Do these basic math problems please

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].

Answer:

(C) X<5

Step-by-step explanation:

Set of all old integers less than 5=(1,3)

X will represent this in inequality :

X<5.

Therefore, the answer is C.

Answer:

The answer to this question is the wall is 45.95 yards tall

Step-by-step explanation:

To solve this, we list out the given variables and the unknowns thus

Height of ball at launch = 8 yards

Distance of net from the ball = 80 yards

Distance of the wall down the path = 75%

Maximum height of the ball= 80 yards

equation of Motion of the ball = parabolic motion =

v² = u² - 2gS

S = 80 - 8 = 72 yards

at maximum height v = 0 thus u² = 2×9.81×72 =1412.64

u = 37.59 m/s

also v = u - gt and again at max height v = 0

Therefore 37.59 = 9.81×t or t = 3.83 s

If the motion of the ball is free of obstruction then time of flight before the ball just reaches the 8 yards off the ground = 3.83×2 = 7.66 seconds

Taking the initial velocity as zero at maximum height and from the equation

S = ut + 0.5×gt² we get, where S is the heigt of the ball from touching the actual field ground which is 80 yards we have

80 = 0.5×9.81×t²

so that t² = 2×80÷9.81 = 16.31 or t = 4.04s

Therefore the total time of flight = 4.04 + 3.83 = 7.87 seconds

if the ball is considered as having a constant horizontal velocity, therefore

at 75% of the way the time it took will be 0.75×7.87 = 5.9 seconds

However time it took  the  ball to reach maximum height and then starts descent = 3.83s, and the time at which the ball is directly over the wall = 2.07 seconds on the second half just after reaching mximum height

Thus at 2.07 seconds the distance trvelled from the maximum height is

S = ut +0.5gt² as before where u = 0

hence S = 0.5×9.81×2.07² = 21.05 yards or (80 -21.05) yards off the ground =  58.95 yards

As stated in the question, the ball cleared the wall by 13 yards therefore the height of the wall is 58.95 - 13 = 45.95 yards

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:

a. 0.26

b. 0.7

c. 0.38

Step-by-step explanation:

For a)

For sponsored studies: 62 %  found the products favorable. The percentage is 0.62.

For non-sponsored studies 46% found the products favorable. The percentage is 0.46

Total probability = P(A) × P (B)

                           = 0.46 × 0.62

                           = 0.2604

For b)

Probability for the food industry = 0.7

For c) 1 - ( 0.62) = 0.38

The overall probability that a participant finds the product favorable is 57.2%. If the product was found favorable, the probability the study was industry-sponsored is 75.8%. If the product was found unfavorable, the probability the study was not industry-funded is 71.7%.

The first step to answer the given question is understand that this is a problem of conditional probability, related to the concept of favorability or unfavorability towards products based on industry-sponsored studies and non-sponsored studies.

To find the overall probability of a participant finding the product favorable, we add the probabilities of a product being favorable under both industry funding and no industry funding, weighted by their respective chances of occurrence.

To work out the conditional probabilities:

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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: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.

Answer:

12%

Step-by-step explanation:

0.9/7.5 *100

= 12%

Answer:

If you`re trying to figure out how many gifts Maggie can get without the fee going over the budget, then I would say that she can buy 3 gifts. 4 times 3 is 12, and when you add 7 it equals 19. Of course, if 3 is pushing it, Maggie can also buy 1 or 2 gifts, and she`d be fine. Hope this helps!

Step-by-step explanation:

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

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

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.

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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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:

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.

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:

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:

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

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:

i am pretty sure it is B

Step-by-step explanation:

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:

(A)  [tex]f=\frac{k}{L}[/tex]

(B) Frequency becomes half.

Step-by-step explanation:

We have been given that the frequency f of vibration of a violin string is inversely proportional to its length L. The constant of proportionality k is positive and depends on the tension and density of the string.

(A) We know that two inversely proportional quantities are in form [tex]y=\frac{k}{x}[/tex], where y is inversely proportional to x and k is constant of proportionality.

Upon substituting our given values, we will get:

[tex]f=\frac{k}{L}[/tex]

Therefore, our required equation would be [tex]f=\frac{k}{L}[/tex].

(B) For part, we have been given that length is twice, so our new frequency will be [tex]f_n[/tex] and new length [tex]L_n[/tex] is [tex]2L[/tex].

Upon substituting [tex]2L[/tex] in our equation as:

[tex]f_n=\frac{k}{L_n}[/tex]

[tex]f_n=\frac{k}{2L}[/tex]

[tex]f_n=\frac{1}{2}\cdot \frac{k}{L}[/tex]

[tex]f_n=\frac{1}{2}\cdot f[/tex]

Upon comparing [tex]f_n[/tex] with [tex]f[/tex], we can see that [tex]f_n[/tex] is half the value of [tex]f[/tex].

Therefore, the frequency of vibration of violin gets half, when we double the length of the string.

The frequency f of a vibrating string is represented by the equation f = k/L, where k is a constant and L is the string's length. If the length of the string is doubled, the frequency of the string's vibration is halved due to the inverse relationship.

(A) Considering the problem describes the frequency f of a vibrating string being inversely proportional to its length L, we can represent this information mathematically with the equation f = k/L, where k is a constant of proportionality. This constant is positive and depends on the tension and density of the string.

(B) Because frequency and length are inversely proportional, if you double the length of the string (L), the frequency (f) will halve. This is due to the inverse relationship, with the longer string taking more time to complete each vibration and thus reducing the frequency of vibration.

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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:

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

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.

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.

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.

The statement 'the number of new clients (x) is 154% of the number of old clients (y)' can be translated into a linear equation in the standard form 'ax + by = c' as '-1.54y + x = 0' or '-154y + 100x = 0' where a, b, and c are constants.

To translate the given statement into a linear equation, we need to interpret the percentages as a ratio. When we say 'the number of new clients (x) is 154% of the number of old clients (y)', it means 'x is equivalent to 1.54 times y'. So, we can write that as a linear equation:

x = 1.54y

However, that's not in the form 'ax + by = c'. To get it into that form, we can write it as:

Or alternately you might prefer, if you wish to avoid decimals:

Where a = -154, b = 100 and c = 0.

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Answer:

Step-by-step explanation:

Answer: the ladder would be long enough.

Step-by-step explanation:

The ladder forms a right angle triangle with the wall of the building and the ground. The length of the ladder represents the hypotenuse of the right angle triangle. The height from the window to the base of the building represents the opposite side of the right angle triangle.

The distance from the bottom of the ladder to the base of the building represents the adjacent side of the right angle triangle.

To determine if a 12 foot ladder is long enough to reach the window, we would apply Pythagoras theorem which is expressed as

Hypotenuse² = opposite side² + adjacent side²

Let h represent the height that the ladder would get to. Therefore

12² = h² + 3²

144 = h² + 9

h² = 144 - 9 = 135

h = √135 = 11.62 feet

Since the height of the window is 10 feet above the ground, the ladder would be long enough.

Answer:

c.square root of 28

Step-by-step explanation: