Use separation of variables to find the general solution to the following differential equation.​

Answer:

0.3788

Step-by-step explanation:

12 people want to teach math at a college.

6 people have a PhD.

5 people have a master's degree.

boss wants to interview 5 people for the job.

chance that all 5 of the people interviewed have a PhD.

total number of ways to select 5 applicants out of 12 is given by the combination:

combinations formula :

nCr = n! / r! * (n – r)!

C(12, 5) = 12! / (5! * 7!) = 792

number of ways to select 3 applicants with a PhD out of the 6 available is:

C(6, 3) = 6! / (3! * 3!) = 20

remaining 2 applicants can be selected from the remaining 6 applicants with a master's degree:

C(6, 2) = 6! / (2! * 4!) = 15

total number of ways to select 5 applicants with 3 having a PhD is:

20 * 15 = 300

P(3 PhDs) = 300 / 792

P(3 PhDs) = 0.3788 (rounded to four decimal places)

So, the probability of selecting five applicants for an interview where all three applicants have a PhD is approximately 0.3788

ChatGPT

The dimensions of cone B is radius, 6.8 units and height 7 units.

The possible dimensions of cone B is calculated as follows;

Volume of cone = ¹/₃πr²h

Volume of cone A = ¹/₃π(6²)(4) = 150.8 units³

Volume of cone B = 487 units³ - 150.8 units³ = 336.2 units³

The dimensions of cone B is calculated as;

¹/₃πr²h = 336.2 units³

r²h = 321

Let the height of cone B = 7, then the radius of the cone is calculated as;

7r² = 321

r² = 321/7

r² = 45.86

r = √45.86

r = 6.8 units

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The maximum value of the objective functions are 2600, 27 and 1980

Given that

Max Z = 8x + 16y

Where the constraints are

3x + 6y ≤ 900

x + y ≤ 200

y ≤ 125

x, y ≥ 0

Plotting the constraints 3x + 6y ≤ 900, x + y ≤ 200 and y ≤ 125 on the same graph, the coordinates of the feasible region are:

(x, y) = (100, 100), (75, 125) and (50, 125)

So, we have

Z = 8(100) + 16(100) = 2400

Z = 8(75) + 16(125) = 2600

Z = 8(50) + 16(125) = 2400

Hence, the maximum value is 2600

Given that

Max Z = 6x + 3y

Where the constraints are

2x + y ≤ 8

3x + 3y ≤ 18

y ≤ 3

x, y ≥ 0

Plotting the constraints 2x + y ≤ 8, 3x + 3y ≤ 18 and y ≤ 3 on the same graph, the coordinates of the feasible region are:

(x, y) = (3, 3), (2.5, 3) and (2, 4)

So, we have

Z = 6(3) + 3(3) = 27

Z = 6(2.5) + 3(3) = 24

Z = 6(2) + 3(4) = 24

Hence, the maximum value is 27

Given the objective function, the constraints and the final simplex tableau

We have the final values to be

Z = 1980

This means that the maximum value is 1980

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Answer:

The resulting linear equation is 3d = 12, so d = 4 and e = 0.

Answer:

1 Mexican Peso = 0.0552 US Dollar Hope this helped! Have a great day! :)

Step-by-step explanation:

Answer:

1 United States Dollar is equivalent or equal to 18.24 (Mexican peso)

Answer:

68.9 degrees

Step-by-step explanation:

To find this we can use the rule of sines.

It states [tex]\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}[/tex]

We will use 56 degrees and its complementary measurement, which is discovered by observing the opposite side from the angle, which is 4. Then, we will find the side that compliments x, which is 4.5. Then we can plug those values into the rule of sines.

[tex]\frac{sin56}{4}=\frac{Sinx}{4.5}[/tex]

Then, we want to get Sin x by itself.

[tex]\frac{sin56}{4}*4.5=sinx[/tex]

Then, we can solve for sin x.

[tex]0.932667269124=sinx[/tex]

finally, we need to take the inverse of sin to find our solution.

[tex]sin^{-1} (0.932667269124)=sin^-^1(sinx)\\x=68.85\\[/tex]

Which can be rounded to 68.9.

Thus, the unknown leg x of the right angled triangle is found as 2.2 ft.

A right triangle is one that has an interior angle of 90 degrees. The hypotenuse, which is also the side of the right triangle that faces the right angle, is its longest side. The height and base make up the two arms of the right angle.

What a Right Triangle Looks Like

For the given right triangle:

Let the unknown length be 'x'.

Using the Pythagorean theorem:

3² = 2² + x²

x² = 3² - 2²

x² = 9 - 4

x² = 5

x = 2.2

Thus, the unknown leg x of the right angled triangle is found as 2.2 ft.

