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work out dy/dx for the gradient of the tangent line and hence find the equation using y = mx + c.
This line will cross the curve again so work with a pair of simultaneous equations (tangent equation) and (curve equation).
The resulting equation in x will look complicated but you have one extra clue ... you know (1/2 , 1/8) lies on both so the expression must factorise with (x-1/2) or (2x-1) as a factor. That leaves a quadratic implies two more solutions. Wait a mo though. If the line makes a tangent at that point then (2x-1) will be a double factor leaving only a linear equation left for the solution you want.
B.
This is one is a bit unclear.
dy/dx will tell you the gradient of the tangent line. But you don't know the release point so call it (a,b) The tangent line goes through (a,b) and you can also write the gradient in terms of a and b. So form the equation of the tangent line (y=mx+c)
You can substitute for b using the original equation. You also want the line to go through (1,0) so you cab use this to get an equation for finding a.
B
I gotta play with this some more.
mathland wrote:Mathegocart wrote:Hello mathland,
yup, that's all correct - meet or exceed is indeed >=.How about that? It was a guess, really. It's important to understand the WHY in math.
I agree. There wasn't much more - you just stated the problem in mathematical form. I.e, 299mg * cups of milk + 261mg * cups of juice (meets or exceeds, i.e greater than or equal to) 1000 mg.
Great stuff.
Thanks. Always look for my questions. I absolutely never get enough mathematics.
The graph of constants is a horizontal straight line so their gradient function is zero. You have treated R as a variable. It isn't. It is the fixed radius of the tube. Little r is the only variable.
So dv/dr = -2kr
Bob
You are right. This is the correct answer. Can you set up part (c)?
Find dW/dt and then put in the given t value.
B
Hello Bob.
Let me see.
dW/dt = d/dt [35,000 − 20t^2]
dW/dt = -40t
What do I do with dW/dt?
hi mathland,
Rate of change is the gradient of the graph so these questions require dv/dr. (k and big R are constants for differentiation purposes)
Once you have done (a) you'll be able to do (b) and (c) by substituting values of r.
Bob
Good Monday to you. I hope you had a great weekend.
Here is my work for part (a).
dV/dr = k(R^2 − r^2)
dV/dr = kR^2 kr^2
dV/dr = 2k - 2k
dV/dr = 0
How does 0 help me solve part (b)?
The French physician Poiseuille discovered that the volume V of blood (in cubic centimeters per unit time) flowing through an artery with inner radius R (in centimeters) can be modeled by V(R) = kR^4 where k = π/(8vl) is constant
(here, ν represents the viscosity of blood and l is the length of the artery).
(a) Find the rate of change of the volume V of blood flowing
through the artery with respect to the radius R.
(b) Find the rate of change when R = 0.03 and when R = 0.04.
(c) If the radius of a partially clogged artery is increased from
0.03 cm to 0.04 cm, estimate the effect on the rate of change
of the volume V with respect to R of the blood flowing
through the enlarged artery.
NOTE: I am not seeking the answer but the set up for all three parts only. I will do the math work.
Thank you.
Live long and love math!
Let T be the line tangent to the graph of y = x^3 at the point (1/2, 1/8).
At what other point Q on the graph
of y = x^3 does the line T intersect the graph? What is the slope of
the tangent line at Q?
NOTE: I am not seeking the answer but the set up only. I will do the math work.
Thank you.
A dive bomber is flying from right to left along the graph of y = x^2.
When a rocket bomb is released, it follows a path that is approximately along the tangent line. Where should the pilot release the bomb if the target is at (1, 0)?
NOTE: I am not seeking the answer but the set up only. I will do the math work.
Thank you.
Water is leaking out of a swimming pool that measures 20 ft by 40 ft by 6 ft. The amount of water in the pool at a time t is W(t) = 35,000 − 20t^2 gallons, where t equals the number of hours since the pool was last filled. At what rate is the
water leaking when t = 2h?
