HELP!
Will give brainliest to whoever does answer this correctly!
As infants grow from a toddler to a young adult, there may be times when they are ill and medication is needed. It is extremely important that medication be administered in the exact dose so the child receives the correct amount. Too little or too much medication could have serious side effects. A popular children’s fever medicine manufacturer recommends the following dosage information to parents and pediatricians.

Answer:

a)Reflexive, not symmetric, transitive

b)Reflexive, not symmetric, transitive

c)Not reflexive, symmetric, not transitive

d)Reflexive, not symmetric, transitive

Step-by-step explanation:

a)

[tex]R=\left \{ (a,b)\epsilon  \mathbb{R} \times \mathbb{R} \mid a \leq b\right \}[/tex]

The relation R is reflexive for

[tex]a\leq a[/tex] for every real number a

it is not symmetric because 0 is less than 1, but 1 is not less than 0

it is transitive

[tex]a\leq[/tex] and [tex] b\leq c\Rightarrow a\leq c[/tex]

So if aRb and bRc, then aRc

b)  

[tex]R=\left \{ (m,n)\epsilon  \mathbb{Z} \times \mathbb{Z} \mid \exists k\in \mathbb{Z} \ni m=kn \right \}[/tex]

R is reflexive because m=1.m for every integer m

R is not symmetric: 2 is a factor of 4, but 4 is not a factor of 2

R is transitive:  if mRn and nRp if m=kn and n=qp, so m=(kq)p and kq is an integer , so mRp

c)

[tex]R=\left \{ (m,n)\epsilon  \mathbb{Z} \times \mathbb{Z} \mid m\neq n\right \}[/tex]

R is obviously not reflexive because all numbers equals themselves

R is symmetric: if a different to b, then b different to a

R is not transitive: 1R2 and 2R1 (because 1 different to 2), but 1 = 1

d)

[tex]R=\left \{ A,B\mid A\subseteq B \right \}[/tex]

R is reflexive for every set A is a subset of itself

R is not symmetric {1,2} is a subset of {1,2,3} but {1,2,3} is not a subset of {1,2}

R is transitive: if A is subset of B and B is subset of C, then A is subset of C

Answer:

a) The z-score for the mileage of the car is -3.16

b) It appears that the car is getting unusually low gas mileage.

Step-by-step explanation:

The z-score formula is given by:

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

In which: X is the mileage per gallon we are going to find the z-score of, [tex]\mu[/tex] is the mean value of this mileage and [tex]\sigma[/tex] is the standard deviation of this value.

a. Find the​ z-score for the gas mileage of your​ car, assuming the advertised claim is correct.

The gas mileage for you car is 16.4 mpg, so [tex]X = 16.4[/tex]

The advertised gas mileage is 20 mpg, so [tex]\mu = 20[/tex]

The standard deviation is 1.14 mpg, so [tex]\sigma = 1.14[/tex]

The z-score is:

[tex]Z = \frac{X - \mu}{\sigma} = \frac{16.4 - 20}{1.14} = -3.16[/tex]

b. Does it appear that your car is getting unusually low gas​ mileage?

The general rule is that a z-score lower than -1.96 is unusually low. So yes, it appears that the car is getting unusually low gas mileage.

To find the z-score for the gas mileage of your car, use the formula z = (x - μ) / σ. A z-score of -3.16 indicates that your car is getting unusually low gas mileage as it is more than 3 standard deviations below the mean.

To find the z-score for the gas mileage of your car, we can use the formula:

z = (x - μ) / σ

where x is the observed mileage, μ is the mean mileage, and σ is the standard deviation.

In this case, since the advertised mileage is 20 mpg, we have:

z = (16.4 - 20) / 1.14 = -3.16

For part b, a z-score of -3.16 indicates that your car is getting unusually low gas mileage as it is more than 3 standard deviations below the mean. Therefore, the answer is Yes.

Answer:

1/2.

Step-by-step explanation:

There are 2 choices and he has to choose 1 , so the answer is 1/2.

