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) = 4x^{2};
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) = -x^{2} + 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) = -x^{2} + 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) = -16t^{2} + 120t + h_{0}
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 h_{0}.
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) = -t^{2} + 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) = -t^{2} + 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) = -2x^{3}+39x^{2}-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.