NOTE: I am not seeking the answer but the set up only. I will do the math work.
Thank you.
The velocity v of a liquid flowing through a cylindrical tube is given by the Hagen–Poiseuille equation
v = k(R^2 − r^2), where R is the radius of the tube, k is a constant
that depends on the length of the tube and the velocity of the liquid at its ends, and r is the variable distance of the liquid from the center of the tube.
(a) Find the rate of change of v with respect to r at the center of
the tube.
(b) What is the rate of change halfway from the center to the wall
of the tube?
(c) What is the rate of change at the wall of the tube?
1. I am not seeking the answer but the set up only for all three parts. I will do the math work.
2. The rate of change is the derivative. Right? I need to clearly understand what is meant by RATE OF CHANGE before moving on with my calculus textbook.
Thank you.
A general formula that I think applies here is y = ab^x.
Let a = 50
Let b = double or 2
We get P = 50(2)^12n. Here 12 represents the months of a year.
A friend told me that the correct answer is P = 50(2)^(n/12).
Is my friend right? If so, why is my set up wrong?
Try plugging in a couple of values into your equation after developing it - it'll help a ton, especially when it comes to these equations modelling real life.
Since the population of animals at Central High School doubles every 12 years, the period is 12. So when n = 12, we should expect the population to double. If we plug n=12 years into your equation, we see the population skyrockets to an eyewatering 1.11*10^45. Clearly not a doubling in a year.
As we want the population of animals to double in this exponential model, we want P=100 when n=12, so let's solve a simple mathematical equation to determine the exponent's power.
Let k be a real-valued number.
population = base*(how much the population increases every period number of years)^(n*k)
100 = 50*2^(12*k) (divide by 50)
2 = 2^(12*k)Obviously, to equalize both sides, k must be 1/12. Hence, your friend is right - the exponential function modelling the population of animals is indeed 50*(2)^(n/12).
A great, detailed reply. Please, look for my questions from now on. I know that Bob is always on the look out for my threads. Maybe you guys can share in terms of reply. If I don't show my work, it simply means that I haven't the slightest idea how to begin to post an answer. Otherwise, I show work or effort. Is there a way to upload pictures on this site? I am thinking geometry,, trigonometry and calculus questions involving a geometric interpretation. You say?
Hi mathland,
you're all good here. 12 bucks times n items is indeed 12n as written, and 7n + 350 is indeed the cost of producing n items.
It certainly feels good to be right.
You're rather close here, but note that the question states that a tax of 8% is applied(i.e, added onto) the room rate. Your equation as currently written would apply a 92% tax on the hotel rate.
So it should be T(x) = 1.08(99.95x) + 5.00.
Where did 1.08 come from?
Hello mathland,
yup, that's all correct - meet or exceed is indeed >=.
How about that? It was a guess, really. It's important to understand the WHY in math.
I think that's correct. The female result will pull up the overall average compared with male alone and the male result will pull down the overall compared with the female alone.
I'd do some algebra here.
Lets say there are M males and F females.
Then we know that total males = 15M and the total females = 19F.
So the overall average = (15M + 19F)/(M+F)
(15M + 15F)/(M+F) < (15M + 19F)/(M+F) < (19M + 19F)/(M+F)
15 < (15M + 19F)/(M+F) < 19
Bob
I guess I was right.
At a primate reserve, the mean age of all the male primates is 15 years, and the mean age of all female primates is 19 years. Which of the following must be true about the mean age m of the combined group of male and female primates at the primate reserve?
1. m = 15 + 19
2. m > 15 and less than 20.
3. 15 < m < 19
4. None of the above.
Let me see.
I say the answer is choice 3. Of course, this is just my guess.
If I am right, what makes choice 3 correct?
A biology class at Central High School predicted that a local population of animals will double in size every 12 years. The population at the beginning of 2014 was estimated to be 50 animals. If P represents the population n years after 2014, write an equation that represents the class model of the population over time?
