The reflector of a flashlight is in the shape of a parabolic surface. The casting has a diameter of 8 inches and a depth of 4 inches.
How far should The light bulb should be placed from the vertex?

Answer:

d. computational intelligence

Step-by-step explanation:

Neural networks, fuzzy systems, and evolutionary computation are all forms of computational intelligence, where systems develop intelligence through an iterative learning process.

Computational intelligence (CI) is a machine intelligence which usually refers to the ability of a computer to learn or perform a specific task from data or experimental observation. This implies the systems have the ability to learn and generalize from examples and develop intelligence through an iterative process. Computer intelligence is also known as soft computing. The major Computational intelligence are Fuzzy logic, Neural networks and Evolutionary Computing. Presently, Computational Intelligence is an evolving field. New additions are evolving in addition to the three major CI. These new CI include soft computing like artificial endocrine networks, artificial life, ambient intelligence, cultural learning and social reasoning.

Neural networks, fuzzy systems, and evolutionary computation contribute to the field of computational intelligence, demonstrating adaptive learning capabilities and continuous improvement. Option d) is the correct answer.

Neural networks, fuzzy systems, and evolutionary computation are all forms of computational intelligence, where systems develop intelligence through an iterative learning process. Concepts such as evolution and adaptation, which originate in evolutionary biology, stretch to the domain of complex systems involving the development of adaptive, non-biological processes that have dynamic learning and creative abilities.

These systems are part of artificial intelligence, where computer systems learn from data to improve over time. These forms of artificial intelligence have been rapidly developing since the 1980s and are often associated with machine learning, suggesting a broad category where systems can adapt, process information, and optimize problems.

Option A

[tex]\cos 3 x=\cos x-4 \cos x \sin ^{2} x[/tex]

Solution:

Given that we have to rewrite with only sin x and cos x

Given is cos 3x

[tex]cos 3x = cos(x + 2x)[/tex]

We know that,

[tex]\cos (a+b)=\cos a \cos b-\sin a \sin b[/tex]

Therefore,

[tex]\cos (x+2 x)=\cos x \cos 2 x-\sin x \sin 2 x[/tex]  ---- eqn 1

We know that,

[tex]\sin 2 x=2 \sin x \cos x[/tex]

[tex]\cos 2 x=\cos ^{2} x-\sin ^{2} x[/tex]

Substituting these values in eqn 1

[tex]\cos (x+2 x)=\cos x\left(\cos ^{2} x-\sin ^{2} x\right)-\sin x(2 \sin x \cos x)[/tex]  -------- eqn 2

We know that,

[tex]\cos ^{2} x-\sin ^{2} x=1-2 \sin ^{2} x[/tex]

Applying this in above eqn 2, we get

[tex]\cos (x+2 x)=\cos x\left(1-2 \sin ^{2} x\right)-\sin x(2 \sin x \cos x)[/tex]

[tex]\begin{aligned}&\cos (x+2 x)=\cos x-2 \sin ^{2} x \cos x-2 \sin ^{2} x \cos x\\\\&\cos (x+2 x)=\cos x-4 \sin ^{2} x \cos x\end{aligned}[/tex]

[tex]\cos (x+2 x)=\cos x-4 \cos x \sin ^{2} x[/tex]

Therefore,

[tex]\cos 3 x=\cos x-4 \cos x \sin ^{2} x[/tex]

Option A is correct

Answer:

24

Step-by-step explanation:

This problem solution involves permutation to calculate number of arrangements of flags. The reason of using permutation in this scenario is that it is mention in the statement that flags are hanging in a particular order and when order is concern we use permutation. According to permutation definition nPr=n!/(n-r)! and so, the unique signals that are made by using 4 flags are 4P3=4!/(4-3)!=24. Here n=total number=4 and r=number chosen=3.

Final Answer:

The correct answer is D. 24.

Explanation:

To solve this problem, we will calculate the number of permutations of 4 distinct flags taken 3 at a time because the order of the flags matters and no flag can be used more than once in a single signal.
The formula for permutations of n items taken k at a time is given by:
P(n, k) = n! / (n - k)!
Where n! represents the factorial of n, which is the product of all positive integers up to n.
In our case, we have n = 4 flags, and we want to take k = 3 flags to form a signal.
First, we evaluate 4 factorial (4!):
4! = 4 × 3 × 2 × 1 = 24
Now, we need to evaluate the factorial of (4 - 3), which is (1!):
1! = 1
Using the permutation formula, we find the number of unique signals:
P(4, 3) = 4! / (4 - 3)!
P(4, 3) = 24 / 1! (since (4 - 3)! is 1!)
P(4, 3) = 24 / 1
P(4, 3) = 24
Therefore, there are 24 unique signals that can be made using 4 different flags.
The correct answer is D. 24.

