A hypothesis test is to be conducted using LaTeX: \alphaα = 0.05. This means:_________.
a) there is a 5 percent chance that a Type II error has been committed.
b) there is a 5 percent chance that the alternative hypothesis is true.
c) there is a maximum 5 percent chance that a true null hypothesis will be rejected.
d) there is a 5 percent chance that the null hypothesis is true.

Answer:

600 in^2.

Step-by-step explanation:

There are 6 faces on a cube so the area we need is:

6 * 10 * 10

= 600 in^2.

Answer:

114.29 ft

Step-by-step explanation:

tan ∅ = Opp/Adj

tan 77 = x/25

x = 25 tan 77

x = 108.29ft

Plus 6ft = 114.29 ft

Answer:

Hence The composition [tex](fog)(x)[/tex] of the given function is [tex]x^2+12x+29[/tex].

Step-by-step explanation:

Given:

[tex]f(x) = x^2+2x-6[/tex]

[tex]g(x)=x+5[/tex]

We need to find [tex](f o g)(x)[/tex].

Solution:

Now we can say that;

[tex](f o g)(x)[/tex] = [tex]f(g(x))[/tex]

[tex](fog)(x) = (x+5)^2+2(x+5)-6[/tex]

Now Applying distributive property we get;

[tex](fog)(x) = (x+5)^2+2\times x+2\times5-6\\\\(fog)(x) = (x+5)^2+2x+10-6\\\\(fog)(x) = (x+5)^2+2x+4[/tex]

Now Solving the exponent function we get;

[tex](fog)(x) = x^2+2\times x\times 5+5^2+2x+4\\\\(fog)(x) = x^2+10x+25+2x+4\\\\(fog)(x) = x^2+12x+29[/tex]

Hence The composition [tex](fog)(x)[/tex] of the given function is [tex]x^2+12x+29[/tex].

The given statement "All occurrences of the letter u in "Discrete Mathematics" are lowercase" is true.

Here's why:

There are no occurrences of the letter "u" in "Discrete Mathematics" at all.

Therefore, the question of whether they are uppercase or lowercase becomes irrelevant due to the absence of the letter itself.

Because the statement involves a vacuous quantification, meaning it deals with an empty set, it automatically becomes true.

In such cases, it doesn't matter what property is being attributed to the empty set because there are no elements for that property to be true or false for.

Answer:

DC = 10.93 cm ,  AC = 9.8 cm

Step-by-step explanation:

From trigonometry;

⇒ Tan 60 = AB/BD

⇒AB = BD Tan 60 ( where BD = 4 cm )

⇒ AB = 6.93 cm

Also, AB=BC , therefore;

⇒ BC = 6.93 cm

⇒ Cos 60 = BD/AD

⇒ AD = BD/ Cos 60 = 4/Cos 60

⇒ AD = 8 cm

From Pythagoras theorem;

⇒ [tex]AC^{2}[/tex] = [tex]AB^{2}[/tex] + [tex]BC^{2}[/tex] = [tex](6.93)^{2}[/tex] + [tex](6.93)^{2}[/tex]

⇒ AC = [tex]\sqrt{96.05}[/tex] = 9.80 cm

⇒ DC = BD + BC = 4 + 6.93

⇒ DC = 10.93 cm

Answer:

August=$350

September=$560

October=$210

November= $280

Step-by-step explanation:

Total amount left to save $1400.

We calculate the amount save each mount since we are already given the percentages.

For the month of August she saves 25% of $1400, we convert the percentage to equivalent cash

[tex]\frac{25}{100}*1400\\=350[/tex]

for the month of August, she saved $350,

Next For the month of September she saves 40% of $1400, we convert the percentage to equivalent cash

[tex]\frac{40}{100}*1400\\=560[/tex]

for the month of September, she saved $560,

Next For the month of October she saves 15% of $1400, we convert the percentage to equivalent cash

[tex]\frac{15}{100}*1400\\=210[/tex]

for the month of October, she saved $210,

total amount saved by October = $350+$560+$210=$1120

Amount saved by November=1400-1120=$280

Answer: 1 year and 4 months

Step-by-step explanation:

The formula for continuously compounded interest is

A = P x e (r x t)

Where

A represents the future value of the investment after t years.

