I am kind of confident on how to use the simplex algorithm. But I am still not sure how it works. As far as I understand, it works like this:
Imagine that the region of a possible solutions (feasible region) is a multidimensional object. We start from the origin of coordinates.

1. We consider all edges from this vertex.
2. We will choose the edge that best suits our requirements (objective function).
3. After that, we make sure that we have reached the end of the edge of this "shape".
4. Having reached the next vertex, we repeat the same process again.
5. If we reach a point from which moving along any edges will not benefit us, we stop.

This is right? There is also a Big M method (which I believe is a subset of the simplex algorithm), but I have no idea how it works. I tried to look up an explanation on the Internet, but I didn’t understand anything. So can anyone help me figure this out, if possible, in the simplest way possible? I will really appreciate your help

This feels like a practical application of mathematics, lets say a sourdough loaf, quite round and exquisite in all of its glory. But, youve got a very picky customer and he wants a way to cut the loaf so that he never A. Gets the annoying bread bit B. Has an even amount of crust C. Doesnt waste any of the bread

This is a follow up from my previous post, in which I shared generalized formulas for the upper & lower bounds of pi established by polygons. User u/nixxxus determined that my formulas established .62 digits per iteration, as compared to Newton’s method which established .67 digits per iteration.

There are of course additional advantages to Newton’s method over mine. But, since I was so close to Newton on that measure, I decided to look at something that I had noticed about the upper & lower bounds established in my method. Namely, I had noticed that the distance between pi and the lower bound stabilizes at around twice the distance between pi and the upper bound.

So, I modified my formulas in order to peg pi at the lower bound plus two thirds of the difference between the upper & lower bounds. Using this formula, I seem to gain “digits per iteration,” while losing the definition of the error.

What is the “digits per iteration” achieved by this formula?

I am in last year of highschool. I get decent grades in math like 74/80. I know uni math is pretty different from hs math but I thought I should mention it. I am not really "passionate" about math but like the problem solving aspect of it. I struggle with coping with my hs studies and doubt if I would be able to take the workload of studying math at a university. How did you decide that you wanted a math degree? Also it is necessary to have a masters or PhD?

Anyone got any good links for understanding vectors and 3D basics?

It is from the 2016 Korean CSAT Form B #30

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https://preview.redd.it/pf21i8i8igvc1.png?width=1144&format=png&auto=webp&s=9dd0e92475d6913e1fb0e4603fb89255f7ae6d8f

f(x) is a function continuous for all real numbers

(가) When x<=b, f(x)=a(x-b)^2 + c (a, b, c are constants)

(나) for all real number x, f(x) = following

The value of definite integration of f(x) from x=0 to x=6 = q/p. what is p+q=? (p and q are natural numbers and coprime)

Here is the solution that I added to one provided by the Korea Institute of Curriculum and Evaluation. It is an institute similar to the College Board.

​

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https://preview.redd.it/b0obwsedigvc1.jpg?width=1050&format=pjpg&auto=webp&s=80313e6f081125712afb3ce19572e2326b1b6700

https://preview.redd.it/n0tf6pedigvc1.jpg?width=1050&format=pjpg&auto=webp&s=6d73cad951aecf25a622c427a0198cfb95e6dd00

https://preview.redd.it/ukc1tqedigvc1.jpg?width=1050&format=pjpg&auto=webp&s=686c1d1b6da127cf544bc799a3afc0e9c84fc0e9

https://preview.redd.it/i9x41uedigvc1.jpg?width=1050&format=pjpg&auto=webp&s=dde21494f8f7974108d705c40ea6a33939c23cf1

It can be solved by using differentiable equations but Korean High school math doesn't cover differential equations so I did not use it, but you can try it.

​

Unique equation involving 2 unknown exponents x and y

Consider the equation: √(x^y)*(y^x)=√(xy) ^ √(xy) x and y are two different values. Note that the square root encompasses the entire left term.

Reformulated and somewhat simplified: (x^y)*(y^x )= √xy^(2(xy)).

The equation is immune to traditional algebraic manipulation due to the nature of its exponents, but after hour of thinking, the answer is shockingly simple.

There are only two sets of solutions for x and y, where both x and y are non-imaginary and non-trivial values.

