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thank you will give brainlist!!!!​

Answer: A) We need 1 1/2 ft of ribbon

B)$1.60 per foot

C)$2.40 is the total

Step-by-step explanation:

So if the length around the jar is 18 inches and we know that there is 12 inches in a foot, and 3 feet in a yard and $ 4.80 per every 3 feet. These become basis numbers. 18in- 12inch= 6 inch. So now we have 1 1/2 feet of ribbon to cover the circumference of the jar. Now we are left to answer the second question how much does it cost per foot? We can divide 4.80 by 3 to see the total cost of a foot of ribbon. That turns to 1.60 per foot. We divide that by two to cover the 6 inches. And the answer is $2.40 for 1 1/2 feet of ribbon

Answer:

Celia make [tex]5\frac{1 }{4}[/tex] servings.

Step-by-step explanation:

Given:

Celia made 3 1/2 cups of rice.

A serving of rice is 2/3 cup.

Now, to find the number of servings Celia make.

Celia made [tex]3\frac{1}{2} =\frac{7}{2}\ cups.[/tex]

So, to get the number of servings we use unitary method:

If 2/3 cup of rice is of 1 serving.

So, 1 cup of rice is of = [tex]1\div\frac{2}{3}servings.[/tex]=[tex]\frac{3}{2}[/tex]

Then 7/2 cup of rice is of  [tex]=\frac{7}{2} \times \frac{3}{2}[/tex]

                                           [tex]=\frac{21}{4}=5\frac{1}{4}.[/tex]

Thus, 7/2 cup of rice is of [tex]5\frac{1 }{4}[/tex] servings.

Therefore, Celia make [tex]5\frac{1 }{4}[/tex] servings.

Ceila made 5.25 or 5 ¹/₄ servings of rice.

Ceila made 3 1/2 cups of rice. In improper fractions this is:

3 1 /2 = 7/2

The number of servings made by Ceila can be found by:

= Number of cups of rice Ceila made / Cups of rice in a serving

= 7 / 2 ÷ 2/3

= 7/2 × 3/2

= 21 / 4

= 5.25 servings

When dividing fractions, you can instead multiply the first fraction by the inverse of the second fraction.

In conclusion, Ceila made 5.25 or 5 ¹/₄ servings of rice.

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Answer:

2

Step-by-step explanation:

There are three numbers on the first spinner.  Two are not even (1 and 3).

So there are 2 possible outcomes that work: 1 and 6 or 3 and 6.

Final answer:

There are two outcomes that meet the criteria of the question: not showing an even number on the first spinner and showing a 6 on the second spinner.

Explanation:

The question asks how many outcomes result in an odd number on the first spinner and show a 6 on the second spinner. The first spinner has three equal sectors labeled 1, 2, and 3, of which two are odd (1 and 3). The second spinner has four equal sectors labeled 3, 4, 5, and 6, but we are only interested in the outcome when the spinner shows a 6.

Since we want the first spinner to show an odd number, we ignore the 2, leaving us with 1 and 3 as the possible outcomes for the first spinner. For the second spinner, we are only interested in the outcome of 6. This means we multiply the two outcomes from the first spinner (1 and 3) by the one outcome from the second spinner (6), resulting in two possible outcomes: (1,6) and (3,6).

Therefore, the correct answer is there are two outcomes that do not show an even number on the first spinner and show a 6 on the second spinner.

Answer:

  Matt is not correct

Step-by-step explanation:

The remainder from division by 9 will always be a number less than 9. The number 17 is not less than 9, so Matt made an error.

_____

Matt may have tried to find the remainder by adding the digits of the dividend:

  2 + 9 + 6 + 0 = 17

Matt needs to continue this process until he gets a single digit:

  1 + 7 = 8

8 is the remainder from the division.

Answer:

96 km/hr

Step-by-step explanation:

Average number of kilometers per hour = 4,608 km / 48 hours

Average number of kilometers per hour = 96 km/hr

Answer:

96

Step-by-step explanation:

IF they are not getting sleep all of their journey then they must faces 96 kilometers per hour but according to their long journey they must want to get sleep and thats way they must spend some time on rest or sleep.

Answer:

C

Lined up with the points

Answer:

So it will take roughly 18.4471905815477 years.