Let me see.
A general formula that I think applies here is y = ab^x.
Let a = 50
Let b = double or 2
We get P = 50(2)^12n. Here 12 represents the months of a year.
A friend told me that the correct answer is P = 50(2)^(n/12).
Is my friend right? If so, why is my set up wrong?
A company’s manager estimated that the cost C, in dollars, of producing n items is C equals 7 n plus 350. The company sells each item for $12. The company makes a profit when total income from selling a quantity of items is greater than the total cost of producing that quantity of items. Write an inequality that gives all possible values of n for which the manager estimates that the company will make a profit?
Let me see.
We are given C = 7n + 350.
Let n = number of items.
Let 12n = total income for selling a quantity of items.
I can replace C with 12n.
I say the inequality is 12n > 7n + 350.
I now need to solve for n.
Yes?
Aaron is staying at a hotel that charges $99.95 per night plus tax for a room. A tax of 8% is applied to the room rate, and an additional onetime untaxed fee of $5.00 is charged by the hotel. Represent Aaron’s total charge, in dollars, for staying x nights?
Let me see.
Let T = Aaron’s total charge for staying a certain amount of nights.
We need: 99.95x.
We are told that a tax of 8% is applied to the room rate plus a one time untaxed fee of $5.
I think the equation is T(x) = 0.08(99.95x) + 5.00.
Yes?
The recommended daily calcium intake for a 20-year-old is 1,000 milligrams (mg). One cup of milk contains 299 mg of calcium and one cup of juice contains 261 mg of calcium. Write an inequalities that represents the possible number of cups of milk m and cups of juice j a 20-year-old could drink in a day to meet or exceed the recommended daily calcium intake from these drinks alone?
Let me see.
299m + 260j >= 1,000
Is this correct?
No Facebook. I am horrified by how intrusive membership of Facebook is. Too much personal information shared widely. The scandal involving Cambridge Analytica and Facebook trying to fix election results just shows how dangerous it is.
Shortly after the "Leave the EU" referendum result the BBC had a Question Time programme and the topic of the vote came up. A member of the audience was not embarrassed to announce that she was a supporter of 'remain' right up to the day of the vote. Then she thought about those stories that the EU has a regulation forbidding bent bananas and decided to switch to 'leave'. That fake news came from social media.
For a special limited reason I did briefly join under a fake name. Despite entering false information about myself, and keeping my data private, I still received Friends Requests from people I know. I cancelled my membership.
Bob
I myself may leave FB soon. I am a Trump guy. FB is a far left social media that has connected me with friends of long ago. Still, FB greatly dislikes Trump. I fear what America will be like after 4 years under the Biden administration. SAT, GRE, AND GMAT word problems this entire weekend. I will show my work. Ready to have fun with math this weekend?
If s = √4.t then ds/dt = √4
If
then
At t = 1, ds/dt = 1
At t = 4, ds/dt = 1/2
Bob
Wonderfully easy. Thanks.
1. Are you a math professor?
No. after getting a maths degree I became a teacher (11-18).
2. If you are a math professor, do you teach math beyond Calculus 3 aka Vector Calculus?
No. I'd have to look up what that is.
3. There is a huge difference between what is called the REGULAR CALCULUS SEQUENCE (CALCULUS l, ll, and lll) and Advanced Calculus. Do you also teach Advanced Calculus? I think the course also goes by the name Tensor Calculus.
Never done tensors.
4. Are you on FB?
What is FB?
5. Do you have your own math website?
Used to when I taught. It was part of my broadband package. When my A level ICT students constructed a website I would publish it live for them on my site, so they could see if it really did as expected and get another student to test it for them. When I switched broadband provider, I had to give up the site.
