Suppose a preliminary screening is given to prospective student athletes at a university to determine whether they would qualify for a scholarship. The scores are approximately normal with a mean of 85 and a standard deviation of 20. If the range of possible scores is 0 to 100, what percentage of students has a score less than 85?

Answer:

[tex]P(t)=80(10^{0.1t})[/tex]

Step-by-step explanation:

The 'y' axis represent log(P), so it may be modeled as a line (or linear function), where its slope is 0.1:

[tex]log(P)=0.1t+C[/tex]

Pow each part of the equation by 10:

[tex]10^{log(P)}=10^{0.1t+C}\\ P=10^{0.1t+C}[/tex]

Evaluate at t=0, where the population is known.

[tex]P(0)=10^{C}=80[/tex]

Applying logarithmic properties:

[tex]P=10^{0.1t+C}=10^{0.1t}*10^{C}[/tex]

So, the final function is:

[tex]P(t)=80(10^{0.1t})[/tex]

The population size over time follows an exponential growth function described by P(t) = 80 * [tex]e^(0.23026t)[/tex], where the slope on a semilog plot of 0.1 is converted to the natural log base to find the growth rate constant.

When dealing with population growth, plotting the data on a semilog plot where the logarithm of the population size is plotted against time can simplify the analysis. If the resulting plot is a straight line with a slope of 0.1, this indicates exponential growth, because plotting an exponential function on a semilog plot yields a straight line.

The general form of an exponential growth function is:

Where:

Given that the slope of the line on the semi log plot is 0.1, this slope represents the constant k in the context of logs with base 10.

To convert this to a natural logarithm base (e), use the conversion factor ln(10):

k = 0.1 * ln(10)

Since ln(10) ≈ 2.3026, we have:

k = 0.1 * 2.3026

≈ 0.23026

Given the initial population size (P0) at time t = 0 is 80, the function describing the population size over time t is:

P(t) = 80 * [tex]e^(0.23026t)[/tex],

No. The sample size is too small to be reliable. If there were many bags of 500 candies that consistently showed a larger-than-marketed amount of green candy, that will be better proof.

The difference between the observed proportion of green candies (0.12) and candy company's claimed proportion (0.10) does not necessarily mean that the manufacturing process is out of control. It could simply be down to natural variation or chance. A definitive conclusion would require a proper statistical test using more data.

In this scenario, we are looking at a statistic known as the proportion. It is calculated by dividing the number of successful outcomes (green candies) by the total number of outcomes (total candies). The candy company claims a proportion of 10%, which is 0.10. However, the observed proportion in your large bag of candies was 12% of 500, which works out to 0.12.

From a statistical standpoint, a slight variation from the claimed percentage is not unusual, as there will always be some random variability due to chance. While the difference between the observed proportion (0.12) and the expected proportion (0.10) might seem significant at first glance, we cannot definitively say from this one sample that the manufacturing process is out of control without conducting a proper hypothesis test — ideally, using additional data from more bags of candy.

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Answer:  The required equation of the line is [tex]x-y=0.[/tex]

Step-by-step explanation:  We are given to to find the equation of a straight line in the standard form Ax + By = C passing through the points (-4, -4) and (-3, -3).

We know that the slope of a line passing through the points (a, b) and (c. d) is given by

[tex]m=\dfrac{d-b}{c-a}.[/tex]

So, the slope of the given line will be

[tex]m=\dfrac{-3+4}{-3+4}=\dfrac{1}{1}=1.[/tex]

Since the line passes through the point (-3, -3), so its equation will be

[tex]y-(-3)=m(x-(-3))\\\\\Rightarrow y+3=1(x+3)\\\\\Rightarrow y+3=x+3\\\\\Rightarrow y=x\\\\\Rightarrow x-y=0.[/tex]

Thus, the required equation of the line is [tex]x-y=0.[/tex]

19 and 42 are coprime, so we can use the CRT right away. Start with

[tex]x=19+42[/tex]

Taken mod 42, we're left with a remainder of 19. Notice that

[tex]19\cdot3\equiv57\equiv15\pmod{42}[/tex]

so we need to multiply the first term by 3 to get the remainder we want.

[tex]x=19\cdot3+42[/tex]

Next, taken mod 19, we're left with a remainder of 4. Notice that

[tex]42\cdot6\equiv252\equiv5\pmod{19}[/tex]

so we need to multiply the second term by 6.