P represents the present value or initial amount invested

r represents the interest rate

t represents the time in years for which the investment was made.

e is the mathematical constant approximated as 2.7183.

From the information given,

P = 8000

r = 3% = 3/100 = 0.03

A = 8327

Therefore,

8327 = 8000 x 2.7183^(0.03 x t)

8327/8000 = 2.7183^(0.03t)

1.040875 = 2.7183^(0.03t)

Taking log of both sides, it becomes

Log 1.040875 = log 2.7183^(0.03t)

0.0174 = 0.03tlog2.7183

0.0174 = 0.03t × 0.434 = 0.01302t

t = 0.0174/0.01302

t = 1.336

0.336 × 12 = 4.032

Therefore, the money waa in the account for 1 year and 4 months

Answer:

Q1:  p = - 33

Q2: d = - 99

Q3: t = - 13

Step-by-step explanation:

Q1: [tex]$ \textbf{-} \frac{\textbf{p}}{\textbf{3}} \hspace{1mm} \textbf{-} \hspace{1mm} \textbf{8} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{3}[/tex]

We solve this taking LCM.

We get: [tex]$ \frac{-p - 24}{3} = 3 $[/tex]

[tex]$ \implies - p - 24 = 9 $[/tex]

[tex]$ \implies \textbf{p} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{- 33} $[/tex]

Q4: [tex]$ \frac{\textbf{d}}{\textbf{11}} \hspace{1mm} \textbf{-} \hspace{1mm} \textbf{4} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{- 13} $[/tex]

Again we proceed like Q1 by taking LCM.

We get: [tex]$ \frac{d - 44}{11} = - 13 $[/tex]

[tex]$ \implies d - 44 = - 13 \times 11 = - 143 $[/tex]

[tex]$ \implies d = - 143 + 44 $[/tex]

[tex]$ \implies \textbf{d} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{- 99} $[/tex]

Q7: 5t + 12 = 4t - 1

We club the like terms on either side.

[tex]$ \implies 5t - 4t = - 1 - 12 $[/tex]

[tex]$ \implies (5 -4)t = - 13 $[/tex]

[tex]$ \implies \textbf{t} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{- 13} $[/tex]

Hence, the answer.

Answer: A. Approximately 0.017 pounds

Step-by-step explanation:

Formula to find the margin of error :

[tex]E=z^*\dfrac{s}{\sqrt{n}}[/tex] , where z* = critical value for confidence interval , s= standard deviation , n= sample size.

As per given , we have

s= 0.15 pound

n= 300

Critical value for 95% confidence = 1.96

Then, Margin of error for 9%% confidence interval will be :

[tex]E=(1.96)\dfrac{0.15}{\sqrt{300}}\\\\=0.0169740979142\approx0.017[/tex]

Hence, they can expect a margin error of 0.017 pound (approximately.)

Thus , the correct answer is A. Approximately 0.017 pounds

The radius of the cone is found by comparing the ratios of heights and radii in a similar triangle. By applying the given values, the radius of the cone is found to be 9 cm.

In mathematics, problems related to finding the dimensions of cones and frustums are common. As the problem mentions, a frustum is created when a smaller, similar cone is removed from a larger cone. To calculate the dimensions of the frustum, we use the properties of similar triangles.

For similar triangles:

Therefore:

Applying the given values, it becomes:

Hence, the radius of the cone is 9 cm.

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Final answer:

The volume of a frustum can be calculated using the formula [tex]\(V = \frac{1}{3}\pi h (r_{1}^{2} + r_{2}^{2} + r_{1}r_{2})\)[/tex]. Given the dimensions provided for the frustum with the larger cone having a radius of 4.5 cm, the frustum a radius of 3 cm, and height of 3 cm, the volume of the frustum is approximately 134.25π cubic centimeters.

Explanation:

To work out the volume of a frustum that is formed by removing a small cone from a larger, similar cone, you can make use of the formula for the volume of a frustum:

V = [tex]\(\frac{1}{3}\pi h (r_{1}^{2} + r_{2}^{2} + r_{1}r_{2})\),[/tex]

where:

h is the height of the frustum,

r₁ is the radius of the larger base (radius of the original cone),

r₂ is the radius of the smaller base (radius of frustum).

