Answer:
1/14 ; 1/35
Step-by-step explanation:
A
6/10 x 5/9 x 4/8 x 3/7
3/5 x 5/9 x 1/2 x 3/7
1 x 1/3 x 3/14
1/14
B
4/10 x 3/9 x 2/8 x 6/7
2/5 x 1/3 x 1/4 x 6/7
2/15 x 3/14
1/5 x 1/7
1/35
The probability that 4 creature cards is pulled is 1/14 and the probability that 3 land cards and 1 creature card is pulled is 4/35
What is probability?Probability is a measure of the likelihood of an event occurring. The probability formula is defined as the possibility of an event happening is equal to the ratio of the number of favorable outcomes and the total number of outcomes. Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes.
Given here there are 4 land cards and 6 creature cards
thus selecting four cards at random from 10 cards is ¹⁰C₄=210
and the event of selecting 4 creature cards from a total of 6 cards is ⁶C₄ =15
Therefore probability of pulling 4 creature cards is = 15/210
= 1/14
The probability of pulling 3 land cards and 1 creature cards is
= ⁴C₃ × ⁶C₁ / ¹⁰C₄
= 4/35
Hence 1/14 and 4/35 are the respective probabilities.
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a sample of 1200 computer chips revealed that 37% of the chips do not fail in the first 1000 hours of their use. the company's promotional literature claimed that more than 34% do not fail in the first 1000 hours of their use. is there sufficient evidence at the 0.05 level to support the company's claim? state the null and alternative hypotheses for the above scenario.
As per the question the required hypothesis are
Null Hypothesis, H0: p=0.72
Alternate Hypothesis, H1 : p ≠ 0.72
The null hypothesis reflects that there will be no observed effect in our experiment. In a mathematical formulation of the null hypothesis, there will typically be an equal sign. This hypothesis is denoted by H0.
The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample.
Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses.
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David has a bank account which pays interest at the rate of 1.5% per year, compounded annually. determine what amount david must have in the bank, given that he would like to draw an annual salary of $32,635.15 from his account at the end of each year for 30 years. round to the nearest cent. a. $783,760.48 b. $979,054.50 c. $1,225,080.50 d. $795,516.88 please select the best answer from the choices provided a b c d
If David wants to withdraw $32,635.15 every year from his account at the end if each year for 30 years, he has to have $783243.6 in the bank account .
Define the term present annuity?The amount of money that would be required today to finance a series consecutive future annuity payments is referred to as the annuity's present value. sum of money received now is worth more than a similar sum at a later time due to the time value of money.We would use the equation for calculating present annuity.
It is written as
PV = R[1 - (1 + r)^- n]
In which.
The investment's present value is represented by PV.R stands for the consistent payments made (also weekly, monthly)The expression r stands for interest rate/interval payments.A total number of repayments is indicated by the number n.Considering the data provided,
r = 1.5/100 = 0.015
n = 30 years
R = $32,635.15
Put the values in the formula,
PV = 32635.15[1 - (1 + 0.015)⁻³⁰]/0.015
PV = 32635.15[1 - (1.015)⁻³⁰]/0.015
PV = 32635.15(1 - 0.64)/0.015
PV = 32635.15 × 24
PV = $783243.6
Thus, amount David must have in the bank presently is $783243.6.
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select the curve generated by the parametric equations. indicate with an arrow the direction in which the curve is traced as t increases. x
As t increases the curve is changing according to the plot points given by the parametric equations.
Define parametric equations?
The coordinates of the points that make up a geometric object, such as a curve or surface, are frequently expressed using parametric equations; in this case, the equations are collectively referred to as a parametric representation or parameterization (alternatively spelled as parametrisation) of the object. It describes a collection of numbers that represent functions with one or more independent variables. The parameters used here are the independent variables. For the representation of the coordinates that make up geometric objects like curves and surfaces, parametric equations are used. Additionally, a curve that is not a function can be drawn using parametric equations.
The equations are
x=1-t²
y=2t-t²,-1≤t≤2
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What is the value of 13(a • b + a2) + 4 if a = 5 and b = −3?
308
134
−74
−126
The value of the expression, 13(a • b + a²) + 4, when a = 5 and b = -3 is: B. 134.
How to Find the Value of an Expression?To find the value of an expression, we have to plug in the respective values for each of the variables given in the expression, then simplify.
Given the expression, 13(a • b + a²) + 4, where:
a = 5
b = -3
To find the value, substitute a = 5 and b = -3 into the expression:
13(5 • -3 + 5²) + 4
13(-15 + 25) + 4
13(10) + 4
= 130 + 4
= 134
The value is: B. 134.
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Jacob knows that 120 means “1 divided by 20.” He uses this to find the decimal equivalent for 120. Enter a digit into each box to complete his work.
The value of the decimal equivalent for 120 will be;
⇒ 0.05
What is Decimal number?
A decimal number is a number that consists of a whole and a fractional part.
Given that;
Jacob knows that 120 means “1 divided by 20.”
Now,
Since, Jacob knows that 120 means “1 divided by 20.”
Hence, The value of 120 by using this will be;
⇒ 1/20 = 0.05
Thus, The value of the decimal equivalent for 120 will be;
⇒ 0.05
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the second angle of a triangle is 28 more than the first angle. the third angle is two times the first. find the three angles.
Pls help
, I really need an answer for this rn
Answer:
Step-by-step explanation:
A health researcher conducted a survey and recorded the percentage of body fat of 20 randomly selected men from two different towns.
Answer: 7 hrs and 15 minutes
Step-by-step explanation: because 15 - 8 = 7 45 - 30 = 15
Tanya has read 3/4 of a book, which is 36 pages. How many pages are in the entire book?
Answer:
There are 48 pages in the whole book.
Step-by-step explanation:
We know that 36 is 3/4 of the book. To find the total number of pages, we have to divide 36 by 3. We know that is 12. Then, we have to multiply 12 by 4 to get the answer, 48.
A house on the market was valued at 376,000. After several years, the value decreased by 9%. By how much did the house value decrease in dollars. What is the current value of the house?
Answer:
Step-by-step explanation:
The price decreases by $288,000*(0.19)=$54,720
The current value is then $233,280.
Select all of the relations that ARE functions.
Select 2 correct answer(s)
Answer:
the first one at the top . but can you put up a better picture ?
Step-by-step explanation:
(d) is it possible for this curve to have a horizontal tangent at points where it intersects the x-axis? explain your reasoning.
No, there cannot be a horizontal tangent at the point where the curve intersects the x-axis.
What is a curve on the graph?A smooth-drawn figure or line with a bend or turn is referred to as a curve.
A circle is an illustration of a curved shape.
The path, also known as the topological arc, is the name of the curve (or just arc).
If a curve is the continuous injective image of an interval or a circle, it is considered simple.
If a curve is determined by a continuous function, in other words.
So, it possible for the curve to have a horizontal tangent:
On the curve, horizontal tangents appear where:
x = -1 and y ≠ -1
y = 0 where the curve crosses the x-axis.
-1² + 2(-1) + 0⁴ + 4.0 ≠ 5
No, the point where the curve crosses the x-axis cannot have a horizontal tangent.
Therefore, no, there cannot be a horizontal tangent at the point where the curve intersects the x-axis.
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Complete question:
Consider the closed curve in the xy-plane given by : x²+2x+y⁴=4y = 5
Is it possible for this curve to have a horizontal tangent at points where it intersects the x-axis? explain your reasoning.
An article in the Washington Post on March 16, 1993 stated that nearly 45 percent of all Americans have brown eyes. A random sample of n=55 C of I students found 11 with brown eyes. Give the numerical value of the statistic p^(P HAT)=
The numerical value of the statistic ^p or sample proportion (p-hat) is 0.2.
^p = X/N where X denotes the number of successes and N denotes the size of the sample. Here, X=55 and N=11.
Sample Proportion (p-hat) of the data:
The sample proportion is a random variable which varies from sample to sample which is not predicted with certainty.Sample proportion is given by the ratio of number of successes in a sample to the size of that sample. It is symbolised as p-hat.the mean of p-hat is exactly (p).Standard Deviation of sample proportion is [tex]\sqrt{[p(1-p)/n]}[/tex].To know more about Sample proportion:
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Find the roots of the equation
Answer: There are not any roots for that equation.
a national business magazine reports that the mean age of retirement for women executives is 63.2. a women’s rights organization believes that this value does not accurately depict the current trend in retirement. to test this, the group polled a simple random sample of 78 recently retired women executives and found that they had a mean age of retirement of 64.2. assuming the population standard deviation is 4.4 years, is there sufficient evidence to support the organization’s belief at the 0.02 level of significance?
The calculated value |Z| = |-2.00722 | = 2.00722 > 2.054 at 0.02 level of significance. So their claim is wrong.