Assuming Ciaran has no preference for white or brown bread, and each choice is equally likely, the probability of choosing brown bread is 1/2 or 50%.

The probability that Ciaran chooses brown bread depends on the assumption that he has no preference and that the choices are equally likely. If the only options available to Ciaran are white bread or brown bread, and each choice is equally likely, then the probability of choosing one over the other is 1 out of the total number of options.

In this case, there are 2 options (white or brown), so the probability that Ciaran will choose brown bread is 1/2 or 0.5, which can also be expressed as a 50% chance.

Answer:

There are two more sides on a hexagon than on a quadrilateral

Step-by-step explanation:

If the hexagon has 6 sides, and the quadrilateral has 4, then 6-4=2

Answer:

[tex]S_{n} = \sum_{k=1}^{n} (2k-1) = n^2[/tex]

Step-by-step explanation:

Let's take a look at the first few odd numbers and their sum.

Lets define [tex]O_{k}[/tex] as the [tex]kth[/tex] odd number as:

[tex]O_{k} = 2k-1[/tex]

So we have:

[tex]O_{1} = 1\\O_{2} = 3\\O_{3} = 5\\O_{4} = 7\\[/tex]

And lets define the sum of all the odd numbers from [tex]k=1[/tex] to [tex]k=n[/tex] as:

[tex]S_{n} = \sum_{k=1}^n O_{k} = \sum_{k=1}^n (2k-1)[/tex]

Lets now check some values of said sum:

[tex]S_{1} = 1\\S_{2} = 1 + 3 = 4\\S_{3} = 1 + 3 + 5 = 9\\S_{4} = 1+3+5+7 = 16[/tex]

We can then observe than the sum up to [tex]n[/tex] equals [tex]n^2[/tex]

Let us then prove that this is the case by Induction.

First of all, we can prove this by an Induction Proof because we are taking all positive Integers. This is, we are working with the set of natural numbers [tex]\mathbb{N}[/tex].

We want to prove that

[tex]P(n) = S_{n} = \sum_{k=1}^n = n^2 \forall n\in \mathbb{N}[/tex]

This is, we want to prove that the sum of all odd numbers from [tex]1[/tex] to [tex]n[/tex] equals [tex]n^2[/tex] for all natural numbers.

Now, in order to prove something by Induction we need to check 2 things:

[tex]1) The\ base\ case . \ The\ statement\ holds\ for\ n=1\\2) The\ inductive\ step.\ Prove\ that\ if\ the\ statement\ holds\ for\ n\ then\ it\ must\ hold\ for\ n+1\\[/tex]

[tex]P(1)[/tex] is immediate:

[tex]P(1) = \sum_{k=1}^1 2k-1 = 1 = 1^2[/tex]

Now let's assume the statement holds for [tex]P(n)[/tex] and let's take a look at [tex]P(n+1)[/tex]

[tex]P(n+1) = \sum_{k=1}^{n+1} 2k-1[/tex]

And we can rewrite it by taking the last term out as:

[tex]P(n+1) = \sum_{k=1}^n 2k-1 \ + 2.(n+1) - 1[/tex]

And by inductive hypothesis we know that [tex]\sum_{k=1}^n 2k-1 = n^2[/tex]

and then:

[tex]P(n+1) = \sum_{k=1}^n 2k-1 \ + 2.(n+1) -1  = n^2 + 2n +2 -1 = n^2 +2n +1 = (n+1)^2[/tex]

And we have the proof we were looking for!

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:

  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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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.

assuming 30 days per month.

[tex]\bf \begin{array}{ccll} pages&days\\ \cline{1-2} 1787&11\\ x&30 \end{array}\implies \cfrac{1787}{x}=\cfrac{11}{30}\implies 53610=11x \\\\\\ \cfrac{53610}{11}=x\implies 4873\frac{7}{11}=x[/tex]

Answer: 4873.64

Step-by-step explanation:

I'm assuming that you're asking how many pages there are in a month. On average, the typical month is 30 days, correct? We can plug this information into proportions.