(round appropriately if needed)

Step-by-step explanation:

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

6000=2000(1+0.06/4)^(4*t)

6000/2000 = (1+0.06/4)^(4*t)

3 = (1+0.06/4)^(4*t)

3 = (1.015)^(4*t)

log(3) = log((1.015)^(4*t))

log(3) = 4*t*log(1.015)

log(3)/(4*log(1.015)) = t

t = log(3)/(4*log(1.015))

t = 18.4471905815477

Answer:

The expression for Area of rectangle is [tex]x^2+16x+63[/tex].

Step-by-step explanation:

Given:

A rectangle has a length that is 4 units longer than the width.

Let the width of the rectangle be 'x' Units.

Length of rectangle will be = [tex]x+4\ Units[/tex]

If the width is increased by 7 units.

New Width of the rectangle = [tex]x+7\ Units[/tex]

Also the length increased by 5 units,

New Length of rectangle will be = [tex]x+4+5 = x+9\ Units[/tex]

We need to write an equivalent expression for area of the rectangle.

Area of rectangle is given by length times width.

framing in expression form we get;

Area of rectangle = [tex](x+7)\times(x+9)[/tex]

On Solving the equation we get;

Area of rectangle = [tex]x^2+9x+7x+63=x^2+16x+63[/tex]

Hence the expression for Area of rectangle is [tex]x^2+16x+63[/tex].

Answer:

add

Step-by-step explanation:

Answer:

C. f(x)=-3(x+7)(x+4)

Step-by-step explanation:

x-intercept means the value of x where the functions touches the x axis.

The function touches x axis at two places, x=-7 and x=-4 ie. both -7 and -4 are the roots of the equation.

And the terms (x+7) and (x+4) are the factors of the function f(x),

Only one such case exists where (x+7) and (x+4) are there as factors, ie. Option C.

Therefore, C.-3(x+7)(x+4).

Answer:

C. f(x)=-3(x+7)(x+4)

Step-by-step explanation:

Got it right on the test

Answer:

8 years

Step-by-step explanation:

x years later type A and type B will have the same height

Type A = Type B: 10  + (8/12)*X  = 6 + (14/12)*X

multiply 12 each side

120 + 8x = 72 + 14x

14x - 8x = 120 - 72

6x = 48

x = 8

check: A: 10 x 12 + 8 x 8 = 184

            B: 6 x 12 + 14 x 8 = 184

The solution to the system of equations is x = -3 and y = 2

To solve the given system of linear equations, use the method of elimination by multiplying the equations to eliminate y, solving for x, substituting the value of x back into one of the equations to solve for y, and checking the solution.

The given system of linear equations is:

x − 7y = -11

5x + 2y = -18

To solve this system of equations, we can use the method of substitution or elimination. I will demonstrate the method of elimination:

The solution to the system of equations is x = -3 and y = 2.

complete question given below:

Liner systems of equations

x−7y=−11

5x+2y=−18

Point is equidistant from points A and B because a point on a perpendicular bisector is equidistant from the endpoints of the segment it makes up.

The angle bisector theorem states that a triangle's opposite side is divided into two halves by an angle bisector that is proportional to the triangle's other two sides.

A point on a perpendicular bisector is equidistant from the endpoints of the segment it intersects, therefore if mACD = 90°, point D is equidistant from points A and B.

Any point on the perpendicular bisector is simply equal distance from both endpoints of the line segment on which it is drawn, according to the perpendicular bisector theorem.

The answer is that point  is equidistant from points A and B because a point on a perpendicular bisector is equidistant from the endpoints of the segment it makes up.

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To prove that point D is equidistant from the jungle gym and monkey bars (points A and B), we would utilise the geometrical concept of congruent triangles. This involves drawing line segments from D to A and D to B, and if those segments are congruent, then point D is equidistant from points A and B. The process would encompass drawing a perpendicular bisector from point D to segment AB.

To demonstrate that point D is equidistant from the jungle gym and the monkey bars, Beth would need to use specific concepts from geometry. First, she would need to identify points A and B as the jungle gym and monkey bars respectively. Point D is the location of the swings.  

Next, she would draw line segments from D to A and D to B. If line segment DA is congruent to line segment DB, this means they have the same length, hence point D is an equal distance from both points A and B (the jungle gym and the monkey bars).

This can be proved through congruent triangles if a perpendicular bisector is drawn from point D to line segment AB. Such concept echoes that a point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment. The resulting triangles would be congruent, meaning every part, including the sides DA and DB, would be the same.

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Answer:

90-32 is the answer

Step-by-step explanation:

Because this is a right angle its total angle in degrees is 90 and if it is split in to then m1 and m2 must equal 90 degrees so if m2 is 32 degrees then m1 would be what's left which is 90-32

Answer:

In summation form [tex]\sum_{n = 1} ^{5} 6(6)^{n - 1}[/tex]

Step-by-step explanation:

Within one hour, the first person gives a stack of flyers to six people and within the next hour, those six people give a stack of flyers to six new people.

