Problem #1
C(x) denotes the cost to produce x items and p(x) denotes the price-demand function in the given economic scenario. In each Exercise, do the following: 1) Find and interpret C(0). 2) Find and interpret C(10). 3) Find and interpret p(5). 4) Find and simplify R(x). 5) Find and simplify P(x). 6)Solve P(x) = 0 and interpret.
*The daily cost, in dollars, to produce x Sasquatch Berry Pies C(x) = 3x + 36, x ≥ 0 and the price-demand function, in dollars per pie, is p(x) = 12 - 0.5x, 0 ≤ x ≤ 24.
Problem #2
* Another Classic Problem: A can is made in the shape of a right circular cylinder and is to hold one pint. (For dry goods, one pint is equal to 33.6 cubic inches.)
(a) Find an expression for the volume V of the can in terms of the height h and the base radius r.
(b) Find an expression for the surface area S of the can in terms of the height h and the base radius r. (Hint: The top and bottom of the can are circles of radius r and the side of the can is really just a rectangle that has been bent into a cylinder.)
(c) Using the fact that V = 33.6, write S as a function of r and state its applied domain.
(d) Use your graphing calculator to find the dimensions of the can which has minimal surface area
Problem #3
In this problem we introduce three widely used measurement scales which involve common logarithms: the Richter scale, the decibel scale and the pH scale. The computations involved in all three scales are nearly identical so pay attention to the subtle differences.
*The pH of a solution is a measure of its acidity or alkalinity. Specifically, pH = - log[H+] where [H+] is the hydrogen ion concentration in moles per liter. A solution with a pH less than 7 is an acid, one with a pH greater than 7 is a base (alkaline) and a pH of 7 is regarded as neutral. (a) The hydrogen ion concentration of pure water is [H+] = 10-7 . Find its pH. (b) Find the pH of a solution with [H+] = 6.3 × 10-13. (c) The pH of gastric acid (the acid in your stomach) is about 0.7. What is the corresponding hydrogen ion concentration?
Problem #4
*According to facebook, the number of active users of facebook has grown significantly since its initial launch from a Harvard dorm room in February 2004. The chart below has the approximate number U(x) of active users, in millions, x months after February 2004. For example, the first entry (10, 1) means that there were 1 million active users in December 2004 and the last entry (77, 500) means that there were 500 million active users in July 2010. Month x 10, 22, 34, 38, 44, 54, 59, 60, 62, 65, 67, 70, 72, 77. Active Users in Millions U(x) 1, 5.5, 12, 20, 50, 100, 150, 175, 200, 250, 300, 350, 400, 500. With the help of your classmates, find a model for this data.