Given that the mean age of retirement for women, executives is 63.2
Given that the size of the sample 'n' = 78
Given that the mean of sample x⁻ = 64.2
Given that the standard deviation 'σ' = 4.4 years
Null hypothesis: H₀: μ ≠ 63.2
Alternative Hypothesis: H₁: μ = 63.2
Test statistic
Z = -2.00722
|Z| = 2.00722 > 2.054 at 0.02 level of significance
The calculated value |Z| = 2.00722 > 2.054 at 0.02 level of significance
Therefore, we reject the hypothesis.
Statistics is the study of research and surveys on numerical data in mathematics. In order to conduct surveys, the hypothesis must be established. There are typically two different kinds of hypotheses. A null hypothesis is one that holds true, whereas an alternate hypothesis is another.
The null hypothesis is a broad assertion or default stance that there is zero happening or nothing happening in probability and statistics. For instance, there is no association between two measured events or a connection between groups.
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Factor 24cd − 36c2d.
A: 12c2d(2 − 6)
B: 12cd(2 − 36c2d)
C: 6cd(4 − 36c2d)
D: 4cd(6 − 9c)
Answer:
D
Step-by-step explanation:
24cd-36c²d = (4×6)c×d-(4×9)c×c×d
Then we take the common numbers and variables
4cd(6-9c) and to check our answer we can distribute 4cd to get:
4cd×6-(4cd×9c)
24cd-36c²d
what is the variance of the number of times a 6 appears when a fair die is rolled 12 times? (enter the final answer in decimal format and round to three decimal places.)
The variance of the number of occasions a 6 appears when a fair die is thrown 12 times is 0.156.
It determines the likelihood of achieving a specific number of successes over a specific number of trials with a consistent success rate using the binomial distribution.
With a 1/6 likelihood of success for each trial, the number of successes in this example is equal to the number of 6s that occur.
The probability of each conceivable value of the quantity of 6s that appear may be calculated using the binomial formula. The likelihood of receiving exactly 0 number of 6s is (12 choose 0) ×[tex](\frac{1}{6})^0[/tex] × [tex](\frac{5}{6}) ^{12}[/tex] = 0.1680. The probability of getting exactly 1 number of 6 is (12 choose 1) × [tex](\frac{1}{6})^1[/tex] × [tex](\frac{5}{6})^{11}[/tex] = 0.4401. The probability of getting exactly 2 numbers of 6s is (12 choose 2) × [tex](\frac{1}{6})^2[/tex] × [tex](\frac{5}{6})^{10}[/tex] = 0.3744. And so on.
The likelihood of receiving 0 6s, for instance, multiplied by the square of the 0's divergence from 2 is 0.1680 × (0 - 2)² = 0.7056. The likelihood of receiving 1 6 when multiplied by the square of the difference between 1 and 2 is 0.4401 × (1 - 2)² = 0.4401. And so on.
To get the anticipated value of the squared departure of the number of 6s that occur from 2, we may finally total up all of these probabilities.
This predicted value is equal to 1.8801 by adding 0.7056, 0.4401, 0.3744, 0.1596, 0.0368, and 0.0036.
The range in the number of 6s that appear after tossing a fair die 12 times is thus 1.8801.
To express the variation in decimal notation, divide the difference by the total number of rolls (12), then round the output to the closest three decimals.
As a consequence, we obtain a variance of 0.1567. The variance in the number of occasions a 6 appears when a fair dice is rolled 12 times is thus 0.1567.
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write the general antiderivative. (use c for the constant of integration. remember to use absolute values where appropriate.) 1 8 x 1 8x 1 8 x dx
The general antiderivative is; (x²/16) + 1/8 ln |x| - 1 / (3 ln × 2 × 8ˣ) + C
Given,
f (x) = x/8 + 1/8x + (1/8)ˣ
We have to find the antiderivative;
Antiderivative;-
In calculus, a differentiable perform F whose spinoff is sufficient to the initial perform f could be an antiderivative, inverse derivative, primitive perform, primitive integral, or integral of a perform f. Symbolically, this could be expressed as F' = f.
Here,
To find antiderivative, we have to integrate the function first;
Then,
∫ f(x) + g(x) dx = ∫ f(x) dx + g(x) dx
∫ f(x/8) dx = x² / 16
∫ f(1/8 x) dx = 1/8 ln |x|
∫ f(1/8ˣ) dx = -1/ (3 ln × 2 × 8ˣ)
That is,
∫ x/8 + 1/8x + 1/8ˣ dx = (x²/16) + 1/8 ln |x| - 1 / (3 ln × 2 × 8ˣ) + C
Therefore,
(x²/16) + 1/8 ln |x| - 1 / (3 ln × 2 × 8ˣ) + C will be the antiderivative.
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What is 8640 minutes, in days? (remember, there are 60 minutes in an hour and 24 hours in a day. ).
As there are 60 minutes in an hour and 24 hours in a day, there are 6 days as well as 144 days in 8640 minutes.
What is division?A number is split in division, which is a straightforward procedure. The simplest way to conceptualize it is as a set of things being distributed among a set of individuals, as in the example given above. In mathematics, division is the process of dividing a number into equal parts and calculating the maximum number of equal parts that may be formed. For instance, dividing 15 by 3 results in the division of 15 into 3 groups of 5 each.
Here,
There are 60 minutes in an hour,
=8640/60
=144 hours
There are 24 hours in a day,
=144/24
= 6 days
There is 6 days as well as 144 days in 8640 minutes as there are 60 minutes in an hour and 24 hours in a day.
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Angles a and b are supplementary. determine the measure of angle a if the measure of angle b is 115.2°. 244.8° 64.8° 25.2° 11.5°
The measure of angle 'a' is 64.8°.
What is supplementary angles?
Two angles are considered to be supplementary if their sums total 180 degrees.
For instance, angle 130° and angle 50° are supplementary angles since the sum of these two angles is 180°.
The given angle is a = 115.2°.
We have to find the angle 'b' supplementary to the angle 115.2°.
Since, sum of supplementary angles is 180 degrees.
⇒ a + b = 180°
115.2 + b = 180°
Subtract 115.2° from both sides,
b = 180° = 115.2°
b = 64.8°
Hence, the measure of angle 'b' which is supplementary to angle 'a' is 64.8°.
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henry scored 85, 76, 84, and 69 on four exams. what must he make on the fifth exam to have an average of 80?
Henry must have to score 86 marks in the fifth exam to make an average of 80 marks.
What is the average?
This is the arithmetic mean and is calculated by adding a group of numbers and then dividing by the count of those numbers. For example, the average of 2, 3, 3, 5, 7, and 10 is 30 divided by 6, which is 5.
Henry scored 85, 76, 84, and 69, and let's suppose he scored 'x' marks in the fifth exam.
Average = sum of all marks/total number of exams
80 =(85 + 76 + 84 + 69 + x)/5
80 = 314 + x/5
80 * 5 = 314 + x
400 = 314 + x
x = 400 - 314
x = 86
Hence, Henry must have to score 86 marks in the fifth exam to make an average of 80 marks.
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Use point-slope form to write the equation of a line that passes through the point
(-12, 15) with slope - 1.
Answer:
y=-1x+3
Step-by-step explanation:
Find the equation of a line given the slope m = -1 and the point P = (-12 , 15)
Now, the y-intercept is b=y1−m⋅x1
b = 15 - (-1) ⋅ (-12) = 3
Finally, the equation of the line can be written in the form y=mx+b.
y = -1x + 3
Slope-intercept form is:
y = -1x + 3
Point-slope form is:
y - (15) = -1 ⋅ ( x - (-12))
If you want to check if right then:
Input the point given it should give you the y output if the equation of the line is correct.
-1x-(12)=12 and then add 3 and you get 15. It is therefore correct (-12,15).
I hope this helps!^^
A new Community Wellness Complex is being built in Del Rio. The perimeter of the rectangular playing field is 414 yards. The length of the field is 9 yards less than triple the width What are the dimensions of the playing field?
Answer:
Length=153 yards
Width=54 yards
Hope this helps! :)
Step-by-step explanation: (153*=306, 54*2=108, 306+108=414)
P=2l+2w
P=414
L=3w-9
W=w
414=2(3w-9)+2w
414=6w-18+2w
414=8w-18
+18 +18
432=8w
/8
54=w
L=3(54)-9
L=162-9
L=153
A random number generator is used to create a list of 150 single-digit numbers. of those 150 numbers, 72 are odd and 78 are even. the number 9 was generated 15 times to the nearest whole percent, what is the experimental probability of an odd number other than 9 being generated? o 38% 0 48% o 52% o 79% 20°f sunny 'o
The experimental probability of an odd number other than 9 being generated is 38%.
