The radius and height of a circular cylinder are changing with time in such a way that the volume remains constant at 1 liter (= 1000 cubic centimeters). If, at a certain time, the radius is 4 centimeters and is increasing at the rate of 1/2 = 0.5 centimeter per second, what is the rate of change of the height?

Answers

Answer 1

Given that radius, `r` = 4 cm, and it is increasing at the rate of `dr/dt = 0.5` cm/s.Also, Volume of cylinder, `degree V = πr²h = 1000 cm³`.

[Given]Differentiating with respect to `t` on both sides, we get: `dV/dt = d/dt(πr²h) = 0`or `d/dt(πr²h) = 0`We can write it as: `2πr(dr/dt)h + πr²(dh/dt) = 0`[∵ Applying product rule of differentiation]

Substituting the given values, we get: `2π(4)(0.5)h + π(4)²(dh/dt) = 0`or `dh/dt = - (2 * 2 * 0.5)h / 16`or `dh/dt = - (1/2) h / 4`or `dh/dt = - (1/8) h`Negative sign indicates that the height of the cylinder is decreasing at the rate of `(1/8)h` cm/s. Hence, the rate of change of the height is `- (1/8)h` cm/s.

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Related Questions

Find the volume of the sphere.
Either enter an exact answer in terms of \piπpi or use \dfrac{22}{7}
7
22

start fraction, 22, divided by, 7, end fraction for \piπpi.

Answers

The volume of sphere is 1437.33 if the given radius is 7 .

What is Surface area of sphere ?

Surface area of sphere can be defined as the product of four , pi and square of radius. And here the value of pi could be 3.14 or 22/7 .

Given ,

Volume of a sphere = 4/3πr³

π = 22/7

r = 7

Volume of a sphere = 4/3πr³

= 4/3 * 22/7 * 7³

= 4/3 * 22/7 * 7 * 7 * 7

= (4 * 22 * 7 * 7 * 7) / (3 * 7)

= 30,184/21

= 1437.3333333333

Volume of a sphere = 1,437.33

Therefore, the volume of sphere is 1437.33 if the given radius is 7 .

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A car travels 22 km due south and then 28 km in a direction 60° east of south. Find the Direction of the car's resultant vector. PLEASE HELP

Answers

A car travels 22 km due south and then 28 km in a direction 60° east of south. Find the Direction of the car's resultant Remember that any can be written as a coordinate like this: (AcosΘ,AsinΘ) where A is the magnitude of the object and Θ is the directional angle. So for the car travelling 20km north, that vector can be written as: (20cos90°, 20sin90°) Then when the car drives northwest at 60 degrees, that vector can be written as: (35cos150°, 35sin150°) (do you see why I it's 150°?) Then, simply calculate each of these coordinates, add them up, and then that resulting coordinate is the coordinate of the resulting displacement vector. (20cos90°, 20sin90°) = (0, 20) (35cos150°, 35sin150°) = (-30.31, 17.50) Displacement vector (x, y) = (-30.31, 37.50) With this, we can calculate the magnitude which is \displaystyle \sqrt{x^2+y^2} x 2 +y 2 . Thus: Magnitude = \displaystyle \sqrt{(-30.31)^2+(37.50)^2}=48.2km (−30.31) 2 +(37.50) 2 =48.2km The resulting directional angle is then \displaystyle \tan^{-1}(\frac{y}{x})+\pitan −1 ( x y )+π. Adding \displaystyle \piπ is only necessary when x < 0 in order to put us in the proper quadrant. Thus: Direction = \displaystyle \tan^{-1}(\frac{37.50}{-30.31})+\pi=2.25\ radians=128.9°tan −1 ( −30.31 37.50 )+π=2.25 radians=128.9° ges ok HOPE THIS HELPS :):0

Solve the multiple choice for number 6

Answers

Answer:

I'm pretty sure it is A

Step-by-step explanation:

find all the expressions that are equal to 4*10^-3

Answers

Answer:

Attached to this answer are some of the ways you could rewrite $4*10^{-3}$

find all the expressions that are equal to 4*10^-3

Graph (X-5)2/25 - (y+3)2/36 = 1.

Answers

The graph of the parabola (x- 5 )²/25 - (y + 3)²/36 = 1 is added as an attachment

How to determine the graph of the parabola

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

(X-5)2/25 - (y+3)2/36 = 1.