6. I, like most students, struggle with word problems, especially SAT, GRE and GMAT word problems. Do I plan to take the SAT, GRE AND GMAT? No. I am 56 years old but the word problems found in test prep books for the SAT, GRE AND GMAT make good practice problems. What is the best way to improve word problems skills? My greatest struggle is converting WORDS to algebraic expressions or equations.
I don't know what any of those acronyms stand for.
MIF has a page on this https://www.mathsisfun.com/algebra/word … lving.html
Here's one of the problems from that page. I'll take you through how I would solve it.
The denominator of a fraction is 3 more than the numerator. If both the numerator and the denominator are increased by 4, the fraction becomes 4/5
What was the original fraction?Let one of the unknowns be x. Then 'translate' the information into expressions with x in.
Let the original numerator be x.
The denominator is 3 more, so it is x+3
Both are increased by another 4, so the new numerator becomes x + 4, and the new denominator becomes x + 7
So now the fraction is (x+4)/(x+7) and we're told that is 4/5. So now I can make and solve an equation.
Multiply by 5(x+7)
Check this works.
Original was 8/11. New is 12/15 = 4/5 Tick.
7. Do you think it is normal for a man my age to have a passion for learning math, even though I am never going to become a math teacher?
Not many would want to do this but I think it's great that you want to.
Bob
Hello Bob. See below.
1. By 11-18 you mean ages 11 to 18, right? This is your teaching age category. Yes?
2. Vector Calculus is known as Calculus 3 in America. It is the Calculus that deals with vectors, points in space, 3D graphs, double and triple integrals, etc.
3. FB = FACEBOOK. Are you on FB?
4. If you are on FB, I can create a math group for us to further discuss mathematics.
5. SAT, GRE AND GMAT are prerequisite exams.
High school students take the SAT to get placed into advanced classes or to earn college credits. The GRE stands for GRADUATE RECORD EXAMINATION. The GRE is taken by students wishing to enter Graduate School. However, most Graduate Schools no longer require the GRE as the demand for more students has become a great need for colleges. The GMAT is taken by people wanting to enter certain business schools. I like the SAT, GRE AND GMAT practice word problems. Very challenging but fun.
6. The practice word problem you posted is easy. The SAT, GRE AND GMAT problems are more involved. I will post a few questions in the coming days, if time allows.
7. I get criticized by friends and family for studying math. Why? What's the big crime? I don't get it. I am not doing anything wrong. I am not hurting anyone.
I am simply learning advanced material that fascinates me. Besides, I am a 56, lonely, family not too close, my adult son living his own life, etc. What else am I to do at 56? Math keeps my memory power in tact, at least I hope it does.
People do not like to see others happy. I had the same problem when I decided to enroll in the Family Radio School of the Bible many years ago. Friends and family criticizing my study time with Scripture. Why?
Bob, I am a loner. Most loners are misunderstood people. I am most happy in my alone time. Sociable people envy loners. Sociable people don't understand why loners are happy when alone. This is a puzzle for the gregarious bunch. Anyway, let me know about FB.
As an object in rectilinear motion moves,
its distance s from the origin at time t is given by the equation
s = s(t) = √4t, where s is in centimeters and t is in seconds.
Find the velocity v of the object at t_0 = 1 and t_0 = 4.s is often used in mechanics for distance to avoid confusion with calculus notation, where 'd' has a special meaning.
If you plot the graph of s against t, you'll get a straight line with gradient √4. The gradient is a measure of velocity as V = D/T so the velocity is √4 at t =1 and t = 4.
It occurs to me that maybe the problem was
You could use limits to get an answer like this:
Let A = (1,2) Let B = [1.1, √(4.4)] calculate the gradient of AB using (difference in y coords) / (difference in x coords).
Then recalculate with B = [1.01,√(4.04)]. Continue, gradually moving B closer and closer to A.
Once you have some calculus this question can be answered more easily using differentiation.
Bob
Thank you. I am familiar with basic differentiation. How can we use differentiation to solve this problem more easily?