Answer:

JH = 8, GH = 12, and GJ = 10.6

Step-by-step explanation:

According to Midsegment Theorem, a segment that connects the midpoints of two sides of a triangle is half the length of the third side.

GH = ½ DE

JH = ½ DF

GJ = ½ EF

DE is 24, so GH = 12.

JH is half of DF.  Since G is the midpoint of DF, DG is also half of DF.  So JH = DG = 8.

GJ is half of EF.  Since H is the midpoint of EF, HE is also half of EF.  So GJ = HE = 10.6.

Answer:

B

Step-by-step explanation:

In this case, we use variables to represent the the cost of the big and the smaller boxes. Let x be the selling price of the big boxes while y be the selling price of the bigger boxes.

The total selling price of the smaller boxes is 20x. The total selling price of the bigger ones is 12y.

Now we know he is making a profit of $1,300

Hence in equation form, all the information above can be represented as:

20x + 12y = 1,300

Answer:

39 feet.

Step-by-step explanation:

E use the Pythagoras theorem:

40^2 = 8^2 + h^2 as h is the height we require.

h^2 = 1600 - 64

h^2 = 1536

h = 39.19 feet

Answer:

Step-by-step explanation:

For the first one, revenue - cost = profit.  We have equations for revenue and profit, so filling in:

[tex]-.3x^2+150x-cost=-.5x^2+250x-300[/tex]

Let's add cost to both sides to make it positive and bring everything on the right over to the left and combine like terms:

[tex].2x^2-100x+300=cost[/tex]

For the second one:

P(x)*T(x) = [tex](x^2-3x-7)(3x)[/tex]

Distribute the 3x into everything inside the parenthesis to get

[tex]3x^3-9x^2-21x[/tex]

For the second part of that problem, C(x)*P(x) = [tex](x-4)(x^2-3x-7)[/tex]

Distribute the x into everything first to get:

[tex]x^3-3x^2-7x[/tex]

The distribute the -4 into everything to get:

[tex]-4x^2+12x+28[/tex]

Combine the like terms and put everything together to get

[tex]x^3-7x^2+5x+28[/tex]

Answer:

Mrs. Allan's Car will use 20 gallons of gas for 560 miles trip.

Mrs. Owen's Car will use 16 gallons of gas for 560 miles trip.

Step-by-step explanation:

Given:

Number of Miles for Mrs Allan car = 224 miles

Number of gallons of gas required = 8

We need to find the number of gallons of gas required for 560 miles for Mrs. Allan's car.

First we will find, in 1 gallons how many miles does Mrs Allan's Car drives.

For 8 gallons = 224 miles

So for 1 gallons = Number of miles in 1 gallon of gas

By using Unitary method we get;

Number of miles in 1 gallon of gas = [tex]\frac{224}{8}= 28\ miles[/tex]

Now we know that;

for 28 miles = 1 gallon of gas is required.

So for 560 miles = Number of gallon of gas required in 560 miles.

Again by using Unitary method we get;

Number of gallon of gas required in 560 miles = [tex]\frac{560}{28}= 20\ gallons[/tex]

Hence Mrs. Allan's Car will use 20 gallons of gas for 560 miles trip.

Also Given:

Number of Miles for Mrs Owen's car = 210 miles

Number of gallons of gas required = 6

We need to find the number of gallons of gas required for 560 miles for Mrs. Owen's car.

First we will find, in 1 gallons how many miles does Mrs Owen's Car drives.

For 6 gallons = 210 miles

So for 1 gallons = Number of miles in 1 gallon of gas

By using Unitary method we get;

Number of miles in 1 gallon of gas = [tex]\frac{210}{6}= 35\ miles[/tex]

Now we know that;

for 35 miles = 1 gallon of gas is required.

So for 560 miles = Number of gallon of gas required in 560 miles.

Again by using Unitary method we get;

Number of gallon of gas required in 560 miles = [tex]\frac{560}{35}= 16\ gallons[/tex]

Hence Mrs. Owen's Car will use 16 gallons of gas for 560 miles trip.