Explain the term experimental probability?A probability that has been established by a series of tests is called an experimental probability. To ascertain their possibility, a random experiment is initiated and iterated over a number of times; each iteration is referred to as a trial. The goal of the experiment is to predict the likelihood that an event will occur or not.For the stated question-
150 numbers altogether.Odd numbers in total: 72There are 15 nines.Other odd numbers outside nine are as follows: 72 - 15 = 57
Probability = (number of odd numbers except nine)]/(total number)
Probability = 57/150
Probability = 0.38.
Thus, the experimental probability of an odd number other than 9 being generated is 38%.
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How many solutions does the equation 5n + 1 = 1 + 4n have
Short Answer:
The given equation has one solution; [tex]n=0[/tex].
Further Explanation:______________________________________________________
"How Can You Determine Whether An Equation Has One Solution, Infinite Solutions, or No Solution?"
One Solution - Some equations have exactly one solution. In these equations, there is only one value for the variable that makes the equation true. You can tell that an equation has one solution if you solve the equation and get a variable equal to a number.Infinite Solutions - Some equations have infinitely many solutions. In these equations, any value for the variable makes the equation true. You can tell that an equation has infinitely many solutions if you try to solve the equation and get a variable or a number equal to itself.No Solution - If a linear equation has the same variable term but different constant values on opposite sides of the equation, it has no solutions.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This being said, lets now go back to the given scenario and determine the amount of solutions it has, if any.
1. We are given the following to solve:
[tex]5n+1=1+4n[/tex]2. First, subtract 1 from both sides.
[tex]5n+1-1=1+4n-1[/tex]3. Simplify.
[tex]5n=4n[/tex]4. Move 4n to the left side, leaving us with our final solution.
[tex]n=0[/tex] ← Final Solution____________________________________________________
Hope this helps!
a manufacturer must test that his bolts are 1.00cm long when they come off the assembly line. he must recalibrate his machines if the bolts are too long or too short. after sampling 121 randomly selected bolts off the assembly line, he calculates the sample mean to be 0.94cm. he knows that the population standard deviation is 0.31cm. assuming a level of significance of 0.05, is there sufficient evidence to show that the manufacturer needs to recalibrate the machines?
There is insufficient evidence to support that bolts are too long or too short.
What is Mean and Standard Deviation?
Mean: Dividing the sum of all values in a data set by the number of values.
Standard Deviation: The standard deviation is a measure that summarizes how far away from the mean each observation is.
Given:
mean: 0.94cm
standard deviation = 0.31cm
Alright, so a manufacturer must test that these bolts are long as they come off the manufacturing line and he must readjust the machines if they're too long or too short. Consequently, sample 121V and sample mean are 0.94 cm. So we want to test whether or not these are actually equal to two, so are no hypothesis is that the mean is too and our alternative hypothesis, because it says that we have to recalibrate if they're too short or too long, is that the mean is not too we could probably go with the left tail test here because the sample mean is under but we'll be safer with a two tail test um just because the standard deviation may sway things.
So what we want to do next is compute the test statistic and that's gonna be a Z score and that's just supplied by X bar minus population mean over standard deviation over squared event. And we're using Z here because we have a high sample size and since we know the population standard deviation.
Hence, There is insufficient evidence to support that bolts are too long or too short.
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consider a function z(x, y) and its partial derivative (∂z/∂y)x. under what conditions is this partial derivative equal to the total derivative dz/dy?
When the (∂z/∂x) is 0, the partial derivative (∂z/∂y) will be equal to the total derivative dz/dy.
What is Total Derivative?
The overall change in the dependent variable as a result of changes in all the independent variables is known as the total derivative of a function of multiple variables.
Given that,
A function z(x, y)and its partial derivative (∂z/∂y) is x.
We know that,
total derivative (dz) = (∂z/∂x) dx + (∂z/∂y) dy
here, (∂z/∂y) = x
So, The dz will be equal to (∂z/∂y) dy only if (∂z/∂x) dx = 0.
hence, (∂z/∂x) should be 0.
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Practice #3
Tell whether each table represents a function.
a. The table represents a function.
b. The domain is given as follows: {1, 2, .., 12 AM, 1, 2, ... , 12 PM).
c. The range is given as follows: {1, 2, 3, ..., 24}.
Is this a function? What are the domain and the range?To obtain whether a relation represents a function, we need to observe if each input is mapped to only one output.