So, in the first 5 hours, the summation of people that receive a stack of flyers not including the initial person will be given by

6 + (6 × 6) + (6 × 6 × 6) + (6 × 6 × 6 × 6) + (6 × 6 × 6 × 6 × 6).

So, in summation form [tex]\sum_{n = 1} ^{5} 6(6)^{n - 1}[/tex]

Therefore, Sigma-Summation Underscript n = 1 Overscript 5 EndScripts 6 (6) Superscript n minus 1, gives the correct solution. (Answer)

We can evaluate how many persons are getting flyer by each person then calculate summation.

Option A: [tex]\sum^5_{n=1}6(6)^{(n-1)}\\[/tex]summation form.

First person gives flyers to 6 people.

Those 6 persons give flyers to 6 new people, thus [tex]6 \times 6 = 36[/tex] people...

And so on five times for five hour as this process is done on hourly basis.

Thus, the summation without including first person, for five hours to count total number of people who received flyers is:

[tex]6 + (6 \times 6) + (6 \times 6 \times 6) + (6 \times 6 \times 6 \times 6 ) + (6 \times 6 \times 6 \times 6 \times 6)[/tex]

or in summation it can be rewritten as:

[tex]\sum^5_{n=1}6(6)^{(n-1)}\\[/tex]

Thus, Option A:  [tex]\sum^5_{n=1}6(6)^{(n-1)}\\[/tex] is the needed summation form.

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The area of a triangle for one of the legs being 3 inches and the hypotenuse being 9 inches is 12.727 square inches

Solution:

Given that to find area of a triangle for one of the legs being 3 inches and the hypotenuse being 9 inches

From given information,

Let "c" = hypotenuse = 9 inches

Let "a" = length of one of the leg of triangle = 3 inches

To find: area of triangle

The area of triangle when hypotenuse and length of one side of triangle is given:

[tex]A = \frac{1}{2} a \sqrt{c^2 - a^2}[/tex]

Where, "c" is the length of hypotenuse

"a" is the length of one side of triangle

Substituting the given values we get,

[tex]A = \frac{1}{2} \times 3 \times \sqrt{9^2 - 3^2}[/tex]

[tex]A =\frac{1}{2} \times 3 \times \sqrt{81-9}\\\\A =\frac{1}{2} \times 3 \times \sqrt{72}\\\\A =\frac{1}{2} \times 3 \times 8.48528\\\\A = \frac{1}{2} \times 25.45584\\\\A = 12.727[/tex]

Thus area of triangle is 12.727 square inches

If [tex]P(4,7)\in G_f[/tex] where [tex]G_f=\{(x,f(x):\forall x\in\mathbb{R}\wedge f(x)\in\mathbb{R}\}[/tex] then [tex]f(x-2)[/tex] will shift the original function by 2 in the right resulting with new point [tex]P'(6,9)[/tex].

Hope this helps.

Work is provided in the image attached.

Final answer:

Using the ratio of 7 girls to 3 boys and knowing that there are 9 boys, we find that there are 21 girls at the movie by setting up a proportion and solving for the number of girls.

Explanation:

To find out the number of girls at the movie, we need to use the ratio given, which is 7 girls to 3 boys or 7:3. Since there are 9 boys, we can set up a proportion to solve for the number of girls (G):

7 girls : 3 boys = G girls : 9 boys

Now we cross multiply and solve for G:

7 * 9 = 3 * G

63 = 3 * G

Divide both sides by 3 to find the number of girls:

G = 63 / 3

G = 21

Therefore, there are 21 girls at the movie.

Answer:

$5903.39

Step-by-step explanation:

This can be solve using compound interest formula. The formula is:

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

Where

F is the future amount (what we are looking for)

P is the present amount (which is 4800)

r is the rate of compound interest per year, in decimal (3% per year, 3/100 = 0.03)

t is the time in years ( t = 7)

Now we substitute these values into the formula and find F:

[tex]F=P(1+r)^t\\F=4800(1+0.03)^7\\F=4800(1.03)^7\\F=5903.39[/tex]

So, Colin would have $5903.39 after 7 years, in his account.