Then by the CRT, we have

[tex]x\equiv19\cdot3+42\cdot6\equiv309\pmod{42\cdot19}\implies x\equiv309\pmod{798}[/tex]

so that any solution of the form [tex]x=798n+309[/tex] is a solution to this system.

[tex]2x\equiv3\pmod{11}[/tex]

Since 3 * 4 = 12 = 1 mod 11, we can multiply both sides by 4 to get

[tex]4\cdot2x\equiv3\cdot4\pmod{11}\implies8x\equiv1\pmod{11}[/tex]

Since 8 * 7 = 56 = 1 mod 11, multiplying both sides by 7 gives

[tex]7\cdot8x\equiv7\cdot1\pmod{11}\implies x\equiv7\pmod{11}[/tex]

Answer:

(A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6

(B) f(x + h) - f(x) = 8xh + 4h² - 6h

(C) [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]

Step-by-step explanation:

* Lets explain how to solve the problem

- The function f(x) = 4x² - 6x + 6

- To find f(x + h) substitute x in the function by (x + h)

∵ f(x) = 4x² - 6x + 6

∴ f(x + h) = 4(x + h)² - 6(x + h) + 6

- Lets simplify 4(x + h)²

∵ (x + h)² = (x)(x) + 2(x)(h) + (h)(h) = x² + 2xh + h²

∴ 4(x + h)² = 4(x² + 2xh + h²) = 4x² + 8xh + 4h²

- Lets simplify 6(x + h)

∵ 6(x + h) = 6(x) + 6(h)

∴ 6(x + h) = 6x + 6h

∴ f(x + h) = 4x² + 8xh + 4h² - (6x + 6h) + 6

- Remember (-)(+) = (-)

∴ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6

* (A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6

- Lets find f(x + h) - f(x)

∵ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6

∵ f(x) = 4x² - 6x + 6

∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - (4x² - 6x + 6)

- Remember (-)(-) = (+)

∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - 4x² + 6x - 6

- Simplify by adding the like terms

∴ f(x + h) - f(x) = (4x² - 4x²) + 8xh + 4h² + (- 6x + 6x) - 6h + (6 - 6)

∴ f(x + h) - f(x) = 8xh + 4h² - 6h

* (B) f(x + h) - f(x) = 8xh + 4h² - 6h

- Lets find [tex]\frac{f(x+h)-f(x)}{h}[/tex]

∵ f(x + h) - f(x) = 8xh + 4h² - 6h

∴ [tex]\frac{f(x+h)-f(x)}{h}=\frac{8xh + 4h^{2}-6h}{h}[/tex]

- Simplify by separate the three terms

∴ [tex]\frac{f(x+h)-f(x)}{h}=\frac{8xh}{h}+\frac{4h^{2} }{h}-\frac{6h}{h}[/tex]

∴ [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]

* (C) [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]

Answer:

260 machines for minimum cost.

Step-by-step explanation:

c(x) = 0.3x^2 - 156x + 26.657

Finding the derivative:

c'(x) = 0.6x - 156

0.6x - 156 = 0   for  maxm/minm cost.

x = 156 / 0.6

= 260

The second derivative  is positive (0.6) so this is a minimum.

Answer:

[tex]x=260\ machines[/tex]

Step-by-step explanation:

Note that we have a cudratic function of negative principal coefficient.

The minimum value reached by this function is found in its vertex.

For a quadratic function of the form

[tex]ax ^ 2 + bx + c[/tex]

the x coordinate of the vertex is given by the following expression

[tex]x=-\frac{b}{2a}[/tex]

In this case the function is:

[tex]C(x) = 0.3x^2 - 156x + 26,657[/tex]

So:

[tex]a=0.3\\b=-156\\c=26,657[/tex]

Then the x coordinate of the vertex is:

[tex]x=-\frac{-156}{2(0.3)}[/tex]

[tex]x=260\ machines[/tex]

Then the number of machines that must be made to minimize the cost is:

[tex]x=260\ machines[/tex]

Answer: 0.1310

Step-by-step explanation:

Given : Mean : [tex]\mu = \text{131 millimeters}[/tex]

Standard deviation :  [tex]\sigma = \text{7 millimeters}[/tex]

Sample size : [tex]n=31[/tex]

To find the probability that the sample mean would differ from the population mean by more than 1.9 millimeters i.e. less than 129.1 milliliters and less than 132.9 milliliters.