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Step-by-step explanation:

We can use the standard normal distribution to calculate probabilities for a normal distribution with mean 33 and standard deviation 4. We just need to standardize the intervals using the formula:

z = (x - mu) / sigma

where x is the specific value in the interval, mu is the mean, sigma is the standard deviation, and z is the corresponding z-score.

a. Between 29 and 37:

z1 = (29 - 33) / 4 = -1

z2 = (37 - 33) / 4 = 1

Using a standard normal distribution table, the cumulative probability of z being between -1 and 1 is approximately 0.6827.

So the probability that a randomly selected x-value from the distribution is between 29 and 37 is approximately 0.6827.

b. Between 33 and 45:

z1 = (33 - 33) / 4 = 0

z2 = (45 - 33) / 4 = 3

The cumulative probability of z being between 0 and 3 is approximately 0.4987.

So the probability that a randomly selected x-value from the distribution is between 33 and 45 is approximately 0.4987.

c. At least 29:

z = (29 - 33) / 4 = -1

The cumulative probability of z being less than -1 is approximately 0.1587. So the probability that a randomly selected x-value from the distribution is at least 29 is approximately 1 - 0.1587 = 0.8413.

d. At most 21:

z = (21 - 33) / 4 = -3

The cumulative probability of z being less than -3 is very close to 0. So the probability that a randomly selected x-value from the distribution is at most 21 is approximately 0.

15.048 deciliters is equivalent to 1 quart.

Conversion is the process of changing a quantity from one unit of measure to another unit of measure using a conversion factor or a formula.

We have:

To convert meters to feet, multiply by 3.28.

To convert liters to gallons, multiply by 0.26.

To convert kilograms to pounds, multiply by 2.20.

To convert deciliters to quarts, divide by 15.048.

Let's use these conversions to convert the given measurement:

15.048 dL = qt

Dividing both sides by 15.048, we get:

1 dL = qt/15.048

Multiplying both sides by 0.946353, which is the number of quarts in a liter, we get:

0.946353 dL = qt/15.048 * 0.946353

Simplifying the right-hand side, we get:

0.946353 dL = qt/15.961

Multiplying both sides by 3.78541, which is the number of liters in a gallon, we get:

3.78541 * 0.946353 dL = 3.78541 * qt/15.961

Simplifying the left-hand side, we get:

3.58702 L = qt/4.22676

Multiplying both sides by 4.22676, we get:

15.1485 L = qt

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Answer:

  $1070.16

Step-by-step explanation:

You want the prorated amount of taxes if the annual amount is $1743.25 and the sale closes August 12, based on a 360 day year.

A 360-day year assumes months are 30 days. The tax charged to the seller will be that for 7 months plus 11 days:

  (7·30 +11)/360 × $1743.25 = $1070.16

The seller will pay $1070.16 of the tax bill.

__

Additional comment

We have presumed the buyer pays the taxes for August 12, the first day they own the property.

Thus, the height of the oak trees of found to be 98 feet.

In maths, addition is the process of adding two or more numbers together. The numbers that added are known as addends, while the outcome of the addition process, or the final response, is known as the sum.

Given data:

Height of the bird from the base of the oak tree: 33 feet.

Height flew by the bird to reach at the top of the tress: 65 feet.

So,

Height of the oak tree = 33 + 65

Height of the oak tree = 98 feet

Thus, the height of the oak trees of found to be 98 feet.

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Okay, here are the steps to solve this problem:

* There are 10 sections on the spinner, 3 of which are green and 7 of which are yellow.

* So there is a 3/10 = 0.3 probability that the spinner will land on green.

* A fair coin has a 1/2 probability of landing on heads.

* For the spinner and coin toss to both have the desired outcome (green and heads), we multiply their individual probabilities:

* 0.3 * 1/2 = 0.15

Therefore, the probability that the spinner lands on green and the coin toss is heads is 0.15.

Answer:

Amount of change: 4,000 - 3,600 = 400

Percent of change: 400/4,000 = 1/10 = 10%

There was a 10% decrease in the number of visitors:

The space sample would include the lotion, spray and gel, with the SPF and there are 15 possible outcomes.

The enumeration of sample space and all potential results can be achieved by duly considering various combinations of SPF and sunscreen variety. It is achievable to list the entire gamut of possibilities when each type of sunscreen is matched with every value of SPF:

The sample space would look like this:

This shows that there are 15 possible outcomes in the sample space.

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The intervals in which the function is decreasing. [tex][-\pi , -\frac{\pi }{3} ], [\frac{\pi }{3}, \pi ][/tex]. Option 3

We're given a function f(x) = 2 sin x - x, which describes a curve on a graph. We want to find the intervals where this curve is decreasing (going down) within the range of -π to π.