Answer:

see the explanation

Step-by-step explanation:

Let

x ---->the number of books she can sell if she comes back on Sunday

y ----> the number of books sold on Saturday

we know that

The number of books sold on Saturday plus the number of books she can sell if she comes back on Sunday must be less than or equal to the initial number of books in the library (30 books)

so

The inequality that represent this situation is

[tex]x+y\leq 30[/tex]

we have

[tex]y=9\ books[/tex]

substitute

[tex]x+9\leq 30[/tex]

solve for x

subtract 9 both sides

[tex]x\leq 30-9[/tex]

[tex]x\leq 21\ books[/tex]

therefore

The maximum number of books she can sell on Sunday is 21

Final answer:

Jim will not have enough money to buy the car after 2 years with a bank account that pays 9.8% interest compounded continuously. The other bank offering 10% interest compounded semiannually is a better deal as Jim will have enough money to buy the car.

Explanation:

To determine if Jim will have enough money to buy a car that costs $12,160 after 2 years with a bank account that pays 9.8% interest compounded continuously, we can use the formula for compound interest:

A = P * e^(rt)

where A is the ending amount, P is the principal (initial amount), e is Euler's number (approximately 2.71828), r is the interest rate, and t is the number of years. Plugging in the values, we have:

A = $10,000 * e^(0.098*2) = $10,000 * e^0.196 ≈ $11,961.34

Therefore, Jim will not have enough money to buy the car.

To determine if the other bank offering 10% interest compounded semiannually is a better deal, we can use the same formula:

A = P * (1 + r/n)^(nt)

where n is the number of times interest is compounded per year. Plugging in the values, we have:

A = $10,000 * (1 + 0.10/2)^(2*2) = $10,000 * (1 + 0.05)^4 = $10,000 * 1.05^4 ≈ $12,265.63

Therefore, the other bank is a better deal as Jim will have enough money to buy the car.

Answer:$864

Step-by-step explanation:

Since Mohammad makes a payment of $96 monthly, the amount paid by Mohammad for 9months will be 9×$96 that's $96 in 9places

9 × $96 = $864

Answer:

462000

Step-by-step explanation:

here,let the punjabi be x

110/7=7260000/x

110x=7260000*7

x=462000

Answer: 462

Step-by-step explanation: I had the question and it said it was 462

Answer:

1) [tex]np\geq 5[/tex]

2) [tex]nq = n(1-p)\geq 5[/tex]

Other conditions that are important are:

3) n is large

4) p is close to 1/2 or 0.5

Step-by-step explanation:

1) Previous concepts  

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".  

2) Solution to the problem

Let X the random variable of interest, on this case we now that:  

[tex]X \sim Binom(n, p)[/tex]  

The probability mass function for the Binomial distribution is given as:  

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]  

Where (nCx) means combinatory and it's given by this formula:  

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]  

In order to apply the normal apprximation we need to satisfy these two conditions:

1) [tex]np\geq 5[/tex]

2) [tex]nq = n(1-p)\geq 5[/tex]

Other conditions that are important are:

3) n is large

4) p is close to 1/2 or 0.5

Answer:

Step-by-step explanation:

Let d be the cost of a drink.  The expression for the cost of the pizza and 4 drinks is

12.75 + 4d

If each drink costs $3, then

12.75 + 4(3) simpifies to

12.75 + 12 which equals 24.75

Answer:

Step-by-step explanation:

I would divide the answer in two parts

part A

If the given equation is correct, then the equation has no answer

Process

let y = 0 then 5x^2 + 2 = 0. We subtract 2 from both sides 5x^2= -2.

Now we divide each side by 5 and apply square root property

x^2=-2/5  --> x=square root(-2/5). Therefore, the equation has no solution. Complex solution.

Part B

If the given equation is not correct (the exercise is badly copied), then the correct exercise would be 5x^2+3x-2=0

We can solve the polynomial using quadratic formula, and we will obtain

x=-1 and x=2/5.

So the answer X cannot be integer.

Answer:

The answer to your question is letter A) cot x

Step-by-step explanation:

                                            [tex]\frac{1 - cot x}{tan x - 1}[/tex]

Remember that cot x = [tex]\frac{1}{tan x}[/tex]

Then

                                             [tex]\frac{1 - \frac{1}{tan x} }{tanx -1}[/tex]

Simplify

                                             [tex]\frac{\frac{tan x - 1}{tan x} }{tan x - 1}[/tex]

                                             [tex]\frac{tan x - 1}{tan x (tan x - 1)}[/tex]

                                             [tex]\frac{1}{tan x}[/tex]

Result

                                            [tex]\frac{1}{tan x} = cot x[/tex]                                                      

The price of one burger is Rs 40 and a cup of ice cream is Rs 20.