[tex]\bf (\stackrel{x_1}{-2}~,~\stackrel{y_1}{-6})\qquad (\stackrel{x_2}{2}~,~\stackrel{y_2}{2}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{2}-\stackrel{y1}{(-6)}}}{\underset{run} {\underset{x_2}{2}-\underset{x_1}{(-2)}}}\implies \cfrac{2+6}{2+2}\implies \cfrac{8}{4}\implies 2[/tex]

The slope of the line passing through the points (−2,−6) and (2,2) is 2, calculated using the slope formula in mathematics. This means there is a rise of 2 on the y-axis for every 1 increase on the x-axis.

Your task is to calculate the slope of a line passing through the points (−2,−6) and (2,2). In mathematics, slope is defined as the 'rise over run' - meaning, the change in the vertical axis (y-axis) for every unit increase in the horizontal axis (x-axis).

Conventionally, the slope (m) is calculated using the formula: m = (y2 - y1) / (x2 - x1). Applying this formula to the given points:

m = (2 - (-6))/(2 - (-2))

m = (2 + 6)/(2 + 2) = 8/4 = 2.

So, the slope of the line that passes through points (-2,-6) and (2,2) is 2; this means there's a rise of 2 on the y-axis for every 1 increase on the x-axis when you move from point (-2, -6) to point (2, 2).

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Answer:

Therefore,

Distance between XY is 878 ft and YB is 524 ft.

Step-by-step explanation:

Given:

BW = 1612 ft

∠ Y = 72°

∠ X = 49°

To Find:

XY = ?

YB = ?

Solution:

In right angle Triangle Δ WBY Tangent identity,

[tex]\tan Y= \frac{\textrm{side opposite to angle Y}}{\textrm{side adjacent to angle Y}}[/tex]

Substituting we get

[tex]\tan 72= \frac{WB}{YB}=\frac{1612}{BY}[/tex]

[tex]\therefore BY=\frac{1612}{3.077}=523.88=524\ ft...(approximate)[/tex]

Similarly,

In right angle Triangle Δ WBX Tangent identity,

[tex]\tan X= \frac{\textrm{side opposite to angle X}}{\textrm{side adjacent to angle X}}[/tex]

Substituting we get

[tex]\tan 49= \frac{WB}{XB}=\frac{1612}{XB}[/tex]

[tex]\therefore XB=\frac{1612}{1.15}=1401.73=1402\ ft...(approximate)[/tex]

Now For

[tex]XB = XY +BY[/tex].............Addition Property

Substituting we get

[tex]1402 = XY +524\\\\\therefore XY=1402-524=878\ ft[/tex]

Therefore

Distance between XY is 878 ft and YB is 524 ft.

Answer:

Step-by-step explanation:

twice a number is increased by 7.....let x represent the number

ur expression would be : 2x + 7

Answer:

It will cost $256542 to prepare the field for next weeks match.

Step-by-step explanation:

The maximum dimensions of a field according to FIFA rules is 110 m by 75 m.

So, the maximum area of a field is (110 × 75) = 8250 square meters.

Now, 1 square meters is equivalent to 1.196 square yards.

So, the maximum area of a field is (8350 × 1.196) = 9867 square yards.

Now, given that it costs to make ready each square yard of a full-size FIFA field, $26 before each game.

Therefore, it will cost $(26 × 9867) = $256542 to prepare the field for next week's match. (Answer)

The maximum revenue occurs at [tex]\( n = 13 \).[/tex]

The correct option is (a).

To maximize the revenue for the tour company, we need to find the value of ( n ) that maximizes the revenue function [tex]\( R = 52n - 2n^2 \).[/tex]

1. Find the derivative of the revenue function:

  The derivative of a function gives us the rate of change of the function. We will find the derivative of the revenue function with respect to ( n ) to find where the revenue function is increasing or decreasing.

Given the revenue function [tex]\( R = 52n - 2n^2 \)[/tex], we'll find its derivative [tex]\( \frac{dR}{dn} \):[/tex]

[tex]\[ \frac{dR}{dn} = \frac{d}{dn}(52n - 2n^2) \][/tex]

[tex]\[ \frac{dR}{dn} = 52 - 4n \][/tex]

2. Set the derivative equal to zero to find critical points:

  To find the critical points, we set the derivative equal to zero and solve for ( n ):

[tex]\[ 52 - 4n = 0 \][/tex]

[tex]\[ 4n = 52 \][/tex]

[tex]\[ n = \frac{52}{4} \][/tex]

[tex]\[ n = 13 \][/tex]

3. Determine the nature of the critical point:

  To determine if ( n = 13 ) is a maximum or minimum, we use the second derivative test. If the second derivative is negative at the critical point, it's a maximum; if positive, it's a minimum.