To find when the function is decreasing, we look at its slope. The slope tells us if the curve is going up or down. We find the slope by taking the first derivative of the function: f'(x) = 2 cos x - 1.

We now have an equation for the slope, f'(x) = 2 cos x - 1. A negative slope means the function is decreasing. So, we want to find where f'(x) is less than 0 (negative).

We set up the inequality: 2 cos x - 1 < 0. We solve it to find the x-values where the slope is negative. The solution is cos x < 1/2.

From the inequality cos x < 1/2, we find the intervals within the range of -π to π where the function is decreasing. These intervals are [-π, -π/3] and [π/3, π].

The above answer is in response to the question below as seen in the picture.

Determine the interval(s) in [tex][-\pi, \pi ][/tex] on

which f(x) =  2 sin x - x

is decreasing.

1. [tex][-\frac{\pi }{3}, \frac{\pi }{3} ][/tex]

2. [tex][-\frac{\pi }{6}, \frac{\pi }{6} ][/tex]

3. [tex][-\pi , -\frac{\pi }{3} ], [\frac{\pi }{3}, \pi ][/tex]

4. [tex][-\pi , -\frac{-2\pi }{3} ], [\frac{2\pi }{3}, \pi ][/tex]

5. [tex][-\pi , - \frac{5x}{6} ], [\frac{\pi }{6}, \pi ][/tex]

6. [tex][-\frac{\pi }{6}, \frac{5\pi }{6} ][/tex]

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A dilation can be defined as a transformation that multiplies the distance between every point in an object and a fixed point, called the center of dilation, by a constant factor called the scale factor.

The lengths of the corresponding segments are given as follows:

Hence the scale factor is given as follows:

k = 4/2

k = 2.

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as you already know, to get the inverse of any expression we start off by doing a quick switcheroo on the variables and then solving for "y", let's do so.

[tex]\stackrel{f(x)}{y}~~ = ~~x^3+9\hspace{5em}\stackrel{\textit{quick switcheroo}}{x~~ = ~~y^3+9} \\\\\\ x-9=y^3\implies \sqrt[3]{x-9}=y=f^{-1}(x)[/tex]

Answer:

  (-4, 2)∪(8, ∞)

Step-by-step explanation:

Given a particle's position is described by x(t) = t³ -6t² -96t, you want the intervals where speed is increasing.

The speed of the particle is the magnitude of its rate of change of position.

The rate of change of position is ...

  x'(t) = 3t² -12t -96 = 3(t² -4t) -96

  x'(t) = 3(t -2)² -108

This describes a parabola that opens upward, with a vertex at (2, -108). It has zeros at x = 2 ± 6 = {-4, 8}.

The magnitude of the speed is shown by the blue curve in the attachment. Between t=-4 and t=8, it is the opposite of the parabola described by the above equation.

The rate of change of speed is the derivative of speed with respect to time. The green curve in the attachment shows the particle's rate of change of speed. Speed is increasing when the green curve is above the x-axis.

Between the point when speed is 0, at t=-4, and when it reaches a local maximum, at t=2, it is increasing. Speed is increasing again after it becomes 0 at t=8.

The intervals of increasing speed are (-4, 2) ∪ (8, ∞).

__

Additional comment

We have made the distinction between speed and velocity. Velocity is the signed rate of change of position. If position is plotted on a number line increasing to the right, then velocity is positive anytime the particle is moving to the right. Velocity is increasing if acceleration is to the right (positive).

Velocity of this particle is increasing on the interval (2, ∞).

Answer:

Step-by-step explanation:

The polynomial f(x) with real coefficients, degree 4, and zeros 6 (multiplicity 2), 3i can be formed as follows:

Since the zero 6 has a multiplicity of 2, it appears twice in the factored form of f(x), i.e., (x-6)(x-6) = (x-6)^2.

The other zero is 3i, which means its complex conjugate, -3i, is also a zero. Therefore, the factored form of f(x) can be written as:

(x-6)^2(x-3i)(x+3i)

Expanding this expression, we get:

f(x) = (x-6)^2(x^2 + 9)

Multiplying this out, we get:

f(x) = x^4 - 12x^3 + 81x^2 - 216x + 324

Therefore, the polynomial f(x) with real coefficients, degree 4, and zeros 6 (multiplicity 2), 3i can be written as:

f(x) = x^4 - 12x^3 + 81x^2 - 216x + 324.

Using the average daily balance, the total balance on the next billing date, April 1 is $5,059.73.

The average daily balance is a credit card method of computing finance charges.

To determine the average daily balance, the sum of the daily balances over your billing cycle is divided by the number of days in the billing cycle.