Given:

Radius of the large cone (r₁) = 4.5 cm,

Radius of the frustum (r₂) = 3 cm,

Height of the frustum (h) = 3 cm.

The height of the original cone (9 cm) is not needed to calculate the volume of the frustum. Plugging the given values into the formula:

[tex]\(V = \frac{1}{3}\pi \times 3 \times (4.5^{2} + 3^{2} + 4.5 \times 3)\)[/tex]

[tex]\(V = \pi (20.25 + 9 + 13.5)\)[/tex]

[tex]\(V = \pi (42.75)\)[/tex]

[tex]\(V = 134.25\pi \text{cm}^{3}\)[/tex]

Therefore, the volume of the frustum is approximately 134.25π cubic centimeters.

The question involves creating numerical expressions to represent the total number of fruits in a basket after adding 4 peaches. By assuming the initial fruit count as x, two expressions are x + 4, representing the total number of fruits, and 2(x + 4), indicating a scenario where the total fruit count doubles after adding the peaches.

The question asks for two new numerical expressions to represent the total number of fruit in the basket, given that Paul added 4 peaches. To create these expressions, we need to assume there was an initial number of fruit in the basket before Paul added the peaches. Let's denote the initial number of fruit as x. Therefore, our two new numerical expressions could be:

These examples show how basic algebraic expressions can be used to represent situations involving changes in quantity.

Answer:

Therefore the team saves $5.18.

Step-by-step explanation:

i) there are two triangular pyramids

ii) the base of each pyramid is an equilateral triangle with an area of 10.8 square feet.

iii) the base of the pyramids are not covered with foil.

iv) since there are two pyramids the total area not covered

    = 10.8 [tex]\times[/tex] 2 = 21.6 square feet

v) the cost of foil = $0.24 per square foot.

vi) the team save altogether by covering only the lateral area of the two pyramids

  =  21.6 [tex]\times[/tex] 0.24 = $5.18

Therefore the team saves $5.18.

Answer: FALSE

Step-by-step explanation: Statistics is the Science of collecting, organising,sumarising,analysing information to reach at a reasonable conclusion or outcome. Statistics also helps to measure levels of confidence in any conclusion or outcome.

Statistics has been applied in several fields like Medicine, pharmacy, Engineering,Biology etc it helps in determining especially in numerical representation the impact of certain conditions or activities or information.

Answer:

3 packeges of Pamphlets and 5 packages of dog biscuits.

Step-by-step explanation:

Peter have to distribute a pamphlet with one dog buscuit to each person they need equal amount of pamphlets and biscuitsby taking LCM of both 12 and 20 theanswer is 60. Peter needs 60 pamphlets and 60 dog biscuits. So he require 3packages of pamphlets  and 5packages of dog biscuits.

Answer:yes

Step-by-step explanation: the answer does represent a function because if you were to put points in the line wherever, and connect them by drawing a line vertically, it would not cross two points.

Juan does not have enough paint to cover shape of a right rectangle prism

Solution:

Given that,

Juan wants to paint something in the shape of a right rectangle prism

From given,

Length = 17 inches

Width = 11 inches

Height = 9 inches

The surface area of prism is given as:

[tex]A=2(wl+hl+hw)[/tex]

Where, "l" is the length and "w" is the width and "h" is the height

Substituting the values we get,

[tex]A = 2(11 \times 17 + 9 \times 17 +9 \times 11)\\\\A = 2(187+153+99)\\\\A = 2 \times 439 \\\\A = 878[/tex]

Thus surface area of prism is 878 square inches

Given that,

He had enough paint to cover 850 sq in

No he cannot paint since surface area is 878 square inches which is greater than 850 square inches

So, he does not have enough paint

Answer:

Step-by-step explanation:

The sum of the angles on a straight line is 180 degrees. Therefore,

Angle XYZ + angle MYZ = 180

Angle XYZ + 115 = 180

Angle XYZ = 180 - 115 = 65 degrees

The sum of the angles in a triangle is 180 degrees. It means that

Angle XZY + angle YXZ + angle MYZ = 180

Therefore,

4k + 5 + 6k + 10 + 65 = 180

4k + 6k + 5 + 10 + 65 = 180

10k + 80 = 180

10k = 180 - 80 = 100

Dividing the left hand side and the right hand side of the equation by 10, it becomes

10k/10 = 100/10

k = 10

Answer:

Students got 3 signature on the first day.

Step-by-step explanation:

We are given the following in the question:

The number of signatures they get each day is 3 times as many as the day before.

Thus, the number of signature forms a G.P with common ratio, r = 3.

The expression 3 to the 6th power represents the number of signatures they got on the sixth day. Thus we can write:

[tex]a_6 = 3^6[/tex]

The [tex]n^{th}[/tex] term in a G.P is given by

[tex]a_n = a_1r^{n-1}[/tex]

where [tex]a_1[/tex] is the first term in the G.p

Putting values, we get,

[tex]3^6 = a_1(3)^{6-1}\\3^6 = a_1(3)^5\\\Rightarrow a_1 = 3[/tex]

Thus, the first term of G.P is 3.

Hence, students got 3 signature on the first day.

Answer:

The equation would be [tex]4x-15=21[/tex].

Step-by-step explanation:

Given;

Total Number of Video games = 21

Let the number of video game Nicholas has be 'x'.

Now Given:

Robert has 5 fewer games than Nicholas.

so we can say that;

number of video game Robert has = [tex]x-5[/tex]

Also Given:

Charlie has twice as many games as Robert.

so we can say that;

number of video game Charlie has = [tex]2(x-5)=2x-10[/tex]

we need to write the equation.

Solution:

Now we can say that;

Total Number of Video games is equal to sum of number of video game Nicholas has, number of video game Robert has and number of video game Charlie has.

framing the equation we get;

[tex]x+x-5+2x-10=21\\\\4x-15=21[/tex]

Hence The equation would be [tex]4x-15=21[/tex].

On Solving we get;

Adding both side by 15 we get;

[tex]4x-15+15=21+15\\\\4x=36[/tex]

Dividing both side by 4 we get;

[tex]\frac{3x}{4}=\frac{36}{4}\\\\x=9[/tex]

Hence Nicholas has = 9 video games

Robert has = [tex]x-5= 9-5=4[/tex] video games

Charlie has = [tex]2x-10 = 2\times9-10=18-10=8[/tex] video games

Answer:

The equation would be  [tex]4x-15=21.[/tex]

Step-by-step explanation:

Given:

Three brothers Charlie Robert and Nicholas have 21 video games Charlie has twice as many games as Robert. Robert has 5 fewer games than Nicholas.

Now, to find the equation.

Let the games Nicholas has be [tex]x.[/tex]

Robert has games [tex]x-5.[/tex]

And Charlie has [tex]2(x-5).[/tex]

Now, the equation is:

[tex](x)+(x-5)+2(x-5)=21.[/tex]

[tex]x+x-5+2x-10=21\\2x-5+2x-10=21\\4x-15=21[/tex]

[tex]4x-15=21.[/tex]

Therefore, the equation would be [tex]4x-15=21.[/tex]

Answer:

The methods that are fair include __A, B and C__ because __D__ is flawed due to increased chances of winning based on one's skill, ___F___ is flawed due to poor randomization, and ___E___ is flawed due to unequal probabilities of winning and losing.

Step-by-step explanation:

A. Place both names in equal amounts into a hat and draw one without looking:

If you place both names in equal amounts into a hat, then the probability of picking any one of the names would be equal.

For example, if you put 20 pieces of paper containing each name in a hat, then,there will be 40 names in the hat. The probability of picking any one or the other randomly would be:

P = 20/40 =1/2

B. Ask a stranger to flip a coin:

A coin has only two faces, head and tail. Hence, if one person picks head and the other picks tail, the probability of landing on either head or tail is given as:

P = 1/2

C. Roll a die and evaluate the outcome as either even or odd:

A die has 6 faces with equal number of faces with equal number of even numbers and odd numbers, 3. Hence, the probability of rolling an even or odd number is given as:

P = 3/6 = 1/2

D. Throw a stone closest to an object:

This method is not fair because then it depends solely on who can throw the farthest i.e. the physical ability of the throwers.