Step-by-step explanation:

Let,

Price of one burger = x

Price of cup of ice cream = y

According to given statement;

x = y+20      Eqn 1

x+2y=80      Eqn 2

Putting value of x from Eqn 1 in Eqn 2

[tex](y+20)+2y=80\\y+20+2y=80\\3y=80-20\\3y=60[/tex]

Dividing both sides by 3

[tex]\frac{3y}{3}=\frac{60}{3}\\y=20[/tex]

Putting y=20 in Eqn 1

[tex]x=20+20\\x=40[/tex]

The price of one burger is Rs 40 and a cup of ice cream is Rs 20.

Keywords: linear equation, substitution method

Learn more about linear equations at:

LearnwithBrainly

The price of a cup of ice cream is Rs 20, and the price of a burger is Rs 40, determined by setting up equations based on the given cost relationship and total cost.

To find the price of a burger and a cup of ice cream, let's use the information given in the problem. Let B represent the cost of a burger, and I represent the cost of a cup of ice cream. We are told that a burger costs Rs 20 more than a cup of ice cream, so we can write this as B = I + 20. Additionally, we know that the total cost of a burger and two ice creams is Rs 80, which can be represented as B + 2I = 80. Now we can substitute the first equation into the second one to find the cost of each item.

Step 1: Substitute B = I + 20 into B + 2I = 80.

Returns: (I + 20) + 2I = 80

Step 2: Combine like terms.

Returns: 3I + 20 = 80

Step 3: Subtract 20 from both sides.

Returns: 3I = 60

Step 4: Divide both sides by 3.

Returns: I = 20

Now we know that a cup of ice cream costs Rs 20, and from the first equation, a burger costs Rs 20 more than the ice cream, so:

Returns: B = I + 20 = 20 + 20 = Rs 40

The price of a burger is Rs 40, and the price of a cup of ice cream is Rs 20.

Answer:

42 minutes is the median amount of time

Step-by-step explanation:

The Median is the middle value of data set.

If there is "odd" number of numbers, and there are n numbers, the median is:

(n/2) + 1 th  number

If there is "even" number of numbers, and there are n numbers, the median is:

Average of n/2th and (n/2) + 1 th number

Here, we have 5 numbers, so the median would be:

(5/2) + 1 th number

That is

3rd number

But, we need to arrange the numbers from least to greatest. Lets write it:

19, 23, 42, 56, 67

The third number is 42, this is the median

Answer:

On a unit circle, the point that corresponds to an angle of [tex]0^{\circ}[/tex] is at position [tex](1, \, 0)[/tex].

The point that corresponds to an angle of [tex]90^{\circ}[/tex] is at position [tex](0, \, 1)[/tex].

Step-by-step explanation:

On a cartesian plane, a unit circle is

The circle crosses the x- and y-axis at four points:

Join a point on the circle with the origin using a segment. The "angle" here likely refers to the counter-clockwise angle between the positive x-axis and that segment.

When the angle is equal to [tex]0^\circ[/tex], the segment overlaps with the positive x-axis. The point is on both the circle and the positive x-axis. Its coordinates would be [tex](1, \, 0)[/tex].

To locate the point with a [tex]90^{\circ}[/tex] angle, rotate the [tex]0^\circ[/tex] segment counter-clockwise by [tex]90^{\circ}[/tex]. The segment would land on the positive y-axis. In other words, the [tex]90^{\circ}[/tex]-point would be at the intersection of the positive y-axis and the circle. Its coordinates would be [tex](0, \, 1)[/tex].

Answer:

Angle of 0º: (1,0)

Angle of 90º: (0,1)

Step-by-step explanation:

A unit circle has a radius of 1.

Therefore, the points on the x- and y-axis are as follows:

    Angle of 0º: (1,0)

    Angle of 90º: (0,1)

    Angle of 180º: (-1,0)

    Angle of 270º: (0,-1)

    Angle of 360º [have now circled back to an angle of 0º]: (1,0)

Answer:

Alex has driven Total of 1400 miles.

Step-by-step explanation:

Given:

Miles Driven by Alex = 588 miles

Percentage mile driven by Alex = 42%

Solution:

Let the total miles driven by Alex be 'x'.

And given that 42% of the total miles he drive for work, which is 588 miles.

That means x multiplied with 42% is equal to 588.

So framing the above sentence in equation form, we get;

[tex]x\times 42\% =588[/tex]

Now we have to remove the percentile.