The finance charge is then the product of the average daily balance multiplied by the APR and the number of days involved, divided by 365 days.

Average daily balance = $1,900

Unpaid balance at month-end = $1,700

APR = 32.4%

The finance charge for the month = $9.73 ($1900 x 32.4% x 30/365)

The total balance on the next billing date, April 1 = $5,059.73 ($5,050 + $9.73)

Thus, on April 1, the next billing cycle, the balance on the card is $5,059.73.

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The distance that the ball has traveled when the height of the ball is 80 feet is either about 9.86 feet or about 3.64 feet.

We are given that;

Velocity= 140feet

Inclination= 45degrees

Function  ​h(x)=(-32x^2/140^2)+x

Now,

To find the distance that the ball has traveled when the height of the ball is 80 feet, we need to solve the equation:

h(x) = 80

Substituting h(x) with the given function, we get:

(-32x2/1402) + x = 80

Multiplying both sides by 140^2 and simplifying, we get:

-32x^2 + 140x - 11200 = 0

Dividing both sides by -32 and simplifying, we get:

x^2 - (35/4)x + 350 = 0

Using the quadratic formula, we get:

x = [ (35/4) ± √( (35/4)^2 - 4(350) ) ] / 2 x ≈ 9.86 or x ≈ 3.64

Using a graphing utility, we can confirm that these are the approximate x-intercepts of the function h(x) - 80.

Therefore, by graphing the answer will be about 9.86 feet or about 3.64 feet.

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The rating of chips whose cost is $1.10 is obtained replacing the value of x on whichever's equation is correct by 1.10.

To calculate the numeric value of a function or of an expression, we substitute each instance of any variable or unknown on the function by the value at which we want to find the numeric value of the function or of the expression presented in the context of a problem.

The problem is incomplete, hence the procedure to estimate the rating of chips whose cost is $1.10 is presented.

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The entries in both columns when matched are

From the question, we have the following parameters that can be used in our computation:

The graph

When the line of best fit is drawn, we have

(8, 0) and (0, 8)

The equation is calculated as

y = mx + c

So, we have

8 = 0 * m + c

0 = 8 * m + c

This gives

c = 8

0 = 8 * m + 8

So, we have

m = -1

This means that the equation is

y = -x + 8

Using the above as a guide, we have the following:

Equation of line of best fit: y = -x + 8

Slope = -1

y-intercept = 8

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Answer:

To convert 130 kph to mph, we can use the conversion factor 1 kph = 0.62 mph:

130 kph × 0.62 mph/kph ≈ 80.6 mph

So the maximum whole-number speed that Hannah can drive without exceeding the speed limit is 80 mph (option D).

Step-by-step explanation:

The direction and speed the plane traveling is at About 84.3° west of north at approximately 502.5 mph. Option C

We need to first  find the distance between points A and C using the distance formula; Distance AC = √((x2 - x1)² + (y2 - y1)²)

If we input the figures as seen in the diagram, it becomes

Distance AC = √((-30 - 20)² + (520 - 20)²)

which is 502.49. if we round it off, it becomes 502.5

We have to find find the angle θ that the plane is traveling using the law of cosines

cos(θ) = (AB² + BC² - AC²) / (2 x AB x BC)

cos(θ) = (500² + 50² - 502.5²) / (2 x 500 x 50)

which is -0.000125

θ = arccos( -0.000125)

θ = 90.0071621563 (in degrees)

Give than the wind is blowing west, the angle should be measured west of north.

180° - 90.01° = 90°

It only mean that the plane is travelling at approximately 84.3° west of north  

The answer is based on the question below;

A plane is set to fly due north, but it is pushes off course by crosswind blowing west. At 1 pm, the plane is located at point A and at 2pm, the plane is located at point C, as shown in the diagram. In what direction and at what speed is the plane traveling?

A. About 5.7° west of north at approximately 500.1 mph.

B. About 5.7° west of north at approximately 502.5 mph

C. About 84.3° west of north at approximately 500.1 mph.

D. About 84.3° west of north at approximately 502.5 mph

Point C coordinates (-30, 520)

Point A (20, 20)

Distance from A to B on a straight course is 500

B to C is 50

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The expression (cscx + cotx)² is the same as csc²x + 2(cscx)(cotx) + cot²x. (option a).

The given expression is (cscx + cotx)². To simplify this expression, we can use the formula for squaring a binomial, which is (a + b)² = a² + 2ab + b². In this case, a = cscx and b = cotx. Therefore, we can substitute these values into the formula to get:

(cscx + cotx)² = csc²x + 2(cscx)(cotx) + cot²x

So the expression (cscx + cotx)² is the same as csc²x + 2(cscx)(cotx) + cot²x.

Hence the correct option is (a).

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