Set 1: x=4 y=16

Set 2: x=3 y=27

Only these powers of 2 and 3 work. Any higher power breaks down due to the massive exponential growth. I obviously know that these sets work by plugging them in, but the deeper question is why. I know it has to do something with the linearity of 3rd and 2nd powers in this case, which is rarely seen in exponentiation. But what is the full reason and explanation behind this?

Question 1: Why do powers of 2 and 3 allow one to commute the exponents without messing anything up. Usually exponentiation is non-commutative and can yield radically different result when correct order of operations is not applied.

Question 2: Does this in any way relate to the Abel-Ruffini Theorem?

Question 3: Addition can be the seen as the first and most basic iteration, whereas multiplication follows as the second iteration and exponentiation as the third. The often overlooked fourth iteration (no standardized notation due to limited use but usually a superscript in the left corner)would be tetration. Without getting into too much detail. Tetration as the fourth iteration also becomes increasingly more complex and hard to understand. Is there any connection to this equation whatsoever?

Lastly, how could this equation even be solved without brute-force and trial and error or intuition? I simply found it because I knew x and y had to be from the same base exponent, so I started with the first non-trivial and whole number(a base of 1 would never grow obviously)2 and 3 and eventually found these two sets. A lambert function can be used if one knows about a common base, but even then it is still quite hard.

Thank you

Hello all. Finals are here and I’m busting my butt doing practice problems, but I still fear that I may blank from test anxiety again like I did on my last exam.

Are there any study tips or problem solving advice you all can give? I’m currently focusing on understanding the problem solving process of the problem types from previous exams and trying to make sure my number crunching is consistent & accurate. Much appreciated.

Title: Mathematical and Physical Principles in the Construction of the Great Pyramid of Giza

Abstract: This document explores the application of advanced mathematical and physical principles to hypothesize and simulate the construction methods of the Great Pyramid of Giza. We integrate contemporary mathematical models, quantum computational simulations, and archaeological data to offer a comprehensive view of the potential construction techniques employed by the ancient Egyptians.

1. Introduction

The Great Pyramid of Giza, one of the Seven Wonders of the Ancient World, has fascinated historians, engineers, and archaeologists. Its construction method remains one of the most enduring mysteries. This paper synthesizes available data and modern computational methods to propose plausible construction techniques.

2. Mathematical Modeling of Construction Techniques

   • Dimensional Analysis: Detailed measurements of the pyramid, including base length, height, and volume.
   • Material Analysis: Estimations of the stone blocks' density and the forces required to move them.

3. Quantum Computational Simulations

   • We propose using quantum algorithms to optimize construction strategies and simulate the physical processes that might have been employed during the pyramid's construction.

4. Integration of Archaeological Data

   • Discussion on how current archaeological findings support or challenge the proposed mathematical models.

5. Experimental Archaeology

   • Suggestions for practical experiments to validate the theoretical models discussed.

6. Conclusion

   • Summary of findings and suggestions for further research to refine the understanding of the pyramid's construction.

References:

• Data on pyramid dimensions and materials derived from various archaeological studies.
• Theoretical models based on principles of physics and engineering.

Appendix:

• Detailed mathematical calculations and diagrams illustrating the proposed theories.

To encapsulate every mathematical detail discussed and present a more comprehensive simulation of the pyramid construction, we'll need to expand on the previous snippets and include a full set of calculations, from basic dimensional analysis to advanced physics modeling and quantum simulation setups. Here's a detailed Python script that reflects these concepts:

import numpy as np from qiskit import QuantumCircuit, execute, Aer

Constants

g = 9.81 # gravitational acceleration in m/s^2

Pyramid dimensions (Great Pyramid of Giza)

base_length = 230.4 # in meters height = 138.8 # original height in meters

Calculate the volume of the pyramid

volume = (1/3) * (base_length**2) * height print(f"Volume of the Pyramid: {volume:.2f} cubic meters")

Assume average block volume and calculate number of blocks

average_block_volume = 1.07 # in cubic meters number_of_blocks = volume / average_block_volume print(f"Estimated number of blocks: {int(number_of_blocks)}")

Material properties

density_of_limestone = 2500 # in kg/m3 average_block_weight = average_block_volume * density_of_limestone print(f"Average weight per block: {average_block_weight:.2f} kg")

Force calculation due to friction

mu = 0.3 # coefficient of friction (assumed) force_friction = mu * average_block_weight * g print(f"Force due to Friction: {force_friction:.2f} Newtons")