1787/11 = x/30

1787 multiplied by 30 is 53610, and that divided by 11 would be 4873.64, when rounded to the nearest hundredth.

I hope that helped!

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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Answer:

It is a chemical change ⇒ 1st answer

Step-by-step explanation:

* Lets explain the statements to solve the problem

- A chemical change occurs when a new substance is formed through

 a chemical reaction

- Ex: cooking an egg

- Change of reaction is the rate of reaction it can be decreases or

  increasing

- A phase change is a change from one state to another without a

 change in chemical composition

- Ex: Condensation: the substance changes from a gas to a liquid

- A physical change, such as a state change or dissolving, but does

 not create a new substance

- Ex: Breaking a glass

* Lets solve the problem

- A piece of toast came out of the toaster very overcooked.

∵ It is like the cooking an egg

∴ It is a chemical change

Answer:

Chemical

Step-by-step explanation:

Answer:

NOR Gate

Step-by-step explanation:

NOR gate is a two input gate.

It is defined as the complement of (X or Y), where X and Y are the inputs of the gate.

X    Y    X+Y   Complement(X+Y)   NOR

1      1        1                  0                       0

1      0       1                  0                       0

0     1        1                  0                       0

0     0       0                  1                        1

Ut is an operator which gives a value of 1 only when the bvoth the inputs are 0.

A truth table for the logical operator NOR is:

P Q NOR

T T F

T F F

F T F

F F T

How to construct a Truth Table?

A truth table is a mathematical table used in logic to show all possible combinations of truth values for the input variables of a Boolean function and their corresponding output.

To construct a truth table for the logical operator NOR, we can use the following table:

P Q NOR

T T F

T F F

F T F

F F T

The NOR operator returns true only when both inputs are false. Otherwise, it returns false.

This can also be expressed as ¬(P ∨ Q), which is logically equivalent to P NOR Q.

Step-by-step explanation:

The frequency of sine function is given by the number of periods in a given range. For example:

This means that, if we want another sine function with frequency 20% higher, we need that function to have a frequency of 1.2 in the interval [tex][0,2\pi][/tex].

To be easier to see we will consider interval [tex][0,10\pi][/tex] instead of [tex][0,2\pi][/tex]. In this interval [tex]sin(x)[/tex] has 5 periods, therefore our new sine function should have 6 periods.

Finally, as we can see in the graph, the function [tex]sin(\frac{6}{5}x )[/tex] (in blue) has a frequency 20% higher than [tex]sin(x)[/tex] (in red). This can be easily seen counting the number of periods between 0 and [tex]10\pi[/tex] for both functions. 5 for [tex]sin(x)[/tex] and 6 for [tex]sin(\frac{6}{5} x)[/tex].

Answer:

 The standard divisor is 22.48.

Step-by-step explanation:

There are a total of 3259 students at the 5 schools. Then dividing that number by the number of teachers (145) we get the "standard divisor" of ...

  3259/145 ≈ 22.48

__

By Hamilton's method, that divisor is used to divide the number of students at each school, and the result is rounded down. This is the initial allocation of teachers to schools. The remainders from the division are examined. Starting with the largest and working down, one additional teacher is assigned until all the unassigned teachers have been assigned.

For this problem, the initial assignment results in 142 teachers being assigned, so there are 3 more that can be allocated. In order, the highest three remainders are associated with the number of students at East, Central, and South. Each of those schools gets one more teacher than the number initially assigned. The final allocation of teachers is highlighted in the attachment.

Answer:

  corresponding angles

Step-by-step explanation:

Corresponding angles are congruent where a transversal crosses parallel lines. Such a geometry has 4 pairs of corresponding angles. The corresponding angles of each pair are congruent.

Answer:

corresponding angles

Step-by-step explanation:

Answer:

It is more convenient to fill up in the United States.

Step-by-step explanation:

We convert a US gallon to liters:

1 Gallon equals 3.78541 liters.

Therefore, 1 US Gallon costs (3.78541) x (1.29) = 4.8831789 Canadian dollars.