For this we have to divide 42 by 100, then we get;

[tex]x\times\frac{42}{100} = 588[/tex]

Multiplying by 100 on both side, using Multiplication Property we get;

[tex]x\times\frac{42}{100}\times 100 = 588\times 100\\\\42x=58800[/tex]

Dividing both side by 42 using Division Property we get;

[tex]\frac{42x}{42}=\frac{58800}{42}\\\\x= 1400 \ miles[/tex]

Hence Alex has driven Total of 1400 miles.

Answer: 1,400

Step-by-step explanation:%

100

=  

part

whole

42

100

=  

588

x

42x = 58,800

x = 1,400

Answer:

  any digit 0 to 9

Step-by-step explanation:

Nothing about your description of the number suggests that the 10s digit depends on the units digit. The tens digit can be any digit 0 to 9.

Check the picture below.

Answer:

0.55 is the probability that Lions will score less than 3 goals.

Step-by-step explanation:

We are given the following in the question:

Number of goals(x):         0         1           2         3          4

             Probability:      0.05     0.15     0.35     0.30     0.25

We have to find the probability that Lions will score less than 3 goals.

[tex]P(X<x) = \displaystyle\sum_{X=0}^{X=x-1}P(x_i)\\\\P(x<3)=P(x=0)+P(x=1)+P(x=2)\\P(x<3)=0.05 + 0.15 + 0.35=0.55[/tex]

0.55 is the probability that Lions will score less than 3 goals.

Final answer:

The probability that the Lions soccer team will score less than 3 goals in a game is 0.55 or 55%.

Explanation:

To find the probability that the Lions soccer team scores less than 3 goals in a game, we need to add up the probabilities of scoring 0, 1, and 2 goals. From Exhibit 5-3, the probability of scoring 0 goals is 0.05, 1 goal is 0.15, and 2 goals is 0.35.

The total probability of scoring less than 3 goals is the sum of these probabilities:

P(0 goals) = 0.05

P(1 goal) = 0.15

P(2 goals) = 0.35

Thus, P(score < 3 goals) = P(0 goals) + P(1 goal) + P(2 goals) = 0.05 + 0.15 + 0.35 = 0.55.

Therefore, the probability that the Lions will score less than 3 goals in a given game is 0.55 or 55%.

Answer:

Step-by-step explanation:

The id attribute requires a unique string to identify the element.

Among the options provided, the attribute of the "an" element that indicates the media type of a linked document is:

C) type="mime-type"

In HTML, when creating links to external resources or documents, the "an" element is typically represented by the "a" anchor tag. This tag often uses the 'type' attribute to specify the media type of the linked document.

For instance, if you have an anchor tag (<a>) linking to a document like a stylesheet, image, or script, you can specify the media type using the 'type' attribute. The 'type' attribute helps the browser interpret the linked document's format or MIME (Multipurpose Internet Mail Extensions) type. An example might look like:

<a href="url_to_linked_document" type="text/css">Link Text</a>

In this example, 'type="text/css"' indicates that the linked document is a CSS file (text/css MIME type), which helps browsers understand how to handle the resource being linked. Therefore, option C) type="mime-type" is the attribute used to indicate the media type of the linked document.

Complete Question:

Identify an attribute of the "an" element that indicates the media type of z linked document.

A) rel-"type"

B) hrefland="lang"

C) type="mime-type"

D) href="url"

Answer:

True

Step-by-step explanation:

Sample size selection is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is important in any empirical statistical study in which the goal is to make inferences about a population from a sample. The actual sample size used in a study is determined based on the time money and technical cost of data collection, and the need to have sufficient statistical accuracy.

Finding a balance between acceptable error and research cost gives rise to concepts like Confidence Interval and Margin of Error.

Answer:  d. 58

The height of the plant in June 1st is 58 centimeters.

Step-by-step explanation:

Given : On August 1st, a plant was 79 centimeters tall.

In June of that year it grew 12 centimeters, then 9 more centimeters in July.

Order of Month = June → July → August

Then , the height of the plant on June 1st would be

Height in August 1st - Height grew in June - height grew in July

= 79-12-9  centimeters

= 67-9 centimeters

= 58  centimeters

Therefore , the height of the plant in June 1st = 58 centimeters

Hence, the correct answer is d. 58

Answer:

  31,536,000

Step-by-step explanation:

There are 3600 seconds in an hour and 24 hours in a day, so ...

 (3600 s/h)(24 h/da) = 86400 s/da

Then in 365 days, there are ...