E. Play a hand of blackjack

This method is flawed because it has unequal probabilities of winning and losing.

F. Ask a random stranger to select:

This method is unfair because it then depends solely on the random picker. The picker could have personal preferences or a bias based on maybe height, beauty, weight, basically anything characteristic. Hence, it is unfair.

Final answer:

Fair methods for deciding who gets the prize include placing both names in a hat, flipping a coin, and rolling a die as they provide equal chances of winning, while throwing a stone, playing blackjack, or asking a stranger can introduce bias or skill, making them unfair.

Explanation:

The methods that are fair include placing both names in a hat, asking a stranger to flip a coin, and rolling a die because these methods provide equal probabilities of winning or losing. Method D: Throwing a stone closest to an object is flawed due to increased chances of winning based on one's skill, which means it's not random. E: Playing a hand of blackjack is flawed due to poor randomization, as the cards dealt can create significantly different chances of winning. F: Asking a random stranger to select is flawed due to unequal probabilities of winning and losing if the stranger holds a bias, even if unintended.

Tossing a coin is a fair way to decide because both outcomes (heads or tails) have equal chances of occurring, which is a 50% chance each way. This is consistent with the principle of randomness and independent events, where the outcomes of past coin tosses do not influence the results of future tosses.

Answer:

There is a strong, negative correlation between the amount of money in Tara's account and the day of the month.

Step-by-step explanation:

r is between -1 and +1.  r values close to -1 are strongly negative.  r values close to +1 are strongly positive.  r values close to 0 are weak.

r = -0.9870 is strongly negative.  So there is a strong, negative correlation between the amount of money in Tara's account and the day of the month.

Final answer:

The correlation coefficient r = -0.9870 shows a strong, negative correlation between the day of the month and the amount of money in Tara's account, meaning as the days pass, her account balance generally decreases. So, the correct statement is 'There is a strong, negative correlation between the amount of money in Tara's account and the day of the month.'

Explanation:

The correlation coefficient, denoted as r, measures the strength and direction of the linear relationship between two variables. Given that the correlation coefficient in the scenario is r = −0.9870, this indicates a strong, negative correlation between the amount of money in Tara's account and the day of the month. This means that as the days of the month increase, the amount of money in Tara's account tends to decrease, and this pattern is quite consistent, considering the high absolute value of the coefficient, which is close to -1.

Yes, they are saying the same thing. In fact, if a ratio equals one, it means that numerator and denominator are equal.

This is the reason why you can always use A=B or A/B, as they are totally equivalent.

Yo sup??

y=-3/x+4

cross multiply

x+4=-3/y

x=-3/y-4

f(y)=-3/y-4

or

f(x)=-3/x-4

=-4x-3/x

The correct answer is option 4

Hope this helps.

Answer:

It is -4x-3/x.

Answer: The answer is A

Step-by-step explanation: Add the coefficient of x to the both sides.

X²+4x=12

X²+4x+4=12+4

X²+4x+4=16

X²+2x+2x+4=16

X(x+2)+2(x+2)=16

(X+2)(x+2)=16

(X+2)²=16

The equivalent equation is  (x+2)²=16

A quadratic equation can be written in the standard form as ax2 + bx + c = 0, where a, b, c are constants and x is the variable. The values of x that satisfy the equation are called solutions of the equation, and a quadratic equation has at most two solutions.

Given here: The expression x²+4x-12=0

Thus simplifying the equation we get

x²+2.2x+4-12-4=0

(x+2)²-4²=0

(x+6) (x-2)=0

x=-6,2

or the equation can also be rewritten as (x+2)²=16

Thus the equivalent equation is  (x+2)²=16

Learn more about quadratic expressions here:

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Here's the polynomial function representing the area of the remaining portion of the square:

A(x) = 9 - 4x + x^2

1. **Initial area:** The original square has a side length of 3 inches, so its initial area is 3 * 3 = 9 square inches.

2. **Removing squares:** When small squares of side length x are cut out from each corner, the remaining shape becomes a smaller square with a side length of (3 - 2x) inches.

3. **New area:** The area of this smaller square is (3 - 2x) * (3 - 2x) = 9 - 6x + 4x^2 = 4x^2 - 6x + 9.