Express the equation properly

So, we have

(x- 5 )²/25 - (y + 3)²/36 = 1

The above expression is a an equation of a conic section

Next, we plot the graph using a graphing tool

To plot the graph, we enter the equation in a graphing tool and attach the display

See attachment for the graph of the function

Answers

Answer:

x>4

Step-by-step explanation:

15x>-60

x>-60/15

x>4

Describe a sequence of transformations for which Triangle B is the image of Triangle A.

Answers

Answer:

its a translation

Step-by-step explanation:

Triangle B slid to the left to get to triangle A

Not my best subject so I'm pretty sure the other person got it right!

If $ x $ is an odd integer, what can you conclude about $ x^{\wedge} 2 ? $ no conclusion can be made $ x^{\wedge} 2 $ is not an integer $ x^{\wedge} 2 $ is even $ x^{\wedge} 2 $ is odd -/1 P

Answers

If $ x $ is an odd integer, the value of $ x^{\wedge} 2 $ will be odd. Hence, the correct answer is: $ x^{\wedge} 2 $ is odd. Explanation: Let's recall the definition of an odd integer: An odd integer is a number that cannot be divided by 2 evenly.

For example: 1, 3, 5, 7, 9, 11, etc. As per the given question, we can assume that x is an odd integer. We can use this information to draw a conclusion about $ x^{\wedge} 2 $..Now, let's calculate the square of an odd integer: An odd integer can be written as (2n + 1), where n is an integer.

So, \[ x = 2n + 1 \]Now, we can calculate the square of x: \[ x^{\wedge} 2 = (2n + 1)^{\wedge} 2 = 4n^{\wedge} 2 + 4n + 1 \]Let's simplify the above expression:\[ x^{\wedge} 2 = 2(2n^{\wedge} 2 + 2n) + 1 \]

Therefore, $ x^{\wedge} 2 $ is an odd integer because it can be expressed in the form of 2q + 1, where q is an integer. Hence, the correct answer is: $ x^{\wedge} 2 $ is odd.

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Write the polynomial in standard form that represents the perimeter of the quadrilateral. 2x-9 x+5

Answers

Answer:

im pretty sure its -2x but i might be wrong

Step-by-step explanation:

Alexandria, Bettina, and Crystal are planning a road trip together. The road trip is 1,100 miles. They plan to drive 50 miles per hour for the entire trip and have agreed on the following conditions. · Alexandria will drive 60% of the road trip. • Bettina will drive 1 mile for every 10 minutes of the road trip. • Crystal will drive the remainder of the road trip. Find the total driving time for the trip. Then determine the amount of time and how many miles each person will drive.

Answers

So this is the sawger those

Find the mass of the one-dimensional object. A wire that is 10 ft long (starting at x=0 ) and has a density function of rho(x)=x2+3x A force of 6 pounds is required to hold a spring stretched 0.5 feet beyond its natural length. How much work (in footpounds) is done in stretching the spring from its natural length to 0.8 feet beyond its natural length?

Answers

From the given information , the mass of the one-dimensional wire is 363.33 pounds.

To find the mass of the wire, we need to integrate the density function along its length. The density function is given by rho(x) = x^2 + 3x, and the wire is 10 ft long.

The mass of an infinitesimally small element dx of the wire is dm = rho(x) * dx. Therefore, the total mass of the wire is given by:

M = ∫[0,10] (x^2 + 3x) dx

Using the power rule of integration, we can evaluate this integral:

M = [(1/3)x^3 + (3/2)x^2] evaluated from 0 to 10

= [(1/3)(10^3) + (3/2)(10^2)] - [(1/3)(0^3) + (3/2)(0^2)]

= (1000/3 + 150) - 0

= 1150/3

≈ 383.33 pounds

Therefore, the mass of the wire is approximately 383.33 pounds.

The mass of the one-dimensional wire is approximately 383.33 pounds.

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Determine the vertex of the function f(x) = 3x2 – 6x + 13.

1. Identify the values of a and b.

a =
and b =

2. Find the x-coordinate of the vertex.

negative StartFraction b Over 2 a EndFraction =

3. Find the y-coordinate by evaluating the function at the x-value found in the previous step.

The vertex is

.

Answers

Answer:

1.)

A=3

B=-6

2.)

3.)

(1,10)

Answer:

1. Identify the values of a and b.

a =

✔ 3

and b =

✔ –6

2. Find the x-coordinate of the vertex.

negative StartFraction b Over 2 a EndFraction =

✔ 1

3. Find the y-coordinate by evaluating the function at the x-value found in the previous step.

The vertex is

✔ (1, 10)

Construct a two-tape Turing machine with input alphabet {a, b, c} that accepts the language {a^i b^i c^i | i > 0 } .

Answers

The Turing machine will accept any input string where the number of 'a's, 'b's, and 'c's are equal and greater than zero.

I will describe a two-tape Turing machine that accepts the language {a^i b^i c^i | i > 0}.

This language consists of strings where the number of 'a's, 'b's, and 'c's are all equal and greater than zero.

The Turing machine uses two tapes: the input tape and the working tape. The input tape contains the input string, and the working tape is used for processing.

Here's the high-level description of the Turing machine:

Tape 1 (Input tape): Contains the input string, delimited by a special symbol '#' at the end.

Tape 2 (Working tape): Used for processing. Initially, it is empty.

State 0: Initialization

Read the input string until you find the symbol '#' on Tape 1.

Move the head of Tape 1 back to the beginning of the string.

State 1: Match 'a's with 'b's

If the current symbol on Tape 1 is 'a' and Tape 2 is empty, write 'a' on Tape 2 and move right on both tapes.

If the current symbol on Tape 1 is 'a' and the symbol on Tape 2 is 'a', write 'a' on Tape 2 and move right on both tapes.

If the current symbol on Tape 1 is 'b' and the symbol on Tape 2 is 'a', replace 'a' on Tape 2 with 'b' and move right on both tapes.

If the current symbol on Tape 1 is 'b' and the symbol on Tape 2 is 'b', move right on both tapes.

If the current symbol on Tape 1 is 'c' and the symbol on Tape 2 is 'b', replace 'b' on Tape 2 with 'c' and move right on both tapes.

If the current symbol on Tape 1 is 'c' and Tape 2 is empty, reject the input.

State 2: Match 'b's with 'c's

If the current symbol on Tape 1 is 'b' and Tape 2 is empty, reject the input.

If the current symbol on Tape 1 is 'b' and the symbol on Tape 2 is 'b', write 'b' on Tape 2 and move right on both tapes.

If the current symbol on Tape 1 is 'c' and the symbol on Tape 2 is 'b', replace 'b' on Tape 2 with 'c' and move right on both tapes.

If the current symbol on Tape 1 is 'c' and the symbol on Tape 2 is 'c', move right on both tapes.

State 3: Check for termination

If the current symbol on Tape 1 is '#' and Tape 2 is empty, accept the input.

If the current symbol on Tape 1 is '#' and there are remaining symbols on Tape 2, reject the input.

Hence, the Turing machine will accept any input string where the number of 'a's, 'b's, and 'c's are equal and greater than zero.

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What is the solution to the inequality 2(x+9/4)>3

Answers

Answer:

-3/4 is the answer

pleasse help me out with this

Answers

Answer:

2 cos (x + pi/2)

Step-by-step explanation:

Of the choices given, this looks like a cos curve that is shifted to the Left by pi / 2 and multiplied to give an amplitude of 2

Combine like terms 3x-7x=

Answers

Answer:

-4x

Step-by-step explanation:

3-7=-4 just add a x

Can someone help me please????

Answers

i think it would be the last one sorry if i’m wrong

the fraction 2/3 is a. less than b. greater than c. eqaul to one. when 2/3 is mulitiplyed by 6,734, the product will be a. eqaul to b. greater than c. less than than 6,732.

Answers

2/3 = 0.667 which is less than 1.

part a answer: a. Less than

part b: when a number is multiplied by a number less than one the product would be less than the original number.

answer part b: c. Less than

Based on this picture, if ∠3 = 50˚then what is m∠8?
SHOW WORK PLEASE

Answers

The answer is that measure of angle eight is 130°

Given g of x equals cube root of the quantity x minus 5, on what interval is the function negative?

Answers

The function g(x) is negative for x < 5. The function g(x) = ³√ (x - 5) is negative on the interval x < 5.

To determine on what interval the function g(x) = ³√ (x - 5) is negative, we need to find the values of x that make g(x) less than zero.
Set the function equal to zero and solve for x:
³√ (x - 5) = 0
Since the cube root of any number is zero only when the number itself is zero, we have:
x - 5 = 0
x = 5
Now we have a critical point at x = 5. We can divide the number line into three intervals:
Interval 1: x < 5
Interval 2: x = 5
Interval 3: x > 5
Test a value from each interval in the original function to determine its sign:
For x = 4 (Interval 1):
g (4) = ³√ (4 - 5) = ³√ (-1) = -1, which is negative.
For x = 5 (Interval 2):
g (5) = ³√ (5 - 5) = ³√ (0) = 0, which is neither positive nor negative.
For x = 6 (Interval 3):
g (6) = ³√ (6 - 5) = ³√ (1) = 1, which is positive.
From the tests, we can conclude that the function g(x) is negative for x < 5. The function g(x) = ³√ (x - 5) is negative on the interval x < 5. The function g(x) = ³√ (x - 5) is negative on the interval x < 5. To find this interval, we first set the function equal to zero and solve for x. Since the cube root of any number is zero only when the number itself is zero, we find x - 5 = 0, which gives us x = 5. This critical point divides the number line into three intervals: x < 5, x = 5, and x > 5.

Next, we test a value from each interval in the original function to determine its sign. For x = 4 (Interval 1), the function evaluates to -1, which is negative. For x = 5 (Interval 2), the function evaluates to 0, which is neither positive nor negative.

Finally, for x = 6 (Interval 3), the function evaluates to 1, which is positive. Thus, we can conclude that the function g(x) is negative for x < 5. The function g(x) = ³√ (x - 5) is negative on the interval x < 5.

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A music group expects to sell a new compact disc (CD) at the rate R(t)=20,000e−0.12t CDs per week, where t denotes the number of weeks since the CD was first released. To the nearest thousand, how many CDs are expected to be sold during the first 12 weeks after the release?

Answers

Answer:

4739 CDs

Step-by-step explanation:

Given the function that models the rate at which the disc is sold expressed as;

R(t)=20,000e^−0.12t

t is the number of weeks since the CD was first released

We are to look for the number of CDs that are expected to be sold during the first 12 weeks after the release. To do this, you will simply substitute t = 12 into the modeled function and get R(12) as shown;

R(t)=20,000e^−0.12t

R(12)=20,000e^−0.12(12)

R(12)=20,000e^−1.44

R(12)=20,000(0.2369)

R(12) = 4738.56

Hence the total number of CDs expected to be sold to nearest thousand is 4739 CDs

Answer:

127,000

Step-by-step explanation:

you take the integral from 0 to 12 of R(t)

What is the solution set to F(x)=x4+2? Using small values, form a list of ordered pairs.
Include all of your work in your final answer. Submit your solution.

Answers

Answer:

If you don't agree or there's an error, let me know so I can edit.

Step-by-step explanation:

f(x) = x⁴ + 2

differentiate

f(x) = 4x³ + 0 =0

x(4x²) = 0

at x= 0 f(x) = 0 points = (0,0)

at x= 1. f(x) = 3 points = (4,3)......

The ordered pair is (4,3).

What is ordered pair?

An ordered pair is a composition of the x coordinate (abscissa) and the y coordinate (ordinate), having two values written in a fixed order within parentheses. It helps to locate a point on the Cartesian plane for better visual comprehension. The numeric values in an ordered pair can be integers or fractions.

Given:

f(x) = x⁴ + 2

Now, differentiate the above function for the small values

f(x) = 4x³ + 0 =0

x(4x²) = 0

When x= 0

f(x) = 0

So, the points = (0,0)

when x= 1

f(x) = 3

so, the points = (4,3).

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The vertices of a triangle are p (-6,1), q (-2,5) and r (8,1) what is the slope of the median of the triangle that passes through point r the vertices of a triangle are p (-6,1), q (-2,5) and r (8,1) what is the slope of the altitude of the triangle that passes through point q

Answers

The slope of the median of the triangle that passes through point R is -1/6. The slope of the altitude of the triangle that passes through point Q is -1.

To find the slope of the median of the triangle that passes through point R, first we find the midpoint of PQ. Midpoint of PQ = (-2 + (-6))/2, (5 + 1)/2 = (-4, 3)

The equation of the line passing through R and the midpoint of PQ is the median of the triangle that passes through point R.

Slope of the line = (y2 - y1)/(x2 - x1) = (1 - 3)/(8 - (-4)) = -2/12 = -1/6

The slope of the median of the triangle that passes through point R is -1/6.To find the slope of the altitude of the triangle that passes through point Q, first we find the equation of the line containing Q and is perpendicular to the line containing PQ.

The slope of the line PQ is (5 - 1)/(-2 + 6) = 4/4 = 1.

Therefore, the slope of the line containing the altitude is -1/1 = -1.

The equation of the line containing the altitude passing through Q is:y - 5 = -1(x + 2) => y = -x + 3

So, the slope of the altitude passing through Q is -1.

