The dimensions that will minimize the cost of constructing this box is length = 56in , depth = 0.13in and height = 46.6inch
According to the question,
Volume of Rectangular box = 350 in³
Let Area of base of box = l × b
Let height of box be h
So , V = lbh
=> 350 = lbh
=> h = 350/lb ------------(1)
To Minimize the cost,
Total Area = base area + front lateral Area
Front lateral Area = lh + (2b + l) h
So, Total area = lb + lh + (2b + l) h
Substituting value of h from equation (1)
=> TA = lb + l( 350/lb) + (2b + l) (350/lb)
Cost function =>
C = 5lb + 10(350/ b) + 3(2b + l)(350/lb)
Solving,
C = 5lb + 3500/b + 2100/l + 1050/b
=> C = 5lb + 4550/b + 2100/l is required cost function
Now, To minimize the cost
We will differentiate Cost function w.r.t a and b
C' = 5b - 2100/l² = 0 ----------(2)
C' = 5a - 4550/b² =0 -----------(3)
Solving both using substitution method
=> b = 420/l²
=> 5lb² = 4550
=> lb² = 910
=>176400/l³ = 910
=> l³ = 176400/910
=> l = 56 in
putting value of a,
b = 0.13 in
Therefore ,
height = 350 / 7.5
=> h = 46.67 in
To know more about Rectangle here
https://brainly.com/question/29123947
#SPJ4
Graph the equation below by plotting the
y-intercept and a second point on the
line. When you click Done, your line will
appear.
3
y=-x+3
Click on the point(s). To change your selection, drag the
marker to another point. When you've finished, click Done.
8 64-2
Done
8
-6-
NA
2
สี
-2
4
-6-
-8-
2 4
6 8
Edit
By plotting y-intercept (0, -5) and any one of the points given in the table we can get the required line.
What is equation of the line?
The equation y = mx + c is the general equation of any straight line where m is the gradient of the line (how steep the line is) and c is the y -intercept (the point in which the line crosses the y -axis).
Equation of the line has been given as,
[tex]y=\frac{3}{2} x-5$$[/tex]
By comparing this equation with the y-intercept form of the equation,
y=m x+b
Slope of the line ' m '[tex]$=\frac{3}{2}$[/tex]
and y-intercept ' b ' = -5
Table for the points to be plotted on a graph will be,
Table for the points to be plotted on a graph will be,
x y
-4 -11
-2 -6
0 -5
2 -4
4 -3
By plotting y-intercept (0, -5) and any one of the points given in the table we can get the required line.
To learn more about equation of the line visit:https://brainly.com/question/21511618
#SPJ1
Work out the bearing of C from A
The bearing of C from A is 60° South.
How does one calculate bearing from A to B?
The bearing from A to B can be determined using interior angle if the bearing from B to A is known. Use the fact that angles in a full turn add up to 360° to determine the bearing of A to B, then subtract the bearing of B to A from 180° to determine the missing interior angle. For instance, B is 050° from A in terms of bearing.When AN elongates the angle will be 180°
Then the angle between A and C:
35° + 85° + x = 180°
120° + x = 180°
x = 180° - 120°
x = 60°
Hence, The bearing of C from A is 60° South.
To know more about angle check the below link:
https://brainly.com/question/25716982
#SPJ1
After a snowstorm in the town of Golden Glen, the morning temperature was
–
10°F. But by the afternoon, the temperature had risen by 17°F.
The change in temperature based on the information is 27°F.
How to illustrate the relationship?It should be noted that temperature simply means the degree of coldness and hotness in a body. In this case, after a snowstorm in the town of Golden Glen, the morning temperature was -10°F. But by the afternoon, the temperature had risen by 17°F.
Let the change in temperature be represented by x.
Therefore, -10 + x = -17
x = 17 + 10
x = 27
The Temperature is 27°F.
