Ask Question, Ask an Expert

+1-415-315-9853

info@mywordsolution.com

Ask Math Expert


Home >> Math

1 Alice in Wonderland Project

Alice followed the rabbit down the hole. While sliding down the rabbit hole, Alice's speed was given by the function

f(x) = 4x2;

where x is in seconds and f(x) represents feet. It took Alice 20 seconds to slide all the way into wonderland. If a human's instantaneous velocity is more than 80 feet per second upon entry into wonderland, he/she will be killed. Did Alice survive the trip into wonderland? Make the following calculations:

1. Compute Alice's average velocity over the intervals [19:5; 20]; [19:9; 20]; [19:99;20], and then estimate Alice's instantaneous rate of change at x = 20.

2. Compute the difference quotient for f(x) and let h = 0; x = 20 (make sure you simplify the difference quotient before plugging in these values).
What does this say about Alice's instantaneous rate of change at x = 20?

2 Tweedle Dee and Tweedle Dum

The Tweedle Dee and Tweedle Dum brothers each own bakeries which sell only scones. Tweedle Dee's bakery is on the west side. Tweedle Dum's bakery is on the east side of wonderland. They are trying to figure out how to make enough money so that they can rent an apartment in the basement of the Queen's castle. Tweedle Dee's monthly profit is given by the function:
P1(x) = -x2 + 120x;
where x is the price he charges per scone.

1. How much money should Tweedle Dee charge for each scone in order to maximize his profit?

2. What is Tweedle Dee's maximum profit? As the economic situation on the east side is different from the west side, Tweedle Dum has a different profit function. Tweedle Dum's monthly profit is given by:
P2(x) = -x2 + 100x;

3. What is the price x that maximizes Tweedle Dum's profit?

4. What is Tweedle Dum's maximum profit? The trouble with stubborn Tweedle Dee and Tweedle Dum brothers is that they insist on charging the exact same price per scone at each of their bakeries. If they charge the amount found in problem one, Tweedle Dee's profit will be maximized but Tweedle Dum's will not. If they charge the amount found in problem two, then Tweedle Dum's profit will be maximized but Tweedle Dee's will not. Their goal is to be able to afford one of the Queen's apartments. The Queen charges 6100 dollars per month for one of the apartments in her basement.

5. What is the price that maximizes the combined profit P1(x) + P2(x)?

6. Can Tweedle Dee and Tweedle Dum afford an apartment in the Queen's basement if they charge the amount found in the previous problem?

3 Alice meets the Queen

The Queen is a big fan of mathematics. The Queen loves discussing projectiles and she orders Alice to use a certain sling shot to shoot a stone straight into the sky. When a person standing at ground level shoots a stone straight into the sky with this sling shot, its height in feet after t seconds is given by:
h(t) = -16t2 + 120t + h0

The -16 is due to gravity, and the number 120 represents the stones initial velocity. The variable h0 represents the persons height. (the equation is based on the assumption that the stone leaves the bow at a height close to the persons actual height). After Alice shoots the stone the Queen orders alice to find out the stones maximum height. Alice , however, has a problem. She has taken the Queens growing potion and no longer knows how tall she is. Help Alice solve the following problems.

1. The length of a human's femur bone is directly proportional to their height, with a proportionality constant of 1/4. Alice's femur is now 20 feet. How tall is Alice?

2. prepare the equation that represents the height of the stone shot by Alice. Plug in her new height for h0.

3. Determine the maximum height of the stone.

4. How long before the stone returns to the height you found in problem 1. (this is how long Alice has to move out of the way to avoid being hit by the stone on its way down)

4 The Queen, Caterpillar and Mad-hatter

The Queen asked Alice to find a quadratic equation that models the population of wonderland but Alice frowned. Alice  complained: Look, Queen, don't you know wonderland is fake and that math has nothing to do with real life? The Queen responded by saying: Alice dear, you are misunderstanding the situation. Math has applications in physics, engineering, medicine and many other areas of life. The concept that you are learning is that when real life situations can be modelled with a quadratic function we can always find either a maximum or minimum. Since Alice was afraid to help, the Queen enlisted the caterpillar to model the population growth of wonderland. Since the Queen loves parabolas, she demanded that the caterpillar model the population with a quadratic equation. She told the caterpillar that it would be off with his head if he didn't get it right. Here is what the caterpillar came up with.

P(t) = -t2 + 292t + 2400;

where, P(t) is the population in millions, and 0 < t < 300 is years since 2010.

1. find out the vertex of P(t)

2. Assuming the model is correct, during what years is the population of wonderland increasing? During what years is the population decreasing?
When will the inhabitants of wonderland become extinct (when will the population be 0)

3. find out the average rate of change of P(t) over three well chosen intervals and then estimate the instantaneous rate of change of P(t) at its vertex.
The Mad Hatter, the clever fellow that he is, has a hunch that the instantaneous rate of change of any parabola will always be zero at the vertex( where the function is neither increasing nor decreasing) . Further, remembering, that there is a relationship between rate of change and the difference quotient, the Mad Hatter has set out series of steps for  finding the vertex of a parabola (and thereby also  finding its max/min). Complete the following problems to  find the vertex of the parabola.