The formula for z-score :-

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

For x = 129.1 milliliters

[tex]z=\dfrac{129.1-131}{\dfrac{7}{\sqrt{31}}}\approx-1.51[/tex]

For x = 132.9 milliliters

[tex]z=\dfrac{132.9-131}{\dfrac{7}{\sqrt{31}}}\approx1.51[/tex]

The P-value= [tex]P(x<-1.51)+P(x>1.51)[/tex]

[tex]=2P(z>1.51)=2(1-P(z<1.15))\\=2(1-0.9344783)\\=0.1310434\approx0.1310[/tex]

Hence, the required probability = 0.1310

The probability that the sample mean would differ from the population mean by more than 1.9 millimeters is approximately 0.1744.

To find the probability that the sample mean would differ from the population mean by more than 1.9 millimeters, we can use the Z-score formula. The Z-score is calculated by subtracting the population mean from the sample mean and dividing by the standard deviation divided by the square root of the sample size. Once we have the Z-score, we can use a Z-table or calculator to find the probability.

In this case, the Z-score is (1.9 - 0) / (7 / sqrt(31)) = 0.935. To find the probability of a Z-score greater than 0.935, we can look up the corresponding area under the normal distribution curve in a Z-table or use a calculator. The probability is approximately 0.1744.

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Answer:

See below.

Step-by-step explanation:

1.   Suppose that the sum is rational  then we can write:

a/b + i = c/d     where i is irrational and by definition a/b and c/d are rational.

Rearranging:

i = c/d - a/b

Now the sum on the right is rational  so 'irrational' = 'rational' which is a contradiction.

So  the original supposition is false and the sum must be irrational.

2. Proof of For all integers m if m is even then 3m + 7 is odd:

If m is even then 3m is even.

Suppose 3m + 7 is even, then:

3m + 7 = 2p  where p is an integer.

3m - 2p  = -7

But 3m and 2p are both even so their result is even  and -7 is odd.

Therefore the original supposition is false because it leads to a contradiction,  so 3m + 7 is odd.

No idea what the cited section's method is, but this ODE is linear:

[tex]\dfrac{\mathrm dy}{\mathrm dx}=4+y-4x+5[/tex]

[tex]\dfrac{\mathrm dy}{\mathrm dx}-y=9-4x[/tex]

Multiply both sides by [tex]e^{-x}[/tex] so that the left side can be condensed as the derivative of a product:

[tex]e^{-x}\dfrac{\mathrm dy}{\mathrm dx}-e^{-x}y=(9-4x)e^{-x}[/tex]

[tex]\dfrac{\mathrm d}{\mathrm dx}\left[e^{-x}y\right]=(9-4x)e^{-x}[/tex]

Integrating both sides gives

[tex]e^{-x}y=(4x-5)e^{-x}+C[/tex]

[tex]\implies\boxed{y(x)=4x-5+Ce^x}[/tex]

Answer: 0.05

Step-by-step explanation:

Given : Interval for uniform distribution : [46.0 minutes, 56.0 minutes]

The probability density function will be :-

[tex]f(x)=\dfrac{1}{56-46}=\dfrac{1}{10}=0.1\ \ , 46<x<56[/tex]

The probability that a given class period runs between 50.75 and 51.25 minutes is given by :-

[tex]P(50.75<x<51.25)=\int^{51.25}_{50.75}f(x)\ dx\\\\=(0.1)[x]^{51.25}_{50.75}\\\\=(0.1)(51.25-50.75)=0.05[/tex]

Hence, the probability that a given class period runs between 50.75 and 51.25 minutes = 0.05

Answer:

2063.9 kg

Step-by-step explanation:

Given,

1 kg = 2.2046 pounds

This can also be written as:

1 pound = 1/2.2046 kg

We have to calculate the value of 4550 pounds in kg up to nearest 10th

Thus,

[tex]4550 pounds= \frac{4550}{2.2046} kg[/tex]

Solving the above equation, we get:

4550 pounds = 2063.8664 kg

Rounding the above result to nearest tenth as:

4550 pounds = 2063.9 kg

Answer:

433 minutes

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.

The nurse is infusing 1.5 ml of solution every x minutes, Since there are a total of 650 ml we can divide the amount the nurse is infusing every x minutes by the total in order to calculate the amount of time it will take the nurse to finish.