Taking the second derivative of ( R ):

[tex]\[ \frac{d^2R}{dn^2} = \frac{d}{dn}(52 - 4n) \][/tex]

[tex]\[ \frac{d^2R}{dn^2} = -4 \][/tex]

Since the second derivative [tex]\( \frac{d^2R}{dn^2} = -4 \) is negative, \( n = 13 \)[/tex] corresponds to a maximum.

4. Verify the endpoints:

  Since the domain of ( n ) is not specified, we need to check if the endpoints of the domain have any maximum revenue. In this case, we don't have any specific domain constraints, so we don't need to check the endpoints.

5. Conclusion:

The maximum revenue occurs at [tex]\( n = 13 \).[/tex]

Therefore, the correct answer is option a) 13.

complete question given below:

A tour company has a ticket price that goes down $2 for every additional person who signs up for a group trip. They charge, per person, 52-2n Where n is the number of people that go on the trip. Their total revenue, R, as a function of the number of people who can go in the trip is R=52n-2n^2. How many people Maximize the revenue for the tour company

a.13

b.26

c.39

d.22

Maximized revenue for the tour company occurs with 13 people on the trip, yielding $676.

To maximize revenue, we need to find the maximum point of the revenue function [tex]\( R = 52n - 2n^2 \)[/tex]. We can do this by finding the derivative of the revenue function with respect to n and then setting it equal to zero to find the critical points. Then, we can determine whether these critical points correspond to a maximum or minimum by analyzing the second derivative.

So, let's start by finding the derivative of the revenue function:

[tex]\[ \frac{dR}{dn} = 52 - 4n \][/tex]

Setting this derivative equal to zero and solving for [tex]\( n \):[/tex]

[tex]\[ 52 - 4n = 0 \][/tex]

[tex]\[ 4n = 52 \][/tex]

[tex]\[ n = \frac{52}{4} \][/tex]

[tex]\[ n = 13 \][/tex]

Now, we need to check whether this critical point corresponds to a maximum or a minimum. To do this, we'll take the second derivative of the revenue function:

[tex]\[ \frac{d^2R}{dn^2} = -4 \][/tex]

Since the second derivative is negative, the critical point [tex]\( n = 13 \)[/tex]corresponds to a maximum.

So, the tour company will maximize its revenue when 13 people go on the trip.

Answer: add over and over again

Step-by-step explanation:How do you get from 0 to 3? you add 3 to 0. Then, how do you get from 3 to 8? you add another 3 but then you a 2 too, so then 8. Lastly, my last example, how to get from 8 to 15. You take five, and then add 2 to it and add that number(then number you get when you add 5+1), 7 to 8.

Answer:9(3+4)

Step-by-step explanation:

GCF=9 27/9=3 36/9=4

The GCF will be 9 and the expression using distributive property will be 9 ( 3 + 4 ) .

An expression 27 + 36 is given in the question.

We have to simplify the expression by using distributive property.

What is the distributive property ?

The distributive property is given by ; A ( B+ C) = AB + AC , where A, B and C are three different values.

As per the question ;

the expression is 27 + 36

Here ;

We need to find the GCF in between 27 and 36.

The greatest common factor in 27 and 36 is 9.

∵ According to the distributive property ;

A ( B + C ) = AB + AC

so ;

the expression in distributive property can be written as ;

(9 × 3) + (9 × 4)

that will be equal to ;

9 ( 3 + 4 )

So ;

27 + 36 = (9 × 3) + (9 × 4) = 9 ( 3 + 4 )

Thus , the GCF will be 9 and the expression using distributive property will be 9 ( 3 + 4 ) .

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Answer: Oliver's goal is to read 45 minutes more next month

Step-by-step explanation:

Firstly, we need to know how much in minutes is the [tex]10\%[/tex] of [tex]450 min[/tex]:

[tex]450 min(10\%)=(450 min)(\frac{10}{100})=45 min[/tex]

This means Oliver wants to read 45 minutes more next month. If we add this number to the 450 minutes he read this month, we have a total of 495 minutes.

Answer:

The goal is 945 minutes

Step-by-step explanation:

Answer:

2x^2-2x+2

Step-by-step explanation:

Solved through synthetic division

If you struggle with synthetic division you can use Math away a free calculator you can use on google.

After completing the synthetic division, the quotient in polynomial form is equal to: B. 2x² - 2x + 2.

Synthetic division can be defined as a simplified and specialized method that is used for the division of a polynomial by another polynomial of the first degree (x - n).