4. Set up and simplify the difference quotient for P(t) = -t2 + 292t + 2400:

5. Let h = 0 in your answer to the previous problem.

6. Let your equation in previous problem equal to zero and solve. What value does this give you?

5 Method to the Mad Hatter's Madness

The Mad Hatter appears to have found a new way of calculating a vertex. This doesn't sit well with the Queen, who loves math, but prefers the easy way of doing things. Yet the Mad Hatter's new way of  finding maximums and minimums may be applicable to more than just quadratic functions. It better work too, because the Queen has come up with a new set of problems and has declared that if no solution can be found, then all of Wonderland will be sent to the guillotine.

The Queen's new problems revolve around the bakery she has opened. The main problem boils down to finding both the best and worst amount of scones (in pounds) to produce. The Queen's profit is given by P(x) = -2x3+39x2-132x  for 0<=x<=12.

1. Find the maximum and minimum values of P(x) by applying the Mad Hatter's technique.

2. What is the worst amount of scones to produce?

3. How much money will the Queen lose when the worst amount of scones are produced?

4. What is the best amount of scones to produce?

5. How much money will the Queen make when the best amount of scones are produced?

6. As a way of checking, pick at least 4 random values of x, 0<=x<=12 and plug them into P(x). Show that none of these are less than the minimum you found or greater than the maximum you found.

Short answer problems

1. The Queen loves quadratic functions because they are easy to work with. However, not every real life situation can be modeled with a quadratic. Give an ex of a real life function which can be modeled with a quadratic. Does this situation have a max or a min?

2. The Queen wanted the caterpillar to model population growth with a quadratic function, but the Queen was actually wrong in her assumption that population growth can be modeled with quadratic functions. In truth, what type of function should be used to model population growth?

3. Do you think it is possible to find the maximum or minimum of a function that is not a quadratic function? Do you have any suggestions how you would go about finding a maximum or minimum of a function that is not a quadratic function?

4. The Mad Hatter's technique is actually beginning level computations which are performed in a Calculus course. Calculus, as stated in the introduction, was not immediately accepted as legitimate mathematics. What do you think is objectionable to the Mad Hatter's techniques? (these objections were eventually overcome)

5. prepare down two strengths of yours.

Math, Academics

  • Category:- Math
  • Reference No.:- M91739

Have any Question? 


Related Questions in Math

Find the amount that a principal of 800 will accumulate in

Find the amount that a principal of $800 will accumulate in 12 years for each given account: a. 7% simple interest b. 7% compounded quarterly c. 7% compounded monthly

A hospitals revenue in millions of dollars is projected to

A hospital's revenue (in millions of dollars) is projected to be R(x) = 9x2 + 7x + 81 and its costs (in millions of dollars) are projected to be C(x) = -2x2 - 10x - 14 where x represents the number of years into the futu ...

Applying key skillsnbspplease respond to the

"Applying Key Skills" Please respond to the following: Describe the key skills you learned in this course. Explain how you expect to use those skills in your future academic, personal, and / or professional life. Recomme ...

The members of a family home evening group at byu-idaho

The members of a family home evening group at BYU-Idaho each recorded the number of hours they spent studying one particular week. Their results are recorded in the following data set. Find the lower quartile (first quar ...

For a normal distribution with mean of 80 and a standard

For a normal distribution with mean of 80 and a standard deviation of 8, find the number of standard deviations the raw score 67 is from the mean (Give answer as a positive value rounded to nearest hundredth).

Probability75 to 150 wordsin this discussion please answer

Probability75 to 150 Words In this discussion, please answer the following question and post your response to the discussion board. Please keep in mind, in order to remain eligible for full credit in the discussion, you ...

List three major disadvantages of internet search engines

List three major disadvantages of internet search engines, then briefly describe other electronic research tools that overcome those shortcomings. Essay of 350-500 words. (Introduction, body and conclusion)

T angle between two forces of 37 n newtons and 55 n is

The angle between two forces of 37 N (Newtons) and 55 N is 36°. Find the magnitude of the resultant force. Do not round until the final answer. Then round to the nearest integer as needed.)

This assignment involves constructing a vision for the

This assignment involves constructing a vision for the future and envisioning the results of its realization. Enabling others to act in realizing the vision is a crucial aspect of leadership. Practice and repetition of t ...

Health maintenance organizations hmos and prospective

Health Maintenance Organizations (HMOs) and prospective payment systems vary widely in quality and costs and employ a variety of methods to control costs. Discuss the financial pros ad cons of HMOs and include a clear di ...

  • 4,153,160 Questions Asked
  • 13,132 Experts
  • 2,558,936 Questions Answered

Ask Experts for help!!

Looking for Assignment Help?

Start excelling in your Courses, Get help with Assignment

Write us your full requirement for evaluation and you will receive response within 20 minutes turnaround time.

Ask Now Help with Problems, Get a Best Answer

Section onea in an atwood machine suppose two objects of

SECTION ONE (a) In an Atwood Machine, suppose two objects of unequal mass are hung vertically over a frictionless

Part 1you work in hr for a company that operates a factory

Part 1: You work in HR for a company that operates a factory manufacturing fiberglass. There are several hundred empl

Details on advanced accounting paperthis paper is intended

DETAILS ON ADVANCED ACCOUNTING PAPER This paper is intended for students to apply the theoretical knowledge around ac

Create a provider database and related reports and queries

Create a provider database and related reports and queries to capture contact information for potential PC component pro

Describe what you learned about the impact of economic

Describe what you learned about the impact of economic, social, and demographic trends affecting the US labor environmen