[tex]\frac{650}{1.5} = 433.33min[/tex]

So we can see that it will take the nurse 433 min to infuse all of the solution.

Answer:

In 433.33 minutes will it take for the medicine to be given.

Step-by-step explanation:

Given : A nurse must infuse 1.5 ml of solution in x minutes and she has 650 ml of solution.

To find : How many minutes will it take for the medicine to be given ?

Solution :

As 1.5 ml of solution infuse in x minutes

i.e. the amount of solution she has = [tex]1.5x[/tex]

Now, she has 650 ml of solution.

which means  [tex]1.5x=650[/tex]

Solving the equation,

[tex]x=\frac{650}{1.5}[/tex]

[tex]x=433.33[/tex]

Therefore, in 433.33 minutes will it take for the medicine to be given.

Answer:

She earns that day $103.5

Step-by-step explanation:

Lets explain how to solve the question

- She is paid $12.50 per hour between 8 am and 4 pm

- After 4 pm she is paid $14.25 per hour

- She works one particular day from 10 am to 6 pm

* Lets distribute her time of work into two part according to her

 payment above

# 1st part from 10 am to 4 pm

∵ She is paid $12.50 per hour between 8 am to 4 pm

∵ She works from 10 am to 4 pm

- Calculate how many hours between 10 am and 4 pm

∵ am and pm not like terms in time, so change the time to 24-hours

∵ 4 pm in 24-hour time is 4 + 12 = 16

∴ She works form 10 to 16

∴ She works for 6 hours (16  - 10 = 6)

∴ She is paid = 12.50 × 6 = $75

# 2nd part after 4 pm to 6 pm

∵ She is paid $14.25 per hour after 4 pm

∵ She works from 4 pm to 6 pm

- Calculate how many hours between 4 pm and 6 pm

∵ Both times are pm

∴ She works for 2 hours (6 - 4 = 2)

∴ She is paid = 14.25 × 2 = $28.5

- Add the two answers of the 1st part and the 2nd part

∴ She earns = 75 + 28.5 = $103.5

* She earns that day $103.5

Answer:

$103.50.

Step-by-step explanation:

From 10 am until 4 pm ( 6 hours) she earns 6 * 12.50 = $75.00.

Then from 4 pm to 6 pm she earns 2*14.25 = $28.50.

Total for that day = 75 + 28.50

= $103.50.

Answer:

(-∞, -1 )∪(3, ∞)

Step-by-step explanation:

Given function,

[tex]f(x) = x^3 - 3x^2 - 9x + 4[/tex]

Differentiating with respect to x,

[tex]f'(x)=3x^2-6x-9[/tex]

For increasing or decreasing,

f'(x) = 0

[tex]\implies 3x^2-6x-9=0[/tex]

By quadratic formula,

[tex]x=\frac{-(-6)\pm \sqrt{(-6)^2-4\times 3\times -9}}{6}[/tex]

[tex]=\frac{ 6\pm \sqrt{36+108}}{6}[/tex]

[tex]=\frac{6\pm \sqrt{144}}{6}[/tex]

[tex]\frac{6\pm 12}{6}[/tex]

[tex]\implies x = 3\text{ or } x = -1[/tex]

In interval (-∞, -1 ), f'(x) = positive,

⇒ f(x) is increasing on (-∞, -1 ),

In interval (-1, 3), f'(x) = negative,

⇒ f(x) is decreasing on (-1, 3),

In interval (3, ∞), f'(x) = positive,

⇒ f(x) is increasing on (3, ∞),

To find the interval where a function is increasing, find the derivative, then set it greater than 0 and solve for x to get the critical points. These points split the domain into intervals which are classified as increasing or decreasing by plugging into the derivative. Increasing intervals are then written in interval notation.

The subject of the question is about finding the interval on which the function f(x) = x^3 − 3x^2 − 9x + 4 is increasing. This involves the concept of determining increasing and decreasing intervals in calculus, which analyzes the slope of the function's graph.

To find the intervals, we first find the derivative of the function, f'(x) which gives us the slopes of the tangent lines at any point x. The function f(x) = x^3 - 3x^2 - 9x + 4 becomes f'(x) = 3x^2 - 6x - 9. Set f'(x) > 0 to figure out when the function is increasing. Solve for x gives the critical points, which splits the domain of f into intervals. Testing these intervals in the derivative equation helps us classify them as increasing or decreasing.

The final step is writing the increasing intervals in the proper notation, usually in the form of (a, b) where 'a' and 'b' are the endpoints of the interval where the function is increasing. If an interval goes to infinity, you would use a notation like (a, ∞).

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Explanation of probabilities for a worthy research proposal approved by two panels, disapproved by two panels, and approved by one panel.

A worthy proposal is approved by both panels:

Probability = (Probability approved by Panel 1) × (Probability approved by Panel 2) = 0.2 × 0.7 = 0.14

A worthy proposal is disapproved by both panels:

Probability = (Probability disapproved by Panel 1) × (Probability disapproved by Panel 2) = 0.8 × 0.3 = 0.24

A worthy proposal is approved by one panel:

Probability = (Probability approved by Panel 1 and disapproved by Panel 2) + (Probability disapproved by Panel 1 and approved by Panel 2) = (0.2 × 0.3) + (0.8 × 0.7) = 0.06 + 0.56 = 0.62

Final answer:

The probability of picking the double-sided quarter is 1/100. The probability of getting heads on the 8th flip given that the previous 7 flips were all heads is still 1/2.

Explanation:

Let's calculate the probability of picking the double-sided (heads) quarter first.

Out of the 100 quarters, only 1 is double-sided. Therefore, the probability of picking the double-sided quarter is 1/100.

Now, let's calculate the probability of getting heads on the 8th flip given that the previous 7 flips were all heads.

The probability of getting heads on a single flip is 1/2, since there are two possible outcomes (heads or tails) and both are equally likely.

Since the flips are independent, the probability of getting heads on the 8th flip given that the previous 7 flips were all heads is still 1/2.

First compute the probability that a bill would exceed $115. Let [tex]X[/tex] be the random variable representing the value of a monthly utility bill. Then transforming to the standard normal distribution we have

[tex]Z=\dfrac{X-100}{10}[/tex]

[tex]P(X>115)=P\left(Z>\dfrac{115-100}{10}\right)=P(Z>1.5)\approx0.0668[/tex]

Then out of 300 randomly selected bills, one can expect about 6.68% of them to cost more than $115, or about 20.

Both the teams are held at their place by the friction between the ground and their feet. If the pulling force as mentioned above exceeds this friction, the losing team experiences a net force and accelerates.

If object A exerts a force on object B, then object B must exert a force of equal magnitude and opposite direction back on the object.

When a team pulls the rope, the rope pulls the team.

This was Newton's third law.

When both teams pull in opposite directions, they often don't exert equal forces; the stronger team exerts a larger force.

So the two forces don't balance each other and the rope accelerates towards the stronger team.

The weaker team experiences a net force pulling them.

This force is the sum of (i)The reaction by the rope as mentioned above and (ii) The friction between the rope and the team members' hands.

Both the teams are held at their place by the friction between the ground and their feet. If the pulling force as mentioned above exceeds this friction, the losing team experiences a net force and accelerates.

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Newton's Third Law explains the forces exerted in a tug-of-war, but the winning team is often determined by factors like friction and the angle of pulling. Greater friction helps a team resist the opposing force, while a lower angle of pulling may provide an advantage in distributing the tug's force.

Even though Newton's Third Law of Motion states that for every action there is an equal and opposite reaction, this does not mean that all forces will cancel each other out. In the context of a tug-of-war, friction and the angle in which the tug is exerted can influence outcomes.

The tug-of-war is not only a battle of forces but also a battle against friction. Each team is pulling as hard as they can in opposite directions, establishing the action-reaction pair. However, their ability to maintain their ground or pull the other team relies significantly on the friction between their feet and the ground. If either team can increase this friction (e.g., by wearing shoes with better grip, or planting their feet more effectively), they can resist the pulling force of the other team, leading to their victory.

Moreover, the angle at which a team pulls the rope can also determine the advantage in a tug of war. Teams that manage to keep the rope lower to the ground distribute the force more efficiently and therefore have a greater chance of winning.