Now we convert the price of a US gallon in Canadian territory to US dollars:

4.8831789 * 0.79 = 3.85771133 US dollars.

Conclusion: A gallon purchased in the United States costs 2.58 US dollars, while a gallon in Canada is equivalent to 3.85771133 US dollars. This way it is more convenient to fill up in the United States.

It would be more economical to fill up the gas tank in the United States before crossing into Canada.

To determine whether it is more cost-effective to fill up the gas tank in the United States or Canada, we need to convert the Canadian gas price into U.S. dollars per gallon. The price of gas in Canada is $1.29 per liter. Since 1 gallon is equal to 3.78541 liters, the price per gallon in Canada would be $1.29 × 3.78541 = $4.88238 CAD. Now, we need to convert this price into USD, knowing the Canadian dollar is worth 79¢ in U.S. currency. Therefore, $4.88238 CAD × 0.79 = $3.85728 USD per gallon.

The price of gas in the United States is $2.58 per gallon. When comparing the two prices, it is clear that $2.58 per gallon in the United States is cheaper than the converted price of gas in Canada ($3.85728 per gallon in USD). Hence, it would be more economical to fill up the gas tank in the United States before crossing into Canada.

Answer:

$293,562.707

Step-by-step explanation:

As for the provided details we know,

Janet needs $51,500 from end of 7th year for upcoming 20 years.

The present value of 20 installments of $51,500 shall be @ 6% from year 7 to year 8.0858

Thus total value = $51,500 [tex]\times[/tex] 8.0858 = $416,418.7

Now the compound interest factor for 6 year @ 6 % = 1.4185

Thus, value to be invested today = $416,418.70/1.4185 = $293,562.707

As this when compounded annually will provide the balance as required at the end of 6 years.

Answer:

We are given that  a manufacturer sells a product as $2 per unit.

Quantity = q units

So, Total revenue = [tex]\text{Cost per unit} \times quantity[/tex]

Total revenue = [tex]2q[/tex]

So, the total revenue function is  [tex]2q[/tex]

Marginal revenue is the derivative of the revenue functions

So, Marginal revenue = [tex]\frac{dR}{dq} =2[/tex]

The marginal revenue function is 2

The constant marginal revenue function mean that the revenue earned by the addition of the output is constant.

Answer:

|-45|

Step-by-step explanation:

In mathematics, the absolute value of a real number is the numeric value of the  number, regardless the sign, either this is positive or negative.

The absolute value function can be definied as:

Using this definition, we have:

|-21| = -(-21) = 21

|14| = 14

|30| = 30

|-45| = -(-45) = 45

Therefore, the expression |-45| has the greatest value.

Answer:

Explained

Step-by-step explanation:

The trunk of a tree grows in two different ways, first in height and second in diameter.Usually tree grows one ring per year in diameter. So, counting the number of rings we can determine the age of a tree. Both height and diameter growth does not occur at the same rate. Tree grows more in height than in their diameter. Mature trees usually grows 1 inch in diameter every year.

Water oak gains 24 inches in height  every year and  1.5 inch growth in diameter annually, meaning if we divide 1.5 inches by 12 months we gets 0.125 inches growth monthly. So a water oak tree needs only 8 months to grow 1 inch in diameter.

Answer:

0.9503

Step-by-step explanation:

First of all, there are some wrong figures in the original text. Because there is a total of 362362 people, the figures should be 86086 (people in the 18-21 age bracket), 43043 (people in the 18-21 age bracket  who respond) and 43043 people in the 18-21 age bracket who refuse to respond. In the same way, because there are 276276 people in the 22-29 age bracket, it should be 18018 and not 1818 who refuse to respond in this subset of people. Now, let's define the following events:

R: a person respond

A: a person belongs to the 18-21 age bracket. So,

The number of people who respond is 43043 + 258258 = 301301, so

P(R) = 301301/362362 = 0.8315

P(A) = 86086/362362 = 0.2376

P(R | A) = 43043/86086 = 0.5

We are looking for P(A∪R) = P(A) + P(R) - P(A∩R),

P(A∩R) = P(R | A)P(A) = (0.5)(0.2376) = 0.1188, so,

P(A∪B) = 0.2376 + 0.8315 - 0.1188 = 0.9503

Answer:

Given differential equation,

[tex]\frac{dy}{dx}+5y=7[/tex]

[tex]\frac{dy}{dx}=7-5y[/tex]

[tex]\implies \frac{dy}{7-5y}=dx[/tex]

Taking integration both sides,

[tex]\int \frac{dy}{7-5y}=\int dx[/tex]

Put 7 - 5y = u ⇒ -5 dy = du ⇒ dy = -du/5,

[tex]-\frac{1}{5} \int \frac{du}{u} = \log x + C[/tex]

[tex]-\frac{1}{5} \log u = \log x + C[/tex]

[tex]-\frac{1}{5}\log(7-5y) = \log x + C---(1)[/tex]

Here, x = 0, y = 0

[tex]\implies -\frac{1}{5} \log 7= C[/tex]

Hence, from equation (1),

[tex]-\frac{1}{5}\log(7-5y)=\log x -\frac{1}{5}log 7[/tex]

[tex]\log(7-5y)=\log (\frac{x}{7^\frac{1}{5}})[/tex]

[tex]7-5y=\frac{x}{7^\frac{1}{5}}[/tex]

[tex]7-\frac{x}{7^\frac{1}{5}}=5y[/tex]

[tex]\implies y=\frac{1}{5}(7-\frac{x}{7^\frac{1}{5}})[/tex]

Answer:

a) The x value of the point where the two equations intersect in terms of a is [tex]x=\frac{40}{4+5a}[/tex]

b) The value of the functions at the point where they intersect is [tex]\frac{10 (28 + 15 a)}{4 + 5 a}[/tex]

c) The partial derivative of f with respect to [tex]x[/tex] is [tex]\frac{\partial f}{\partial x} = -5a[/tex] and the partial derivative of f with respect to [tex]a[/tex] is [tex]\frac{\partial f}{\partial x} = -5x[/tex]

d) The value of [tex]\frac{\partial f}{\partial x}(3,2) = -10[/tex] and [tex]\frac{\partial f}{\partial a}(3,2) = -15[/tex]

e) [tex]\upsilon_1=-\frac{3}{4} = -0.75[/tex] and [tex]\upsilon_2=-\frac{3}{4} = -0.75[/tex]

f) equation [tex]\upsilon_1 = \frac{-5a\cdot x}{70-5ax}=\frac{ax}{ax-14}[/tex] and [tex]\upsilon_2 = \frac{-5a\cdot a}{70-5ax}=\frac{a^2}{ax-14}[/tex]

Step-by-step explanation:

a) In order to find the [tex]x[/tex] we just need to equal the equations and solve for [tex]x[/tex]:

[tex]f(x,a)=g(x)\\70-5xa = 30+4x\\70-30 = 4x+5xa\\40 = x(4+5a)\\\boxed {x = \frac{40}{4+5a}}[/tex]

b) Since we need to find the value of the function in the intersection point we just need to substitute the result from a) in one of the functions. As a sanity check , I will do it in both and the value (in terms of [tex]a[/tex]) must be the same.

[tex]f(x,a)=70-5ax\\f(\frac{40}{4+5a}, a) = 70-5\cdot a \cdot  \frac{40}{4+5a}\\f(\frac{40}{4+5a}, a) = 70 - \frac{200a}{4+5a}\\f(\frac{40}{4+5a}, a) = \frac{70(4+5a) -200a}{4+5a}\\f(\frac{40}{4+5a}, a) =\frac{280+350a-200a}{4+5a}\\\boxed{ f(\frac{40}{4+5a}, a) =\frac{10(28+15a)}{4+5a}}[/tex]

and for [tex]g(x)[/tex]:

[tex]g(x)=30+4x\\g(\frac{40}{4+5a})=30+4\cdot \frac{40}{4+5a}\\g(\frac{40}{4+5a})=\frac{30(4+5a)+80}{4+5a}\\g(\frac{40}{4+5a})=\frac{120+150a+80}{4+5a}\\\boxed {g(\frac{40}{4+5a})=\frac{10(28+15a)}{4+5a}}[/tex]

c) [tex]\frac{\partial f}{\partial x} = (70-5xa)^{'}=70^{'} - \frac{\partial (5xa)}{\partial x}=0-5a\\\frac{\partial f}{\partial x} =-5a[/tex]

[tex]\frac{\partial f}{\partial a} = (70-5xa)^{'}=70^{'} - \frac{\partial (5xa)}{\partial a}=0-5x\\\frac{\partial f}{\partial a} =-5x[/tex]

d) Then evaluating:

[tex]\frac{\partial f}{\partial x} =-5a\\\frac{\partial f}{\partial x} =-5\cdot 2=-10[/tex]

[tex] \frac{\partial f}{\partial a} =-5x\\\frac{\partial f}{\partial a} =-5\cdot 3=-15[/tex]

e) Substituting the corresponding values:

[tex]\upsilon_1 = \frac{\partial f(3,2)}{\partial x}\cdot \frac{3}{f(3,2)} \\\upsilon_1 = -10 \cdot \frac{3}{40}  = -\frac{3}{4} = -0.75[/tex]

[tex]\upsilon_2 = \frac{\partial f(3,2)}{\partial a}\cdot \frac{3}{f(3,2)} \\\upsilon_2 = -15 \cdot \frac{2}{40}  = -\frac{3}{4} = -0.75[/tex]

f) Writing the equations:

[tex]\upsilon_1=\frac{\partial f (x,a)}{\partial x}\cdot \frac{x}{f(x,a)}\\\upsilon_1=-5a\cdot \frac{x}{70-5xa}\\\upsilon_1=\frac{-5ax}{70-5ax}=\frac{-5ax}{-5(ax-14)}\\\boxed{\upsilon_1=\frac{ax}{ax-14} }[/tex]

[tex]\upsilon_2=\frac{\partial f (x,a)}{\partial x}\cdot \frac{a}{f(x,a)}\\\upsilon_2=-5a\cdot \frac{a}{70-5xa}\\\upsilon_2=\frac{-5a^2}{70-5ax}=\frac{-5a^2}{-5(ax-14)}\\\boxed{\upsilon_2=\frac{a^2}{ax-14} }[/tex]

There are no natural numbers that satisfy the equation x + 2 = 1. Therefore, using the roster method, the set is empty.

The question requires us to use the roster method to write the set of natural numbers x that satisfy the equation x + 2 = 1. Natural numbers, by definition, are counting numbers starting from 1. They are non-negative and do not include zero. So, if we try to find a natural number x that satisfies the equation x + 2 = 1, we see that x would need to be -1 (since -1 + 2 equals 1). However, -1 is not a natural number. Therefore, there are no natural numbers that satisfy the equation, so the set is empty.

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The set of natural numbers x that satisfy x + 2 = 1. The correct answer is EMPTY.

To solve the equation  x + 2 = 1  for natural numbers x  we would first try to isolate x subtracting 2 from both sides of the equation:

x + 2 - 2 = 1 - 2

x = -1

 However, natural numbers are defined as the set of positive integers, starting from 1 and increasing indefinitely.

Since the solution to the equation x = -1 is not a positive integer, it does not belong to the set of natural numbers. Therefore, there is no natural number x that satisfies the equation x + 2 = 1  

Since there are no elements that satisfy the condition, the set is empty. Hence, the correct representation of the set using the roster method is EMPTY.

Answer:

[tex]h(t)=-\dfrac{11}{3}t[/tex]

Step-by-step explanation:

A snorkeler dives for a shell on a reef. After entering the water, the diver decends [tex]\frac{11}{3}[/tex] ft in one second.

