 months.

The sequence of increases is: 6, 10, 14, 18, ..., up to March 2017.

This is an arithmetic sequence where the first term [tex]a = 6[/tex] and the common difference [tex]d = 4[/tex].

The [tex]n^{th}[/tex] term of an arithmetic sequence is given by: [tex]a_n = a + (n-1)d[/tex]

For March 2017 (14 months after January 2016): $a_{14} = 6 + (14-1) * 4 = 6 + 52 = 58 people.

Sum of an Arithmetic Sequence: The sum of the first [tex]n[/tex] terms of an arithmetic sequence is given by: [tex]S_n = \{n}{2} \times (2a + (n-1)d)[/tex]

Here, [tex]a = 6[/tex], [tex]d = 4[/tex], and [tex]n = 14[/tex].

[tex]S_{14} = \{14}{2} \times(2 \times 6 + (14-1) \times 4)[/tex]

[tex]S_{14} = 7 \times (12 + 52) = 7 \times 64 = 448[/tex] people moved in from February 2016 to March 2017.

Total Occupants at the End of March 2017:

Initial occupants: 12

Additional in January 2016: 6

New occupants from February 2016 to March 2017: 448

Total = 12 + 6 + 448 = 466 people