MAPLE Project #1: Analyzing a Function of Two Variables

MAPLE functions must be shown for each problem to receive credit.

The area A, in square inches, of a computer monitor can be expressed by the following function:

A = (ad^{2}) / (a^{2 }+ 1)

where **a** is the aspect ratio of the monitor, expressed as a fraction ( [width/height] or [height/width] ), and **d** is the manufacturer’s advertised monitor length (always in *diagonal* inches).

**a). **Graph this function in both 3D wire frame mode and contour map mode. Set the lower bound of **a** and **d** to zero, as negative values are not meaningful for any variable in this function. Label the **a** and **d** axes for both graphs. Show only the 10, 20, 30, and 40 square-inch area contours for the contour graph.

**b). **Use Maple to evaluate this function at the three implied points:

An advertised 16-inch monitor with a 5:4 aspect ratio

An advertised 17-inch monitor with a 16:9 aspect ratio

An advertised 17-inch monitor with a 4:3 aspect ratio

Rank these monitors from smallest area to largest area.

**c). **Graph the function for an arbitrary fixed **d** (e.g. 20). The result will be a 2D graph of one variable.

**d). **Explain what you notice about this graph. Graph the function many times with varying fixed **d**s if necessary to recognize a pattern.

**e). **Use Maple to find a meaningful critical point with respect to only **a**, and determine if it is a maximum or minimum:

Part e) Hints:

Take the partial derivative of the function with respect to **a**, using Maple.

Set this partial derivative equal to zero, and, using Maple, solve for **a**.

**f). **What have you discovered about the variable **a** in this function, from parts c), d), and e)?

**g). **What does this tell you about the ideal aspect ratio for a computer monitor, should you wish to maximize screen area, for any diagonal length?

**h). **Your local retailer carries only two brands of HDTV: Sany and Somsung. Sany produces various sizes, but always with an aspect ratio of 16:9 widescreen. Somsung models are all 4:3, but they also come in many different sizes. You plan on watching just as much Standard Definition programming as High Definition programming, so any two televisions with equal screen area have equal utility.

The question is, how much larger (diagonally) than a Sany does a Somsung have to be such that each television has the same area?

Part h) Hints:

Your answer should be a percentage (or fraction) of diagonal length (e.g. “The Somsung must be

200% larger than the Sany.”), because no one constant diagonal increase will match **any** two

competing TVs.

Maple’s ability to solve complex equations makes this easy once you know what you’re looking

for. Try setting up one equation for each brand, setting them equal to each other, and using

Maple’s built-in solve() command to solve for some variables in terms of other variables. What

values can be fixed in these equations?

**Bonus: **Derive an equation that expresses the total cost of making an LCD screen with respect to aspect ratio and diagonal length: It costs $10 to run the machine that produces the screens for as long as it takes to make one screen, and $20 for any screen’s plastic backing. One pixel costs $0.00005 to produce. The resolution of any screen produced is 72 ppi (horizontal and vertical pixels per inch).

Another Problem concerns about profit optimization.

A local grocer sells different types of products which are partial substitutes. The price of one affects the demand of the others. The market demands in terms of prices (Pc for chicken and Pb for beef) are given by:

**Chicken:**

Qc = 155 - 3.7*Pc + 2.3*Pb

**Beef:**

Qb = 132 - 2.8*Pb + 1.25*Pc

What can be said about the sensitivity of prices among chicken and beef?

These items have a function of cost defined by:

C= .2*Qc^2 + .23*Qb^2 + 15.2

If the prices are fixed, what is the maximum profit (=income - cost) that can be made?

What is the rate of change as the chicken price, Pc, increases?