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Which inequality represents this situation?
Players must practice at least 45 minutes a day.

A. t 45
D. t ≥ 45

Answers

Answer:

Step-by-step explanation:

The purchase of a diamond ring is a large expense, requiring advanced planning. To estimate how much you would need to spend on a diamond, a random sample of 351 diamonds is taken from the website AwesomeGems.com on July 28, 2005. The sample mean cost per carat is $6242.40 and the sample standard deviation cost is $2895.41.
The 95% confidence interval for the population mean is ($5938.42, $6546.32). Select the correct interpretation of this confidence interval.

Answers

The correct interpretation of this confidence interval is that, based on the random sample of 351 diamonds taken from AwesomeGems.com on July 28, 2005, with a confidence level of 95%, the true population mean cost per carat of diamonds purchased from the website is estimated to fall between $5938.42 and $6546.32.

This means that if we were to repeat the sampling process many times and construct confidence intervals using the same method, about 95% of these intervals would contain the true population mean cost per carat. It is important to note that this statement is about the population mean, not any particular diamond's cost, and that this interval only applies to diamonds purchased from AwesomeGems.com on July 28, 2005.

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In one serving of oatmeal the ratio of the number of ounces of cats to the number of
ounces of dried fruit is 4 to 7

Drag the numbers into the table to show how many ounces of oats and dried frutar
needed for different numbers of servings

24
14
28
8
49
42
Oats (ounces)
Dried Frut (ounces)
2 services

Answers

By knowing the ratio of ounces of oats to ounces of dried fruit, we will complete the given table, which you can see below:

Fruit (ounces) Oats (ounces)

24 13.7

14 8

28 16

8 4.6

49 28

42 24

Working with ratios:

We know that the ratio of the number of ounces of oats to the number of ounces of dried fruit is 4 to 7.

This means that if we have x ounces of dried fruit, then we have (4/7) ounces of oats.

So for each value in the given table, we just need to multiply it by 4/7.

Fruit (ounces) Oats (ounces)

24 24*(4/7) = 13.7

14 14*(4/7) = 8

28 28*(4/7) = 16

8 8*(4/7) = 4.6

49 49*(4/7) = 28

42 42*(4/7) = 24

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What is the area of the figure shown below.

Answers

It’s 64. You can cut it into a rectangle and triangle. For the rectangle, you multiply 6 by 8 to get an area of 48. For the triangle, you multiply 8 by 4, then divide it by 2 to get 16. You add the area of the triangle and rectangle, then you get the area of the entire shape which is 64. (48+16=64)

when a biased coin is tossed, heads is three times as likely to come up as tails. rearrange the following procedure in the correct order to find the probability that should be assigned to the outcome of heads and tails?

Answers

The probability of heads is 3 times that of tails, so the probability of heads is 0.75 and the probability of tails is 0.25.

Assign a probability of 0.5 to heads

Assign a probability of 0.25 to tails

When a biased coin is tossed the probability of heads is three times as likely to come up as tails.

To find the probability that should be assigned to the outcome of heads and tails the first step is to calculate the probability of heads.

Once this is done,

The probability of heads can be assigned a value of 0.5 and the probability of tails can be assigned a value of 0.25.

After that,

The probability of heads and tails can be calculated accordingly.

The probability of heads and tails when a biased coin is tossed, the first step is to calculate the probability of heads.

Once this is done the probability of heads can be set to 0.5 and the probability of tails can be set to 0.25.

With these probabilities assigned the probability of heads is three times that of tails.

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A TV costs R14 000 you made a 15% deposit and pay the remainder over a 3 year period at 10% interested p.a

Calculate the deposit amount? and Determine the remainder to be paid over the 3 years

Answers

Answer:

deposited rs. 2100 of a tv and rs. 3570 paid remainder after 3 years

A piece of wood is cut into three different sized pieces. The ratio of the three pieces is 4:5:6. If the actual length of the wood is 45 inches, then what is the length of the short side?

Answers

Answer:

Step-by-step explanation:

So 4:5:6 = 45

4+5+6 = 15

45/15 = 3

Since 4 is the smallest, 4*3 = 12.

Hope this works!

Answer:

Step-by-step explanation:

This is a question worth knowing how to do. It is prime candidate for a test question.

Equation

Let the unit be x

therefore

4x + 5x + 6x = 45

Solution

15x = 45

x = 45/15

x = 3

The shortest piece is 4*3 = 12

The middle piece = 5*3 = 15

The longest piece = 6*3 = 18

Total 45

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