Let t be the time passed after entering the water, in seconds, and h(t) be the position of the snorkeler under the water, in feet.  

The initial position of the snorkeler was 0 feet under the water.

An equation that models the divers position with respect to time is

[tex]h(t)=0-\dfrac{11}{3}t\\ \\h(t)=-\dfrac{11}{3}t[/tex]

Here the position is negative, because the diver decends (he deepens under the water)

Answer:

(a) P( A ∩ B )=0.5 is not possible.

(b) 0.7

(c) 0.3

(d) 0.3

(e) 0.4

Step-by-step explanation:

Given information: The alphabet A and B represents the following events

A : Individual has a Visa credit card.

B: Individual has a MasterCard.

P(A)= 0.6 and P(B)=0.4.

(a)

We need to check whether P( A ∩ B ) can be 0.5 or not.

[tex]A\cap B\subset A[/tex] and [tex]A\cap B\subset B[/tex]

[tex]P(A\cap B)\leq P(A)[/tex] and [tex]P(A\cap B)\leq P(B)[/tex]

[tex]P(A\cap B)\leq 0.6[/tex] and [tex]P(A\cap B)\leq 0.4[/tex]

From these two inequalities we conclude that

[tex]P(A\cap B)\leq 0.4[/tex]

Therefore, P( A ∩ B )=0.5 is not possible.

(b)

Let [tex]P(A\cap B)=0.3[/tex]

We need to find the probability that student has one of these two types of cards.

[tex]P(A\cup B)=P(A)+P(B)-P(A\cap B)[/tex]

Substitute the given values.

[tex]P(A\cup B)=0.6+0.4-0.3=0.7[/tex]

Therefore the probability that student has one of these two types of cards is 0.7.

(c)

We need to find the probability that the selected student has neither type of card.

[tex]P(A'\cup B')=1-P(A\cup B)[/tex]

[tex]P(A'\cup B')=1-0.7=0.3[/tex]

Therefore the probability that the selected student has neither type of card is 0.3.

(d)

The event that the select student has a visa card, but not a mastercard is defined as

[tex]A-B[/tex]

It can also written as

[tex]A\cap B'[/tex]

The probability of this event is

[tex]P(A\cap B')=P(A)-P(A\cap B)[/tex]

[tex]P(A\cap B')=0.6-0.3=0.3[/tex]

Therefore the probability that the select student has a visa card, but not a mastercard is 0.3.

(e)

We need to find the probability that the selected student has exactly one of the two types of cards.

[tex]P(A\cap B')+P(A\cap B')=P(A\cup B)-P(A\cap B)[/tex]

[tex]P(A\cap B')+P(A\cap B')=0.7-0.3[/tex]

[tex]P(A\cap B')+P(A\cap B')=0.4[/tex]

Therefore the probability that the selected student has exactly one of the two types of cards is 0.4.

Answer: About 16 years

Step-by-step explanation:

The formula to find the compound amount if compounded continuously is given by :-

[tex]A=Pe^{rt}[/tex], where P is Principal amount, r is the rate of interest ( in decimal) and t is time ( in years).

Given : P= $1000   ;    r= 4.6%=0.046

let t be the time it will take to double the amount, the  we have

[tex]2(1000)=(1000)e^{0.046\times t}[/tex]

Dividing 1000 both sides, we get

[tex]2=e^{0.046 t}[/tex]

Taking natural log on each side, we get

[tex]\ln2=\ln(0.046\times t)\\\\\Rightarrow\ 0.6931=0.046t\\\\\Rightarrow\ t=\dfrac{0.6931}{0.046}=15.0673913043\approx16\text{ years}[/tex]

Hence, it will take about 16 years to double the amount.

Answer:

y = x'

z = y'

z' = x + t

Step-by-step explanation:

Hi!

You need to define two new variables y and z:

y = x'

z = y'

Then:

z = y' = x''

z' = x''' = x + t

Now you have a system of 3 equations with only first derivatives