Learn more about temperature on:
brainly.com/question/24746268
#SPJ1
After a snowstorm in the town of Golden Glen, the morning temperature was -10°F. But by the afternoon, the temperature had risen by 17°F. What was the change in temperature?
what is equivalent to (32−−√5)13
Answer:
Your answer is 416 + 13 [tex]\sqrt{5}[/tex].
Step-by-step explanation:
Let the discrete random variable X have the geometric distribution with parameter p. (a) Give a real-life example in which the geometric distribution can be applied. (b) Use the definition of the expected value to show that: E[X] = 1/p. (c) Explain why it makes sense that the expected value of X is inversely proportional to p.
a) A discrete random variable is a variable that can take only a countable number of distinct values, such as 0, 1, 2, 3, 4, and so on.
b) Examples of discrete random variables are the number of children in a family, the number of people who go to the cinema on Friday nights, etc.
c) E(x) = 1/p
Discrete Random Variable:
A discrete random variable can be defined as a type of variable whose value depends on the numerical outcome of some random phenomenon. Also called a random variable. Discrete random variables are always easily countable integers. A probability mass function is used to describe the probability distribution of a discrete random variable.
Discrete random variables are used to quantify the results of random experiments. A discrete random variable takes on an infinite number of possible outcomes. In general, discrete random variables can be counted as 0, 1, 2, 3, 4, ...
Geometric Distribution :
The geometric variate is the variate that specifies the number of consecutive failures before the first success in Bernoulli trials. The probability of success of a Bernoulli trial is given by p and the probability of failure is 1 - p.
The Geometric Random Variable can be written as X ~ G(p).
The probability mass function is P(X = x) = (1 - p)ˣ⁻¹ p
Learn more about Discrete Random Variable:
https://brainly.com/question/17238412
#SPJ4
an engineer says a pipe should be 7/10 centimeters long. The pipe is 9/10 centimeter long. How much of the pipe needs to be cut off? write an equation.
Answer: x = 9/10 - 7/10
Step-by-step explanation:
Which of the following equations have only one solution? Select all correct answers. x 2 - x - 6 = 0 5 x 2 + 20 x + 20 = 0 9 x 2 - 25 = 0 4 x 2 + 4 x = 0 x 2 + 6 x + 9 = 0
Answer: wutttttttttt
Step-by-step explanation: i jus need points
Write a quadratic function in standard form whose graph has a vertex of (2, 6) and passes through the point (4, 10) .
use the data above to test the claim that marble color preference and club membership are related, as follows: (2 pts) carefully state the hypotheses. Hypothesis is
H0: marble color preference and income class are independent
H1: marble color preference and income class are dependent
The Null Hypothesis, H0 is "Marble color preference and income class are independent" and the Alternate Hypothesis, H1 is "Marble color preference and income class are dependent".
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.
We can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypotheses. However, the hypotheses can also be phrased in a general way that applies to any test.
We are given that to test the claim that marble color preference and club membership are related.
Thus, the Null Hypothesis, H0 is "Marble color preference and income class are independent" and the Alternate Hypothesis, H1 is "Marble color preference and income class are dependent".
To learn more about hypotheses, visit brainly.com/question/13025783
#SPJ4
What is the value of u
u=______°
Answer:
u=94 Hope that helps. Mark as brainliest it helps
How do I solve this problem. Please explain step by step if u could
0.25 + (-3) =
Answer:
0.25 − 3
=0.25 + −3
= −2.75
( a number line to help to)
Step-by-step explanation:
a) Determine the series of the given function. In the first box after the summation symbol, type in -1 or 1 indicating whether the series is alternating or not.
b) Write out the sum of the first four nonzero terms of the series representing this function.
c) Determine the interval of convergence. The outside boxes require the endpoints and the inside boxes
a) The given function is [tex]$\frac{x^2-2x+2}{(x-1)^2(x+2)}$[/tex]. The series of this function is [tex]$\sum_{n=0}^{\infty}(-1)^n(n+1)x^n$[/tex]. Therefore, the series is alternating (-1).
b) The sum of the first four nonzero terms of the series is [tex]$-1x^1+2x^2-3x^3+4x^4$.[/tex]
c) The interval of convergence is [tex](-2, 1)[/tex]. This is because the denominator [tex]$(x-1)^2(x+2)$[/tex] has a zero at [tex]$x=-2$[/tex] and a zero at [tex]$x=1$[/tex], and the function is undefined at these two points. Therefore, the interval of convergence is the open interval [tex]$(-2, 1)$[/tex].
Convergence is the process of two or more different entities coming together and becoming one. In mathematics, it can refer to a sequence converging to a limit, or the process of a function approaching a finite value as the number of trials approaches infinity. In technology, convergence can refer to the combining of different media types into a single medium, such as the combination of audio, video, and text into a multimedia presentation.
To learn more about limit, visit:
brainly.com/question/8533149
#SPJ4
In this triangle, what is the value of x?
Enter your answer, rounded to the nearest tenth, in the box.
Answer:
70
Step-by-step explanation:
David says that a triangle with side measures 9 cm, 12 cm, and 17 cm is a right triangle. Susie says it is not. Who is correct? Explain
your reasoning.
12 cm
17 cm
9 cm
Answer: Susie is correct.
Step-by-step explanation:
Susie is correct, because the Pythagorean theorem does not work on this triangle. Since the Pythagorean theorem only works on right triangles, we can conclude that Susie is correct, and that this triangle is NOT a right triangle.
a²+b² = c²
a = 9
b = 12
c = 17
9² + 12² = 17²
==> 81 + 144 = 289
==> 225 = 289, This is not true
Therefore this triangle is not a right triangle.
Eagle Outfitters is a chain of stores specializing in outdoor apparel and camping gear. They are considering a promotion that involves mailing discount coupons to all their credit card customers. This promotion will be considered a success if more than 10% of those receiving the coupons use them. Before going national with the promotion, coupons were sent to a sample of 100 credit card customers.
Develop hypotheses that can be used to test whether the population proportion of those who will use the coupons is sufficient to go national.
The null hypothesis set up for the hypothesis test of the population parameter is given by H₀: p ≥ 0.1 .
We would set up the hypothesis test.
For the null hypothesis,
H₀: p ≥ 0.1
For the alternative hypothesis,
Hₐ: p < 0.1
This is a two-tailed test.
Taking into account the population proportion, p = 0.1 q = likelihood of failure = 1 - p
q = 1 - 0.1 = 0.9
b) In light of the sample,
P = x/n = sample proportion
Where x = the number of successes, n = the number of samples, and p = 13/100 = 0.13
The test statistic, the z score, would be calculated as z = (P - p)/pq/n z = (0.13 - 0.1)/(0.1 0.9)/100 = 1.
The determined test statistic is 1 for the right tail and - 1 for the left tail.
Since α = 0.05, the critical value is calculated using the normal distribution table.
α₂ = 0.51/2 = 0.025 on the left
The z score for an area to the left of 0.025 is - 1.96
α₂ = 1 - 0.025 = 0.975 for the right
1.96 is the z score for an area to the right of 0.975.
The test statistic must be less than - 1.96 or larger than 1.96 in order to reject the null hypothesis.
We cannot reject the null hypothesis since - 1 > - 1.96 and 1 1.96.
As a result of the hypothesis test results, Eagle Outfitter should take the promotion national.
To learn more about null hypothesis visit:
https://brainly.com/question/16261813
#SPJ4
One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor. (a) Write a differential equation that is satisfied by y. (Use k for the constant of proportionality.) dy dt = ky 1-y. (b) Solve the differential equation. Assume y(0) = C. y = 1-ce-kt +1 (c) A small town has 1300 inhabitants. At 8 AM, 100 people have heard a rumor. By noon half the town has heard it. At what time will 90% of the population have heard the rumor? (Do not round k in your calculation. Round the final answer to one decimal place.) hours after the beginning
(a) The differential Equation that is satisfied by y is dy/dt = k × y × (1 - y) ,
(b) Solution of the differential equation assuming y(0) = c is y = [tex]\frac{1}{(\frac{1-c}{c})e^{-kt} +1 }[/tex] .
In the question ,
Part (a)
let the fraction of people who heard the rumor is = y
So , the fraction who have not heard the rumor is = 1 - y .
the rate of rumor spread is ⇒ dy/dt = k×y(1 - y)
dy/y(1-y) = k.dt ...where k is the constant of proportionality .
So , the differential equation is ..
dy/dt = k × y × (1 - y)
Part (b)
So , 1/y(1-y) = 1/y + 1/(1 - y) ....equation(1)
integrating equation(1) , we get
∫dx/(1 + ax) = ㏑(1 + ax)/a ,....where a is the constant
㏑y + ㏑(1-y)/(-1) = kt + d ,.....where d is the constant
By using , ㏑a - ㏑b = ㏑(a/b) and taking exponential . we get ,
y/(1 - y) = c₁[tex]e^{k\times t}[/tex]
for t = 0 and y(0) = c
solving further , we get
c₁ = c/(1 - c)
So , y = (1-y)c₁[tex]e^{k\times t}[/tex]
y(1 + c₁[tex]e^{k\times t}[/tex]) = c₁[tex]e^{k\times t}[/tex]
y = c₁[tex]e^{k\times t}[/tex]/(1 + c₁[tex]e^{k\times t}[/tex])
taking c₁[tex]e^{k\times t}[/tex] common , and substituting the value of c₁ we get ,
the solution as , y = [tex]\frac{1}{(\frac{1-c}{c})e^{-kt} +1 }[/tex] .
Therefore , (a) the differential equation is dy/dt = k × y × (1 - y) and
(b) the solution is y = [tex]\frac{1}{(\frac{1-c}{c})e^{-kt} +1 }[/tex] .
The given question is incomplete , the complete question is
One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor.
(a) Write a differential equation that is satisfied by y. (Use k for the constant of proportionality.)
(b) Solve the differential equation. Assume y(0) = C.
Learn more about Differential Equation here
https://brainly.com/question/15188890
#SPJ4
Need Awnser asap
Right Awnser gets branliest
Answer:
x1,y16 is the rate of change
Step-by-step explanatI subtracted 13-32 which gives me 16, then I added 16 to 32 to be sure and it gave me 48 so the change on the Y axis is going up by 16 and x axis is going up by 1
I’m stuck on this question
Answer: I believe the answer is triangles b and e
Step-by-step explanation:
Translations preserve side length and the do not rotate they only move right and left and/or up and down. Therefor f is out because the sides get larger and a is out because it would be a reflection of the xaxis and d is out because it would a reflection over the yaxis. C and g are both rotations. This is why only B and E are translations of triangle y.
I hope this helps and makes sense! Let me know!
sketch the region enclosed by the given curves (within the given bounds, if provided) and find its area.
The area enclosed by the curves f(x) and g(x) is 1/12
Integral calculus can be used to compute the area between two curves, which is the region between two intersecting curves.
When we are aware of the equation for two curves and the locations of their intersections, integration can be utilized to determine the area under the curves.
Let our given curves be :
f(x) = x² and g(x) = x³ within the interval [0,1]
Formula to calculate area under these curves is
=> A = [tex]\int\limits^{c}_{a} {|f(x) - g(x)| \, dx[/tex]
Interval is given from 0 to 1
Therefore ,
=> A = [tex]\int\limits^{1}_{0} {[x^2 - x^3]} \, dx[/tex]
Integrating,
=> [tex][\frac{x^3}{3} - \frac{x^4}{4}]^{1}_{0}[/tex]
=> 1/3 - 1/4
=> 1/12 is required area
The given question is incomplete So, I've answered the question in general
To know more about Area under curve here
https://brainly.com/question/15122151
#SPJ4
The enrollment at MSU is described by the function
f(x) = 250x + 6000, where x is the number of years since 2010.
I. Find the enrollment in 2016.
II. In what year will the enrollment reach
10,000?
1) The enrollment in 2016 will be 7500.
2) In 2026 year the enrollment reach 10,000.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The function f(x) represents the number of enrollment.
Defined as;
⇒ F(x) = 250x + 6000
Where x represents year since 2010.
(1) Now for finding the enrollment in 2016;
Put x = 2016 - 2010 = 6 in the function
⇒ F(6) = 250x6 + 6000
= 7500
Thus, The required number of enrollment = 7500
(2) Now we have to find the year in which enrollment reach 10,000;
i.e f(x) = 10,000
=> 250x + 6000 = 10000
=> 250x = 4000
=> x = 16
Thus, The required year = 2010 + 16
= 2026 answer.
1) The enrollment in 2016 will be 7500.
2) In 2026 year the enrollment reach 10,000.
Learn more about the mathematical expression visit:
brainly.com/question/1859113
#SPJ1
Find the Probability of, A King, ace, jack of clubs or queen of diamonds appears in drawing a single card from a well shuffled ordinary deck of 52 cards.
The Probability of, a ace, jack of clubs, King or queen of diamonds appears in drawing a single card from a well shuffled ordinary deck of 52 cards is 4/13.
As per the given data,
we need to find out the probability of, King, ace, jack of clubs or queen of diamonds appears in drawing a single card from a well shuffled ordinary deck of 52 cards.
We know that,
Probability(Event) =Number of Favorable Outcomes/Total number of
Outcomes = x/n.
Total number of Outcomes is 52 (given)
So, we will calculate Number of Favorable Outcomes as follows:
The total no. of King in deck of card is 4
The total no. of queen in deck of card is 4.
The total no. of ace in deck of card is 4.
The total no. of jack in deck of card is 4.
Therefore, the number of favorable outcomes is 4+4+4+4= 16
Now, putting values in the above stated formula of probability, we get:
Probability= 16/52=4/13
Therefore, the probability of pulling a King, Ace, Jack of Clubs, or Queen of Diamonds from a 52-card standard deck that has been properly shuffled is 4/13.
For more such questions on Probability:
https://brainly.com/question/25870256
#SPJ4
An SRS of 20 third grade children is selected in Chicago and each is given a test to measure their reading ability. We are interested in a 90% confidence interval for the population mean score. In the sample, the mean score is 64 points, and the standard deviation is 12 points. The margin of error associated with the confidence interval is
O 4.64 points.
O 5.62 points.
O 2.68 points.
The margin or error associated with the confidence interval is 5.62 points
The sample size of the third grade children = 20
The confidence interval for the population mean score = 90%
The mean score = 64 points
The standard deviation = 12 points
The degree of freedom = The sample size - 1
= 20 - 1
= 19
The critical value of t at degree of freedom 19 = ±2.093
The margin error = Critical value of t × [tex]\frac{s}{\sqrt{n} }[/tex]
Substitute the values in the equations
The margin error = 2.093 × [tex]\frac{12}{\sqrt{20} }[/tex]
= 2.093 × 2.68
= 5.62 points
Therefore, the margin error is 5.62 points
Learn more about margin error here
brainly.com/question/29101642
#SPJ4
Using the above graphic, it takes about _____ years to transition from using no oil to consuming 100 million barrels per day whereas it takes about______ years to transition from using 100 million to 200 million barrels per day.
A. 60, 10
B. 10, 60
C. 120, 50
D. 50, 120
It takes about 120 years for the graph to reach 100 million barrels per day whereas it takes 50 years to go from 100 million to 200 million.
As per the question the left axis indicates the amount of oil in the earth in trillions of barrels.
The right axis indicates the global consumption rate of oil in millions of barrels per day.
To the left of the red vertical line are model results that approximate reality whereas to the right are model-based predictions of the future.
The bottom axis is time in years.
The graph attached at the end of the solution.
Let the number of years of transition from using no oil to consuming 100 million barrels per day be a.
Let the number of years of transition from using 100 million to 200 million barrels per day be b.