Also, you should use only the coefficients of the divided polynonial and its divisor and change the sign of the constant term in the divisor, in order to replace subtraction with addition.

Next, we would set up the synthetic division as follows:

-3 | 2       4         -4         6

   |_________________

      ↓     -6          6        -6

   |_________________

      2      -2          2         0

Quotient = 2    -2     2    ⇒  2x² - 2x + 2.

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Answer:

$130 - ($20+$35) is the equation

The equation to represent the total amount spent on t-shirts and hoodies at the River Bluff High School store is 20t + 35h = 130, where t is the number of t-shirts and h is the number of hoodies purchased, and they total $130.

To represent the situation in which t-shirts cost $20 each and hoodies cost $35 each, and a total of $130 is spent, we can introduce variables to represent the quantity of t-shirts and hoodies purchased. Let's use t to represent the number of t-shirts and h to represent the number of hoodies. The equation for this situation in standard form, where Ax + By = C, would be:

20t + 35h = 130,

where 20 is the cost of one t-shirt, 35 is the cost of one hoodie, and 130 represents the total amount spent on t-shirts and hoodies combined.

Answer:  Diameter is increasing as [tex]\frac{1}{3}[/tex] centimeter per hour.

Step-by-step explanation:

Alright, lets get started.

The formula for volume of sphere is given as V : [tex]\frac{4}{3}\pir^3[/tex]

The volume of a sphere is increasing at a rate of [tex]6\pi[/tex] cubic centimeters per hour.

It means : [tex]\frac{dV}{dt}=6\pi[/tex]

[tex]V=\frac{4}{3}\pi r^3[/tex]

Taking derivative with respect to t

[tex]\frac{dV}{dt}=\frac{4}{3}\pi \times 3r^2 \frac{dr}{dt}[/tex]

[tex]6\pi=4\pi r^2 \frac{dr}{dt}[/tex]

at the instant the radius of the sphere is 3 centimeters, means

[tex]\frac{dr}{dt}= \frac{6}{4 \times 3^2}[/tex]

[tex]\frac{dr}{dt}=\frac{1}{6}[/tex]

As [tex]radius = \frac{diameter}{2}[/tex]

[tex]\frac{1}{2}\frac{dD}{dt}=\frac{1}{6}[/tex]

[tex]\frac{dD}{dt} =\frac{1}{3}[/tex]

Hence diameter is increasing as [tex]\frac{1}{3}[/tex] centimeter per hour.   : Answer

Hope it will help :)

The volume of a sphere is increasing at a rate of 6π cubic centimeters per hour. The rate at which its diameter is increasing with respect to time at the instant with the radius of the sphere 3 centimeters is: [tex]\mathbf{\dfrac{1}{3}}[/tex]

Option A is correct.

The volume of a sphere can be represented by using the formula:

[tex]\mathbf{V = \dfrac{4}{3}\pi r^3}[/tex]

Now, by differentiation, if we differentiate the rate at which the volume is increasing with time, we have:

[tex]\mathbf{\dfrac{dV}{dt} = \dfrac{4}{3}\pi r^3 \ \dfrac{dr}{dt}}[/tex]

[tex]\mathbf{\dfrac{dV}{dt} = 4 \pi r^2 \ \dfrac{dr}{dt}}[/tex]

Given that:

Replacing the values into the differentiated equation, we have:

[tex]\mathbf{6 \pi= 4 \pi (3)^2 \ \dfrac{dr}{dt}}[/tex]

[tex]\mathbf{\ \dfrac{dr}{dt}=\dfrac{6 \pi}{4 \pi (3)^2}}[/tex]

[tex]\mathbf{\ \dfrac{dr}{dt}=\dfrac{6 }{4 \times 9}}[/tex]

[tex]\mathbf{\ \dfrac{dr}{dt}=\dfrac{1 }{6}\ cm/sec}[/tex]

∴

[tex]\mathbf{\dfrac{1}{2} \dfrac{dr}{dt} =\dfrac{1}{6} }[/tex]

[tex]\mathbf{ \dfrac{dr}{dt} =\dfrac{1}{6} \times \dfrac{2}{1}}[/tex]

[tex]\mathbf{ \dfrac{dr}{dt} =\dfrac{1}{3}}[/tex]

Therefore, we can conclude that the rate at which its diameter is increasing with respect to time at the instant with the radius of the sphere 3 centimeters is  [tex]\mathbf{\dfrac{1}{3}}[/tex]

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