This table gives the conversion of hours in the AM-PM format to hours in the military format. Each hour in the AM-PM format has only one equivalent hour in the military format, that is, each input is mapped to a single output, hence the table in fact represents a function.
The domain of a function is the set that contains all the input values of the function, hence for this problem the domain is the set of hours in the AM-PM format.
The range of a function is the set that contains all the output values of the function, hence for this problem the range is the set of hours in the military format.
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write a equivalent expression ( x + 3) +4y
The expression equivalent to the given expression will be x+y+3(1+y).
What is an expression?It is defined as the combination of constants and variables with mathematical operators.
It is given that the expression is,( x + 3) +4y
We have to write the equivalent expression for the given expression,
Expressions that function identically despite having different appearances are called equivalent expressions. If two algebraic expressions are equivalent, then they both have the same result when the same value of the variable is entered.
=( x + 3) +4y
=x+3+4y
=x+3+y+3y
=x+y+3+3y
=x+y+3(1+y)
Thus, the expression equivalent to the given expression will be x+y+3(1+y).
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outside temperature over a day can be modeled as a sinusoidal function. suppose you know the temperature is 65 degrees at midnight and the high and low temperature during the day are 76 and 54 degrees, respectively. assuming t is the number of hours since midnight, find an equation for the temperature, d, in terms of t.
Outside temperature over a day can be modeled as a sinusoidal function. suppose you know the temperature is 65 degrees at midnight and the high and low temperature during the day are 76 and 54 degrees, respectively. assuming t is the number of hours since midnight, the temperature, d, in terms of t. The Equation is :
D = - 11 sin(π/12) + 65
The range is 76 - 54 = 22
so, a = 11
we know that the period is 24 hours
So, 2π/k=24
⇒ 24k = 2π
⇒ k = π/12
So far we have:
D = 11 sin(π/12)(t) + 65
Giving us a range from 76 to 54
But obviously the temp after midnight would decrease, whereas our function has it increasing to 76 when t = 6 (6:00 am)
We could do a phase shift, or more simply, just flip the function to
D = -11sin(π/12)t + 76
Checking some values
t = 0 , D = -11sin0 + 65 = 65
t=6 , (6:00 am) D = -11 sin (π/2) + 65 = 54
t = 12 (noon), D = -11 sin π + 65 = 65
t = 18 , (6:00 pm) , D = -11sin 3π/2 + 65 = 76
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water is leaking out of an inverted conical tank at a rate of 8 , 900 8,900 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. the tank has height 11 11 meters and the diameter at the top is 5.5 5.5 meters. if the water level is rising at a rate of 24 24 centimeters per minute when the height of the water is 3 3 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.
the rate at which water is being pumped into the tank in cubic centimeters per minute. is 1508527.582 cm³/min
The net rate of flow dV/dt = flow rate in - flow rate out
Let flow rate in = k. Since flow rate out = 8900 cm³/min,
dV/dt = k - 8900
Now, the volume of a cone V = πr²h/3 where r = radius of cone and h = height of cone
dV/dt = d(πr²h/3)/dt = (πr²dh/dt)/3 + 2πrhdr/dt (since dr/dt is not given we assume it is zero)
So, dV/dt = (πr²dh/dt)/3
Let h = height of tank = 11 m, r = radius of tank = diameter/2 = 3/2 = 1.5 m, h' = height when water level is rising at a rate of 21 cm/min = 3.5 m and r' = radius when water level is rising at a rate of 21 cm/min
Now, by similar triangles, h/r = h'/r'
r' = h'r/h = 3.5 m × 1.5 m/12 m = 5.25 m²/12 m = 2.625 m = 262.5 cm
Since the rate at which the water level is rising is dh/dt = 21 cm/min, and the radius at that point is r' = 262.5 cm.
The net rate of increase of water is dV/dt = (πr'²dh/dt)/3
dV/dt = (π(262.5 cm)² × 21 cm/min)/3
dV/dt = (π(68906.25 cm²) × 21 cm/min)/3
dV/dt = 1447031.25π/3 cm³
dV/dt = 4545982.745/3 cm³
dV/dt = 1515327.582 cm³/min
Since dV/dt = k - 6800 cm³/min
k = dV/dt - 6800 cm³/min
k = 1515327.582 cm³/min - 6800 cm³/min
k = 1508527.582 cm³/min
So, the rate at which water is pumped in is 1508527.582 cm³/min
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