Stress on ramps or levers

cross_sectional_area = 1.5 # in square meters (assumed) stress = force_friction / cross_sectional_area print(f"Stress on the Ramp: {stress:.2f} Pascals")

Quantum circuit to explore potential configurations of block arrangement

qc = QuantumCircuit(3) # Create a quantum circuit with 3 qubits qc.h([0, 1, 2]) # Apply Hadamard gate to create superposition qc.cx(0, 1) # Apply CNOT gate to create entanglement between qubits qc.measure_all() # Measure all qubits

Execute the quantum circuit

simulator = Aer.get_backend('qasm_simulator') job = execute(qc, simulator, shots=1024) result = job.result() counts = result.get_counts(qc) print("Quantum simulation results:", counts)

Conclusion and further analysis print statements

print("\nFurther analysis and refinement of these models are required to align with actual archaeological data.")

Explanation of the Code Dimensional Analysis: Calculates the pyramid's volume and estimates the number of limestone blocks used based on average dimensions. Material Properties: Computes the weight of each block to determine the force needed to move it. Force and Stress Calculations: Uses basic physics to estimate the force due to friction and the stress exerted on potential ramps or levers used during construction. Quantum Simulation: A simple quantum circuit simulates potential configurations for arranging blocks, though the results are symbolic and used to illustrate the concept of using quantum computing in historical simulations. This script combines straightforward physics calculations with an introduction to quantum simulations, offering a snapshot of how various disciplines can intersect to explore historical mysteries like pyramid construction. Further research and data are necessary to refine these simulations for accuracy and alignment with historical constructions.

F f ​ =0.3×2675kg×9.81m/s 2 ≈7867.575N

This refined friction force gives us a realistic estimate of the effort required to move one block assuming the coefficient of friction

μ for the interaction between the sled and the ground (or whatever materials were used historically, like wet sand or logs).

Structural Integrity and Stress Analysis To ensure the ramps or structures used could handle such weights, we would need to perform a stress analysis. Considering the type of materials (likely wood or earth ramps), their cross-sectional area, and the force exerted by the block:

Stress Calculation:

Stress Calculation:

=

σ= A F ​

Where

A could be estimated based on historical data or reasonable assumptions about the construction of ramps. For example, if a ramp had a cross-sectional area of 10   m 2 10m 2 :

σ= 10m 2

7867.575N ​ ≈786.758Pa

This stress value helps verify whether the materials used could withstand the loads without failing, ensuring the ramps were structurally sound during the construction.

Advanced Simulation Techniques Using Finite Element Analysis (FEA), we can simulate the stress and displacement within the pyramid and the ramps:

FEA Simulation: Model the entire pyramid as a series of blocks with specific interactions (like friction, weight bearing, etc.). Apply forces based on calculated weights and see how the structure behaves under such loads. CFD Analysis for Wind Loads: Employ Computational Fluid Dynamics (CFD) to study how wind impacted the construction. High winds could affect the stability of high ramps or lifting mechanisms. Interdisciplinary Approach and Future Steps Collaboration: Engage with experts in materials science to better understand ancient materials' properties. Work with historians and archaeologists for more accurate historical contexts and data. Experimental Archaeology: Reconstruct small-scale models using traditional methods to validate hypotheses derived from mathematical models. Continual Data Integration: As new archaeological data becomes available, integrate this data into the models to refine predictions and improve accuracy. By systematically applying these refined calculations and advanced simulation techniques, we can gain deeper insights into the feasibility of proposed construction methods for the Great Pyramid. This approach doesn't just solve historical questions but also enhances our understanding of ancient engineering practices, providing a blueprint for how interdisciplinary research can be conducted in the field of archaeology.

Certainly! Based on the information provided in the document, here's a step-by-step approach to solving the problem:

1. Dimensional Analysis: Calculate the volume of the pyramid using the formula for the volume of a pyramid: ( \text{Volume} = \frac{1}{3} \times \text{base length}^2 \times \text{height} ).

   • Volume of the Pyramid: ( \text{Volume} = \frac{1}{3} \times (230.4 , \text{m})^2 \times 138.8 , \text{m} )
   • Estimated number of blocks: Divide the volume by the average block volume (1.07 cubic meters).

2. Material Properties: Calculate the weight of each block using the density of limestone and the average block volume.

   • Average weight per block: ( \text{Average weight per block} = \text{average block volume} \times \text{density of limestone} ).

3. Force and Stress Calculations: Estimate the force due to friction and stress on ramps or levers.

   • Friction force: ( \text{Force due to Friction} = \mu \times \text{average block weight} \times g ), where ( \mu ) is the coefficient of friction (assumed) and ( g ) is the gravitational acceleration.
   • Stress on the Ramp: ( \text{Stress} = \frac{\text{Force due to Friction}}{\text{cross-sectional area}} ), where the cross-sectional area could be estimated based on historical data or assumptions about ramp construction.

4. Advanced Simulation Techniques: Utilize Finite Element Analysis (FEA) to simulate stress and displacement within the pyramid and ramps, and Computational Fluid Dynamics (CFD) to study wind impacts on construction stability.

5. Interdisciplinary Approach: Collaborate with experts in materials science, history, and archaeology for accurate data integration and validation through experimental archaeology.

This systematic approach combines mathematical modeling, physics principles, and advanced simulation techniques to gain deeper insights into the construction methods of the Great Pyramid of Giza.

Is there a theorem that proves or disproves that all sequences or series can be represented in terms of the nth term? For example 1,2,3…♾️ is just n, and 2,4,6,8,10….♾️ is 2n

The Konan permutation cycle presents an intriguing problem in the field of combinatorics and permutation theory. Named after its unique method of element reordering, the Konan permutation cycle rearranges elements in a list according to a distinctive pattern, cycling through configurations until returning to the original arrangement.

Description of the Permutation Cycle: The Konan permutation cycle operates on an ordered list of n elements, starting with the list in its natural order (e.g., [1, 2, 3, ..., n]). The permutation proceeds by the following rule: the first element remains in place, the second element moves to the leftmost position, the third element moves to the right of the first element, the fourth to the left of the second, and so on, alternating left and right insertions with each subsequent element.

Mathematical Challenges: The intriguing aspects of this cycle lead to several mathematical challenges:

Upper Bound Proof: Prove that the number of steps required to complete the cycle for any number n never exceeds n. Lower Bound Proof: Demonstrate that the cycle length is always at least ⌈log2​(n)+1⌉, where n is the number of elements. We also encourage participants to derive a general mathematical formula that predicts the cycle length based on the number of elements.

Example: Konan Permutation Cycle for 12 Elements:

Initial: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] Step 1: [12, 10, 8, 6, 4, 2, 1, 3, 5, 7, 9, 11] Step 2: [11, 7, 3, 2, 6, 10, 12, 8, 4, 1, 5, 9] Step 3: [9, 1, 8, 10, 2, 7, 11, 3, 6, 12, 4, 5] Additional steps... Returns to initial configuration after 10 steps. ·

Why This Matters: A proof or disproof of this hypothesis could contribute to deeper understanding in the fields of algebra, combinatorics, and theoretical computer science, providing insights into the behavior of complex permutations and their cycles. Moreover, it could shed light on the properties of permutation groups and their applications in various mathematical and real-world contexts.

How to Participate: We welcome submissions of proofs, discussions, and any theoretical or empirical insights related to this challenge. Contributions can be submitted in the form of articles, preprints, or detailed posts on platforms dedicated to mathematical exploration. Collaborative efforts through forums and discussion groups are also encouraged to foster a deeper collective understanding of the Konan permutation cycle.

This challenge not only serves as a stimulating mathematical puzzle but also as an opportunity for deeper research into permutation theory and cyclic orders.

My motivation for this question is I am super passionate about math having been reading all the past quanta magazine math articles since inception, reading about mathematicians and what they do on the AMS and other websites, watching podcasts of mathematicians etc but not actually doing any math itself. I think about this all the time. I am 37 with a small child working in industry so going to do a math degree at this point is next to impossible. My plan closer to retirement would be to do a math degree and if I am at least ok to get into any math phd program then a phd as well.

I know that Richard Smullyan got his phd at age 40, Preda Mihilescu at 42, Jean Birman around 41. Are there people who got phd’s in math past age 50 or even more 60? If so, did any of them not have any math degrees until late in life?

I need help to grasp the following:

Suppose we have a family of sets with an index set named I. Why is it that our index set I can't equal the empty set when defining the intersection of the sets in our family but for the union of those sets it doesn't matter? The annotation in my script says that if the index set would be empty the intersection would have to contain every object. How do I come to that conclusion?

I'm happy about every helping thoughts. Thanks in advance! :)

Ramanujan summation is tipping my brain apart because of this one simple thing

Someone please explain to me: In the Ramanujan Summation how are we allowed to shift on the order of one of the required number series. Ive seen it described as “2B” where B represents 1-2+3-4… but when most people does 2B they say (1-2+3-4) + (0+1-2+3). I believe this is a fallacy as, in my view, youre now saying B_0 - B_1 and then B_1 - B_2. What am I missing?

hey, I am not a mathematician nor do I find mathematics particularly interesting, more like a tool to help me in some aspects of my life. I know the basics that high schoolers know about math, but nothing else. I am not gonna lie, the teachers that I had, have made me see math with bitterness, but I have heard many times from people that they started later in life reading/learning math by themselves and they understood the beauty of them etc. Even the famous Carl Jung has said something similar. Probably there are people like them in the sub. I wanna give math a try, because I would like to find a beauty in them or in the end just to gain more knowledge in life. So, my first "request" is to ask you what made you fall in love with mathematics or what changed your opinion about them. Next, the problem with that is that I am not sure I understand the term learning by myself in its meaning. Every day I don't need higher mathematics, or those that I need I already know, so I have no idea how do you find math things or problems to study, how you select them from others, which are more "beginner-friendly" etc. Last but not least, how do you teach yourself math? Probably many reading this will laugh, but how do you understand them? Or their proofs? Or even writing new proofs? Don't you need a teacher (a good one) to explain to you things? I know my question is trivial, if not comical, but as I said in the beginning I would like to abolish the negative stereotypes that I have of math and if it's possible to find a deeper meaning into them.

P.S. sorry English is not my first language. Lol even foreigners, how do you understand mathematical terminology in English?

He called me and said I was one of his best students. Given my credentials and degree, he said, there is a position here for you during the fall. I wanted to cry and felt so happy but I currently have a job. I do not know what I will decide but I am proud of myself. So many men in my life who told me and doubted me by saying, "you don't look like the type girl who is a math major." "haha, are you sure." In classes and in life. One told me, "you only get straight A's because the professor thinks you're pretty and you're a girl." I graduated summa cum laude from my hard work and dedication. My brain works in mysterious ways.

Hi everyone! I am currently a junior in high school, and I will be offering a free Algebra II Prep Course in the Summer of 2024. If anyone is interested, here is the link to my instagram post with all of the information:

https://www.instagram.com/p/C54j37fR3QR/?utm_source=ig_web_button_share_sheet&igsh=MzRlODBiNWFlZA==

Lmk if you have any questions! :)

Iv got my log log model and I’m trying to find out what happens if in this case the distance of water (independent ) magically decreased by half and how that would effect price per (dependent). So far Iv got my B for water then

(E^B* (either ln(0,5) or 0.5))-1 to find the % change ge in price per that is caused by the 50% decress in water distance

but for the life of me don’t know witch

I suppose there are three mini-questions, here:

1. What mathematical model do they use to determine the probability that, e.g., horse A comes in first, horse B second, horse C third … ?
2. What data do they use to build that model (e.g., running times from the past x years? age and sex of the horse?)?
3. How do they prevent “runs on the bank” in case an improbable event with huge odds occurs? Do they have to have 100% cover with the losing tickets? Are they insured? Etc.

I've been doing some reading on Lie Groups/Lie Algebras, and I'm having trouble understanding why Lie Algebras are defined the way they are.

Lie Groups (I think) make sense in that we're looking for a way of representing ''continuous symmetries'' in a sense, and this is why we require the additional structure of a differentiable manifold on top of the existing group structure. And of course we'd want our group operations to also be smooth. This all feels very intuitive.

First, should I just forego trying to understand Lie Algebras outside of the context of Lie Groups? As in, is it substantially more intuitive to just abuse Lie's Third Theorem the whole way through and always view Lie Algebras as the tangent space for some Lie Group?

In this context, it makes sense why a Lie Algebra should be a vector space (since every point is just a vector with origin at the identity at the Lie Group). However, a lot of the other properties aren't so easy to see.

Why do we require that a Lie Algebra have this Lie Bracket that is alternating, bilinear, and satisfies the Jacobi identity? What is the point of the Jacobi Identity?

I've seen some posts referencing how this measures how not commutative the corresponding Lie Group is, but I don't understand why. I also don't get the relationship to flows

Any clarifying thoughts would be greatly appreciated :D tyty

Hello everyone,

I recently applied for a professional certificate (3 months) at UCD for "mathematics for statistics and data analysis". My background is a bachelor in psychology and an MSc in neuroscience, and this program states it is made for people with no degree level mathematical background. The two main aspects that it deals with is basic algebra and calculus (both from an introductory level).

Nevertheless, when I got accepted in the program they specified in their email that people should only apply if "they are happy with their current level in math", whatever that may mean. Now, given my (almost zero) math background, do you think this course is doable? Just to specify, I have last dealt with math in highschool and haven't really used it or remember anything ever since (besides statistics for my degrees).

I obtained this formula after a month long exercise in which I used a logical arrangement of circles to obtain the plot points for the dodecagon, and then bisected one side, found the equation for the line that passes through the bisection point & origin, and then found the point at which the bisecting line intersects with the original circle. Then, for each polygonal wedge with its midpoint, I calculated upper & lower bounds of pi by relating the area of the polygon with the area of the circles which intersect with the polygon's vertices & side-midpoints (the vertical circle must have an area greater than the polygon's, and the bisecting circle must have an area less than the polygon's). The pattern of the formula emerged in the math for the 48-sided polygon, and held true through the 384-sided polygon.

Computers cannot distinguish thhe result of these formulas after n=15 & 16 for upper & lower bound formulas respectively.

Hi there. I have a bachelors in math, a bachelors in art, and a weird brain that likes to doodle constructions.

Helpful Graph edit: points should be ordered ABC clockwise.

I was working with a triangle inscribed in a circle, let's say △ABC.

I constructed the perpendicular bisector of each side, AB, BC, AC.

I marked the point on each bisector on the portion that had not gone through the triangle (opposite the circumcenter) where it intersected the circle, constructing △A'B'C'.

I then repeated the process for △A'B'C', constructing △A''B''C''.

I repeated the process until △A^5 B^5 C^5 (I know it isn't correct formatting but it was easier)(6 triangles).

It seems that as the process is continued, the resulting triangles approach being equilateral triangles.

Is this a known phenomenon?

Thank you.

Next year I will be completing my undergraduate degree in Applied Mathematics with the hope of continuing on to grad school. However, I do not feel that the program I am in currently for my undergraduate degree is sufficient enough to get into a PhD program. My math program has so far consisted of 3 sections of calculus, an ODE course, linear algebra, real analysis, abstract algebra, and an advanced engineering mathematics course. I have also gotten close with the math faculty and am currently taking a PDE course as an independent study since no one else at the school is required to (or wants to) take it. I also am working on a research project with one of the professors over the summer. The main worry I have is that I have not taken enough math courses compared to other students at other math programs, which will put me at a disadvantage when applying for grad school. I have also been working as a peer tutor on campus at our support center, an in house tutor for the math department, and I'm going to be a grader for next semester ODE course.

I just don't feel like any of this is enough compared to what I see other universities offer and what I am expected to know for grad school. Any advice on what to I could do to help my chances would be appreciated.

Hi guys, I'm seeking a fast way (doesnt have to be cheap) to learn math from ground zero to PhD level.
I always had problems with mathematics in school and never saw a real reason to learn more than I have to use outta it.

Now, I'm a few ages older and I could easily see how being math ninja could help me in my career.
So, the time to fix the mistake has come.

Do you have any recommendations other than Uni?
Thank you in advance.

EDIT:

I'm eternally grateful for your engagement in the thread. Thats way more than I expected! :))

I got a 42 on my first midterm and I'm pretty sure I got another 42 on the 2nd midterm. Is it possible for me to ace the final and pass the class? C- starts at 67. Both my midterms are 20% and final is 30%. I looked at the grade calculator and I will need at least a 95 on my final to do this. I know it's possible but I just need some advice like how can I practice harder calc 2 problems that will help my problem solving for test problems because currently the problems I practice don't help on tests. My final is in 2 weeks and I really don't want to take calc for a 3rd time. If you don't have advice but a success story provide them below since they help too.

Edit: I realized i spend too much time on Reddit instead of practicing, probably another reason why I failed my midterm.

34 gallon fuel tank, with a 15 gallon per hour fuel pump.

Not accounting for the fuel used while driving.

Does this mean that the pump will fully recirculate the 34 gallons in 1.133 per hour.