From the given graph, we can see that initial consumption rate is low, and it takes 120 years for the graph to reach 100 million barrels.
But it takes 50 years to go from 100 million to 200 million.
This is known as exponential growth.
Therefore, the value of a is 120 and the value of b is 50.
For more questions on exponential growth
https://brainly.com/question/11487261
#SPJ4
A bag contains 3 red, 6 blue, and 7 yellow marbles. What is
the probability of drawing 2 marbles of different colors out of
the bag?
The probability of drawing 2 marbles of different colors would be 0.317.
What is probability?
Probability is a measure of the likelihood of an event occurring. It is typically expressed as a fraction or a decimal between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain to occur.
In the scenario you described, a bag contains 3 red, 6 blue, and 7 yellow marbles. If you draw 2 marbles out of the bag, the probability of drawing 2 marbles of different colors is:
Number of ways to draw 2 marbles of different colors: (3 red marbles) x (6 blue marbles) + (3 red marbles) x (7 yellow marbles) + (6 blue marbles) x (7 yellow marbles) = 18 + 21 + 42 = 81
Total number of ways to draw 2 marbles: 16 (4 marbles of each color)
Probability of drawing 2 marbles of different colors: 81/256 = 0.317
Hence, the probability of drawing 2 marbles of different colors would be 0.317.
To learn more about the probabilities, visit:
https://brainly.com/question/24756209
#SPJ1
The probability of drawing 2 marbles of different colors would be 0.317.
What is probability?
Probability is a measure of the likelihood of an event occurring. It is typically expressed as a fraction or a decimal between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain to occur.
In the scenario you described, a bag contains 3 red, 6 blue, and 7 yellow marbles. If you draw 2 marbles out of the bag, the probability of drawing 2 marbles of different colors is:
Number of ways to draw 2 marbles of different colors: (3 red marbles) x (6 blue marbles) + (3 red marbles) x (7 yellow marbles) + (6 blue marbles) x (7 yellow marbles) = 18 + 21 + 42 = 81
Total number of ways to draw 2 marbles: 16 (4 marbles of each color)
Probability of drawing 2 marbles of different colors: 81/256 = 0.317
Hence, the probability of drawing 2 marbles of different colors would be 0.317.
To learn more about the probabilities, visit:
brainly.com/question/24756209
#SPJ1
Is 5x-8+7y=y-6 linear or nonlinear
Answer:
Step-by-step explanation:
The equation 5x-8+7y=y-6 is linear because it contains only terms with the variables x and y raised to the power of 1. In a linear equation, the highest power of any variable is 1. Nonlinear equations contain exponents that are higher than 1 on one or more variables.
5. Add the polynomials.
(3k² + 2k - 3) + (- 2k² + 8k + 11).
k² + 10k + 8 is the sum of (3k² + 2k - 3) + (- 2k² + 8k + 11).
Define addition.The act of combining two or more items is known as addition. The process of computing the sum of two or more numbers in mathematics is called addition. It is a fundamental mathematical procedure that is frequently used in daily life. When we work with money, figure out our food bills, or figure out the time, we frequently employ addition. Mathematicians employ the action of addition to combine numbers. The sum of the given numbers is the outcome of addition, or the total of the inputted numbers. For instance, when we add the numbers 2 and 3, we get the number 5. Here, we added the two numbers 2 and 3 to obtain the result, which is 5.
Given
Polynomials
(3k² + 2k - 3) + (- 2k² + 8k + 11)
Adding the polynomials with same exponents
3k² + 2k - 3 - 2k² + 8k + 11
3k² - 2k² + 2k + 8k - 3 + 11
k² + 10k + 8
k² + 10k + 8 is the sum of (3k² + 2k - 3) + (- 2k² + 8k + 11).
To learn more about addition, visit:
https://brainly.com/question/8265353
#SPJ1
Mr. White is renting a small truck for one week and a few additional days d. He does not have to pay a per mile fee. Evaluate the expression 460+140d to find how much he will pay for a 10-day rental.