4. **Simplifying:** We can rearrange this expression to get a simpler polynomial function: A(x) = 9 - 4x + x^2.

Therefore, A(x) = 9 - 4x + x^2 represents the area of the remaining portion of the square after the small squares are cut out, as a function of the side length x of the removed squares.

Answer: the amount of money in the account that earns 10% interest is $500

the amount of money in the account that earns 12% interest is $900

Step-by-step explanation:

Let x represent the amount of money in the account that earns 10% interest.

Let y represent the amount of money in the account that earns 12% interest.

There is $400 more in the account that pays 12% than there is in the other account. This means that

y = x + 400

The formula for determining simple interest is expressed as

I = PRT/100

Where

I represents interest paid on the loan.

P represents the principal or amount taken as loan

R represents interest rate

T represents the duration of the loan in years.

Considering the account that earns 10% interest.

P = x

R = 10

T = 1 year

I = (x × 10 × 1) = 0.1x

Considering the account that earns 12% interest.

P = y

R = 12

T = 1 year

I = (y × 12 × 1) = 0.12x

If the total interest for a year is $158, it means that

0.1x + 0.12y = 158 - - - - - - - - - - -1

Substituting y = x + 400 into equation 1, it becomes

0.1x + 0.12(x + 400) = 158

0.1x + 0.12x + 48 = 158

0.22x = 158 - 48 = 110

x = 110/0.22 = 500

y = x + 400 = 500 + 500

y = 900

0.75 hours more is needed to practice to meet his goal

Solution:

Given that,

Goal per week of Joel = [tex]4\frac{1}{2} = \frac{2 \times 4 + 1}{2} = \frac{9}{2}[/tex]

From given,

The list below shows the number of hours Joel has a practiced so far for the week

[tex]Monday = 1\frac{1}{2}\ hours = \frac{3}{2}\ hours[/tex]

[tex]Wednesday = 1\frac{1}{4}\ hours = \frac{5}{4}\ hours\\\\Thursday = 1\ hour[/tex]

How many more hours does he need to practice this week to meet his goal

Find the difference

Hours needed = Goal per week of Joel - (monday + wednesday + thursday)

[tex]Hours\ needed = \frac{9}{2} - (\frac{3}{2} + \frac{5}{4} + 1)\\\\Hours\ needed = 4.5-(1.5+1.25+1)\\\\Hours\ needed = 4.5 - 3.75 = 0.75\\\\Hours\ needed = 0.75 = \frac{3}{4}[/tex]

Thus 0.75 hours more is needed to practice to meet his goal

Joel needs to practice \( \frac{3}{4} \) hours more this week to meet his goal.

First, we need to calculate the total number of hours Joel has practiced so far. We will add the hours practiced on Monday, Wednesday, and Thursday.

[tex]Monday: \( 1 \frac{1}{2} \) hours = \( 1 + \frac{1}{2} \) hours = \( \frac{3}{2} \) hours[/tex]

[tex]Wednesday: \( 1 \frac{1}{4} \) hours = \( 1 + \frac{1}{4} \) hours = \( \frac{5}{4} \) hours[/tex]

[tex]Thursday: \( 1 \) hour = \( \frac{4}{4} \) hours (to keep the denominator consistent)[/tex]

Now, let's add these hours together:

[tex]\( \frac{3}{2} + \frac{5}{4} + \frac{4}{4} = \frac{6}{4} + \frac{5}{4} + \frac{4}{4} = \frac{15}{4} \) hours[/tex]

Joel's goal is to practice [tex]\( 4 \frac{1}{2} \)[/tex] hours per week, which is [tex]\( 4 + \frac{1}{2} \) hours = \( \frac{8}{2} + \frac{1}{2} \) hours = \( \frac{9}{2} \) hours.[/tex]

To find out how many more hours Joel needs to practice, we subtract the total hours he has already practiced from his goal:

[tex]\( \frac{9}{2} - \frac{15}{4} \)[/tex]

To subtract these fractions, we need a common denominator, which is 4:

[tex]\( \frac{18}{4} - \frac{15}{4} = \frac{3}{4} \) hours[/tex]

Therefore, Joel needs to practice [tex]\( \frac{3}{4} \)[/tex] hours more to meet his weekly goal.