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Answer:

17 cm

Step-by-step explanation:

In the above figure the question has been explained

the perpendicular from circle on the chord divides chord into two halves

let r be radius of circle in triangle NOY

NO= PO-PN = r-2 and YO=r(radius) YN=8cm

applying pythagorus theorem

[tex]r^{2} =8^{2} +(r-2)^{2}[/tex]

after rearranging and solving the equation we get

r=17 cm

therefore the radius of the triangle is  17 cm

note: on observing we find that the sides of triangle are pythagorian triplet 17, 15 and 8

The radius of the circle is calculated using the Pythagorean theorem for the right triangle formed by the perpendicular bisector, chord, and radius. By finding the length of the radius from the center to the midpoint of the chord and using NP as 2 cm, we calculate the radius to be approximately 8.246 cm.

The student has asked us to calculate the radius of the circle where a perpendicular bisector of a chord XY cuts through the chord at point N and the circle at P, with XY measuring 16 cm and NP measuring 2 cm. To solve this we can apply the theorem stating that in a circle, the radius bisecting an angle at the center is perpendicular to the chord that subtends the angle and bisects this chord. Since NP is half the distance of XY, we can conclude that N is the midpoint of XY, making XD (XD being the length from one end of the chord to its perpendicular bisector) equal to 8 cm. Considering triangle FNP, where F is the center of the circle, FN is the radius, and NP is 2 cm, we can use the Pythagorean theorem to find the radius FN:

FN2 = NP2 + FN2

FN2 = 82 + 22

FN2 = 64 + 4
FN2 = 68
FN = √68
FN ≈ 8.246 cm
Therefore, the radius of the circle is approximately 8.246 cm.

Answer:

Explanation:

The little red square indicates that the angle C measures 90°, which means that ABC is a right triangle.

The opposite side to angle C is the hypotenuse of the triangle. Then, the hypotenuse measures 9.8 in.

The requested measure, m ∠ ABC, is the measure of the angle B.

With respect to angle B you have:

Thus, the trigonometric ratios for angle B are:

Finally, to find the measures of angle B, you just need to find the inverse functions, which are:

Answer:A&E

Step-by-step explanation:

JUST TOOK TEST

Answer: a) The probability is approximately = 0.5793

b) The probability is approximately=0.8810

Step-by-step explanation:

Given : Mean : [tex]\mu= 62.5\text{ in}[/tex]

Standard deviation : [tex]\sigma = \text{2.5 in}[/tex]

a) The formula for z -score :

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

Sample size = 1

For x= 63 in. ,

[tex]z=\dfrac{63-62.5}{\dfrac{2.5}{\sqrt{1}}}=0.2[/tex]

The p-value = [tex]P(z<0.2)=[/tex]

[tex]0.5792597\approx0.5793[/tex]

Thus, the probability is approximately = 0.5793

b)  Sample size = 35

For x= 63 ,

[tex]z=\dfrac{63-62.5}{\dfrac{2.5}{\sqrt{35}}}\approx1.18[/tex]

The p-value = [tex]P(z<1.18)[/tex]

[tex]= 0.8809999\approx0.8810[/tex]

Thus , the probability is approximately=0.8810.

Answer:

Step-by-step explanation:

(A-B)U (A∩B)  [   We know that difference of A and B is A∩B')

A∩B')U(A∩B)    

A∩(B'UB)      (Using Distributive law)

A∩(U)            [ Union of a set and its complement is always a universal set]

A                        Hence proved

Answer:

75.

Step-by-step explanation:

Let x be the original price. 20% = 0.2. The price with the discount is 60, that is

x-x*0.2 = 60

x(1-0.2) = 60 using common factor,

x (0.8) = 60

x = 60/0.8

x = 75.

So, the original price is $75.

The original price of an item discounted 20% with a sale price after the discount of $60 is $75.

A discounted price is the marked-down price of an item.

The discounted price represents the selling price after reducing it with the discount.

Discount rate = 20%

Discounted price = $60

Original price = $75 ($60/1-20%)

Thus, the original price of an item discounted 20% with a sale price after the discount of $60 is $75.

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[tex]\vec F[/tex] is conservative if we can find a scalar function [tex]f[/tex] such that [tex]\nabla f=\vec F[/tex]. This would require

[tex]\dfrac{\partial f}{\partial x}=5y^2z^3[/tex]

[tex]\dfrac{\partial f}{\partial y}=10xyz^3[/tex]

[tex]\dfrac{\partial f}{\partial z}=15xy^2z^2[/tex]

Integrate both sides of the first PDE with respect to [tex]x[/tex]:

[tex]f(x,y,z)=5xy^2z^3+g(y,z)[/tex] (*)

Differentiate both sides of (*) with respect to [tex]y[/tex]:

[tex]\dfrac{\partial f}{\partial y}=10xyz^3=10xyz^3+\dfrac{\partial g}{\partial y}\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)[/tex]

Differentiate both sides of (*) with respect to [tex]z[/tex]:

[tex]\dfrac{\partial f}{\partial z}=15xy^2z^2=15xy^2z^2+\dfrac{\mathrm dh}{\mathrm dz}\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C[/tex]

So we have

[tex]f(x,y,z)=5xy^2z^3+C[/tex]

and so [tex]\vec F[/tex] is indeed conservative.