In linear equation, the amount of money paid for the 10-day rental will be $ 1860.
What is a linear equation?
A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept.
The variables in the preceding equation are y and x, and it is occasionally referred to as a "linear equation of two variables."
The amount of money for a 10-day rental will be calculated as,
A = 460+140d
A = 460+(140 x 10)
A = 460 + 1400
A = 1860
Therefore, the amount of money paid for the 10-day rental will be $ 1860.
Learn more about linear equation
brainly.com/question/11897796
#SPJ1
find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius r.
The volume of the largest right circular cylinder is [tex]=\frac{4\pi r^3}{3\sqrt{3} }[/tex] cu. unit
Now, According to the question:
The given sphere is of radius R.
Let h be the height and r be the radius of the cylinder inscribed in the sphere.
We know that:
Volume of cylinder
V = [tex]\pi R^2h[/tex] .....(1)
In right Triangle OBA
[tex]AB^2 + OB^2 = OA^2[/tex]
[tex]R^2 + \frac{h^2}{4} = r^2[/tex]
So, [tex]R^2 = r^2 - \frac{h^2}{4}[/tex]
Putting the value of [tex]R^2[/tex] in equation (1), We get
V = [tex]\pi (r^2 - \frac{h^2}{4} )h[/tex]
V = [tex]\pi (r^2h - \frac{h^3}{4} )[/tex] ....(2)
dV/dh = [tex]\pi (r^2 - \frac{3h^2}{4} )[/tex] .....(3)
For, Stationary point, dV/dh = 0
[tex]\pi (r^2 - \frac{3h^2}{4} )[/tex] = 0
[tex](r^2 - \frac{3h}{4} )[/tex] => [tex]h^2 - \frac{4r^2}{3}[/tex] => [tex]h - \frac{2r}{\sqrt{3} }[/tex]
Now, [tex]\frac{d^2V}{dh^2} = \pi (-\frac{6}{4}h )[/tex]
[tex][\frac{d^2V}{dh^2}]_a_t_h_=_\frac{2r}{\sqrt{3} }[/tex] = x[-3/2 , [tex]2r/\sqrt{3}[/tex]]< 0
Volume is maximum at h = 2r/[tex]\sqrt{3}[/tex]
Maximum volume is :
[tex]= \pi (r^2.\frac{2r}{\sqrt{3} }- \frac{1}{4}.\frac{8r^3}{3\sqrt{3} } )[/tex]
[tex]=\pi (\frac{2r^3}{\sqrt{3} }-\frac{2r^3}{3\sqrt{3} } )[/tex]
[tex]=\pi (\frac{6r^3-2r^3}{3\sqrt{3} } )[/tex]
[tex]=\frac{4\pi r^3}{3\sqrt{3} }[/tex] cu. unit
Learn more about Volume of Cylinder at:
https://brainly.com/question/16134180
#SPJ4
You brought popular game on sale for $20 and want to sell it on eBay. You want to mark up the toy 60%. What did you sell it for?
A building has a height of 125 meters and a length of 80 meters. On a scale drawing of the building, the height is 25 centimeters. What is the length of the building on the scale drawing in centimeters?
The length of the building on the scale drawing is found as 16 centimeters.
Describe the term scale factor?The ratio of the scale of an original thing to a new object that is a representation of it but of a specific widths is known as a scale factor (bigger or smaller).Scale factor ratios can be written as a colon, 1:2, or as a fraction, 12. A ratio calculates the difference between two values.However, that ratio isn't a scale factor, thus you could not build a ratio for left-handed pupils to all students.For the given data in the question-
Dimensions of the building;
Height of 125 meter length of 80 meters.Dimensions of the drawing of the building;
height is 25 centimetersLet the length be 'x'.Thus, taking ratios of Height to length
125/80 = 25/x
On simplification;
x = 80 x 25 / 125
x = 16
Thus, the length of building on the scale drawing is found as 16 centimeters.
To know more about the scale factor, here
https://brainly.com/question/25722260
#SPJ1