Answer:

i am pretty sure it is B

Step-by-step explanation:

Answer:

Step-by-step explanation:

1.

cot x sec⁴ x = cot x+2 tan x +tan³x

L.H.S = cot x sec⁴x

       =cot x (sec²x)²

       =cot x (1+tan²x)²     [ ∵ sec²x=1+tan²x]

       =  cot x(1+ 2 tan²x +tan⁴x)

       =cot x+ 2 cot x tan²x+cot x tan⁴x

        =cot x +2 tan x + tan³x        [ ∵cot x tan x [tex]=\frac{ \textrm{tan x }}{\textrm{tan x}}[/tex] =1]

       =R.H.S

2.

(sin x)(tan x cos x - cot x cos x)=1-2 cos²x

 L.H.S =(sin x)(tan x cos x - cot x cos x)

          = sin x tan x cos x - sin x cot x cos x

           [tex]=\textrm{sin x cos x }\times\frac{\textrm{sin x}}{\textrm{cos x} } - \textrm{sinx}\times\frac{\textrm{cos x}}{\textrm{sin x}}\times \textrm{cos x}[/tex]

           = sin²x -cos²x

           =1-cos²x-cos²x

           =1-2 cos²x

           =R.H.S

3.

1+ sec²x sin²x =sec²x

L.H.S =1+ sec²x sin²x

         =[tex]1+\frac{{sin^2x}}{cos^2x}[/tex]                       [[tex]\textrm{sec x}=\frac{1}{\textrm{cos x}}[/tex]]

         =1+tan²x                        [tex][\frac{\textrm{sin x}}{\textrm{cos x}} = \textrm{tan x}][/tex]

         =sec²x

        =R.H.S

4.

[tex]\frac{\textrm{sinx}}{\textrm{1-cos x}} +\frac{\textrm{sinx}}{\textrm{1+cos x}} = \textrm{2 csc x}[/tex]

L.H.S=[tex]\frac{\textrm{sinx}}{\textrm{1-cos x}} +\frac{\textrm{sinx}}{\textrm{1+cos x}}[/tex]

       [tex]=\frac{\textrm{sinx(1+cos x)+{\textrm{sinx(1-cos x)}}}}{\textrm{(1-cos x)\textrm{(1+cos x})}}[/tex]

      [tex]=\frac{\textrm{sinx+sin xcos x+{\textrm{sinx-sin xcos x}}}}{{(1-cos ^2x)}}[/tex]

     [tex]=\frac{\textrm{2sin x}}{sin^2 x}[/tex]

      = 2 csc x

    = R.H.S

5.

-tan²x + sec²x=1

L.H.S=-tan²x + sec²x

        = sec²x-tan²x

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

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

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

        =1

Answer:

Each boy gets approximately Rs 13.09                                              

Step-by-step explanation:

We are given the following in the question:

Total money = Rs 144

Let x be the amount each girl gets and y be the amount each boy gets.

The money is distributed among 2 boys and 3 girls. Thus, we can write:

[tex]3x + 2y = 144[/tex]

Also, each girl gets three times as much as each boy gets.

Thus, we can write:

[tex]x = 3y[/tex]

Substituting these values, we get,

[tex]3(3y) + 2y = 144\\11y = 144\\\\y =\dfrac{144}{11} \approx 13.09\\\\x = 3(\dfrac{144}{11}) \approx 39.27[/tex]

Thus, each boy gets approximately Rs 13.09

Answer: each boy should get $39.27

Step-by-step explanation:

Let x represent the amount that each boy should get.

Let y represent the amount that each girl should get.

The total amount of money that would be distributed among the 2 boys and 3 girls is 144 Rs. This is expressed as

2x + 3y = 144 - - - - - - - - - - - -1

The money is shared in such a way that each girl gets three times as much as each boy gets. This is expressed as

y = 3x

Substituting y = 3x into equation 1, it becomes

2x + 3 × 3x = 144

2x + 9x = 144

11x = 144

x = 144/11 = 13.09

y = 3x = 3 × 13.09

y = $39.27