Answer:

Option A (134.7mm)

Step-by-step explanation:

Let's find the distance, but first we need to remember that the distance between two points with coordinates (Xa,Ya) and (Xb,Yb) is defined by:

[tex]distance = \sqrt{(Xb-Xa)^{2} + (Yb-Ya)^{2}  }[/tex]

From the situation we notice that:

Xb=31.45 and Xa=65.35, as well as:

Yb=-55.50 and Ya=74.88

Using the previous equation we have:

[tex]distance = \sqrt{(31.45-65.35)^{2} + (-55.50-74.88)^{2}  }[/tex]

[tex]distance = \sqrt{(-33.9)^{2} + (-130.38)^{2}  }[/tex]

[tex]distance = \sqrt{1149.21 + 16998.9444}[/tex]

[tex]distance = \sqrt{18148.1544}[/tex]

[tex]distance = 134.7151mm[/tex]

In conclusion, the distance between points (65.35,74.88) and (31.45,-55.50) is 134.7151mm, which is option A (134.7mm).

Answer:

The value of g(4) is 2.

Step-by-step explanation:

Given,

The one-to-one functions g is defined as,

g = {(2,4), (4, 2), (5,-1), (8, -9)}

The element of a function is written in the form of order pair where,  the first element is input value and second element is the output value.

Since, under the function g the output value of input value 2 is 4,

That is, we can write,

g(2) = 4

Hence, the value of g(4) is 2.

The correct statements regarding the sets A and B is:

        II.

[tex]A\bigcap B=\{13,15\}[/tex]

We are given set A as:

[tex]A=\{3,4,13,15\}[/tex]

and set B as:

[tex]B=\{13,15,21,25,30,37\}[/tex]

and the three statements are given by:

I.

[tex]A\bigcup B=\{12,4,21,25,30,37\}[/tex]

II.

[tex]A\bigcap B=\{13,15\}[/tex]

III.

[tex]A\subset B[/tex]

i.e. from the given sets we have:

[tex]A\bigcup B=\{3,4,13,15,21,25,30,37\}[/tex]

i.e.

[tex]A\bigcap B=\{13,15\}[/tex]

All the elements of A are contained in B.

But from the given sets we see that:

3,4 belongs to A but they do not belong to B.

Hence, A is not a subset of B.

Answer:

$333.17

Step-by-step explanation:

Use the compounding formula

[tex]A(t)=P(1+r)^t[/tex]

where A(t) is the amount at the end of the compounding,

P is the initial deposit,

r is the interest rate in decimal form, and

t is the time in years.

Filling in our info:

[tex]A(t)=180(1+.08)^8[/tex]

Simplify a bit to

[tex]A(t)=180(1.08)^8[/tex]

Raise 1.08 to the 8th power and get

A(t) = 180(1.85093021) and then multiply to get

A(t) = $333.17

[tex]\bf \textit{Pythagorean Identities} \\\\ sin^2(\theta)+cos^2(\theta)=1 \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]

[tex]\bf tan(\theta )+cot(\theta )=sec(\theta )csc(\theta ) \\\\[-0.35em] ~\dotfill\\\\ tan(\theta )+cot(\theta )\implies \cfrac{sin(\theta )}{cos(\theta )}+\cfrac{cos(\theta )}{sin(\theta )}\implies \cfrac{sin(\theta ) sin(\theta )+cos(\theta )cos(\theta )}{cos(\theta )sin(\theta )} \\\\\\ \cfrac{sin^2(\theta )+cos^2(\theta )}{cos(\theta )sin(\theta )}\implies \cfrac{1}{cos(\theta )sin(\theta )}\implies \cfrac{1}{cos(\theta )}\cdot \cfrac{1}{sin(\theta )}\implies sec(\theta )csc(\theta )[/tex]