  (365 da)(86400 s/da) = 31,536,000 s

_____

No rounding is needed.

There are 31,536,000 seconds in a year. This is calculated by multiplying 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 365 days in a year.

To calculate the number of seconds in a year, start with the known values of time:

We multiply all these values together to get the number of seconds in a year.

60 seconds/minute * 60 minutes/hour * 24 hours/day * 365 days/year = 31,536,000 seconds/year

To give the answer to the nearest 1000 seconds, we would round it to 31,536,000.

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Answer:  The required solution is [tex]y=50e^{0.1386t}.[/tex]

Step-by-step explanation:

We are given to solve the following differential equation :

[tex]\dfrac{dy}{dt}=ky~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

where k is a constant and the equation satisfies the conditions y(0) = 50, y(5) = 100.

From equation (i), we have

[tex]\dfrac{dy}{y}=kdt.[/tex]

Integrating both sides, we get

[tex]\int\dfrac{dy}{y}=\int kdt\\\\\Rightarrow \log y=kt+c~~~~~~[\textup{c is a constant of integration}]\\\\\Rightarrow y=e^{kt+c}\\\\\Rightarrow y=ae^{kt}~~~~[\textup{where }a=e^c\textup{ is another constant}][/tex]

Also, the conditions are

[tex]y(0)=50\\\\\Rightarrow ae^0=50\\\\\Rightarrow a=50[/tex]

and

[tex]y(5)=100\\\\\Rightarrow 50e^{5k}=100\\\\\Rightarrow e^{5k}=2\\\\\Rightarrow 5k=\log_e2\\\\\Rightarrow 5k=0.6931\\\\\Rightarrow k=0.1386.[/tex]

Thus, the required solution is [tex]y=50e^{0.1386t}.[/tex]

Answer:

[tex]\frac{1}{5}\ln(2)=k[/tex]

Solution without isolating [tex]y[/tex]:

[tex]\ln|y|=\frac{1}{5}\ln(2)x+\ln(50)[/tex]

Solution with isolating [tex]y[/tex]:

[tex]y=50 \cdot 2^{\frac{1}{5}x}[/tex]

Step-by-step explanation:

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

We will separate the variables so we can integrate both sides.

Multiply [tex]dx[/tex] on both sides:

[tex]dy=ky dx[/tex]

Divide both sides by [tex]y[/tex]:

[tex]\frac{dy}{y}=k dx[/tex]

Now we may integrate both sides:

[tex]\ln|y|=kx+C[/tex]

The first condition says [tex]y(0)=50[/tex].

Using this into our equation gives us:

[tex]\ln|50|=k(0)+C[/tex]

[tex]\ln|50|=C[/tex]

So now our equation is:

[tex]\ln|y|=kx+\ln(50)[/tex]

The second condition says [tex]y(5)=100[/tex].

Using this into our equation gives us:

[tex]\ln|100|=k(5)+\ln(50)[/tex]

[tex]\ln(100)=k(5)+\ln(50)[/tex]

Let's find [tex]k[/tex].

Subtract [tex]\ln(50)[/tex] on both sides:

[tex]\ln(100)-\ln(50)=k(5)[/tex]

I'm going to rewrite the left hand side using quotient rule for logarithms:

[tex]\ln(\frac{100}{50})=k(5)[/tex]

Reducing fraction:

[tex]\ln(2)=k(5)[/tex]

Divide both sides by 5:

[tex]\frac{\ln(2)}{5}=k[/tex]

[tex]\frac{1}{5}\ln(2)=k[/tex]

So the solution to the differential equation satisfying the give conditions is:

[tex]\ln|y|=\frac{1}{5}\ln(2)x+\ln(50)[/tex]

Most likely they will prefer the equation where [tex]y[/tex] is isolated.

Let's write our equation in equivalent logarithm form:

[tex]y=e^{\frac{1}{5}\ln(2)x+\ln(50)}[/tex]

We could rewrite this a bit more.

By power rule for logarithms:

[tex]y=e^{\ln(2^{\frac{1}{5}x})+\ln(50)}[/tex]

By product rule for logarithms:

[tex]y=e^{\ln(2^{\frac{1}{5}x} \cdot 50)}[/tex]

Since the natual logarithm and given exponential function are inverses:

[tex]y=2^{\frac{1}{5}x} \cdot 50[/tex]

By commutative property of multiplication:

[tex]y=50 \cdot 2^{\frac{1}{5}x}[/tex]