Solving Complex Word Problems: A Comprehensive Guide

Dealing with complex word problems can often seem daunting, but breaking them down step by step can simplify the process and lead to accurate solutions. This article will explore two such problems and provide a detailed breakdown of the steps needed to solve them. We will cover optimization in mathematics, basic physics, and insurance calculations, offering insights on how to approach these problems systematically.

Word Problem 1: Optimal Piece Cutting for Minimal and Maximal Combined Area

This problem involves cutting a 16-inch wire into two pieces to create a square and a rectangle. The goal is to find the smallest and largest possible combined area of the two shapes.

Step 1: Set up the equations

Let's denote the lengths of the two pieces of wire as s and r. We know that:

sr 16

To shape s into a square, we have:

S s/4

The area of the square is:

Areasquare S2 s2/16

To shape r into a rectangle with a width of 1 inch, we have:

L 1/2r - 2

The area of the rectangle is:

Arearectangle 1L (1/2r - 2)/2 14 - s/2

The total area is:

T s2/16 14 - s/2

With the constraint r 16 - s, substitute into the total area equation:

T s2/16 14 - (16 - s)/2

Step 2: Use Calculus to Find the Min/Max

To find the minimum and maximum values of T, use calculus:

dT/ds s/8 - 1/2

Set dT/ds to 0 to find the critical points:

s/8 1/2

s 4 inches

Substitute s 4 into the equation for T:

Minimum value of T 42/16 14 - (16 - 4)/2 1 10/2 6 square inches

To find the maximum value, set s 16:

Maximum value of T 16 square inches

Word Problem 2: Physics: Object Propulsion and Height Calculation

This problem involves an object being propelled straight upward with an initial velocity of 48 feet per second. The height h in feet after t seconds is given by the equation:

h -16t2 48t

The question asks after how many seconds the object hits the ground. At ground level, h 0:

-16t2 48t 0

This can be factored as:

4t(-4t 12) 0

Setting each factor to 0, we get:

t 0 (initial time) and t 3 seconds

Step 3: Interpret the Height Function

A negative value of h has no physical meaning in this context, so we only consider the positive root t 3 seconds to be valid.

Word Problem 3: Insurance Calculations and Legal Liabilities

This problem involves an insurance accident scenario and the calculations of coverage amounts.

Step 1: Understand the Insurance Coverage

The insurance policy is 50/100/10, meaning:

Up to 50,000 USD for a single injured person Up to 100,000 USD for all people injured in the accident Up to 10,000 USD for property damage

Step 2: Calculate Insurance Payments

There is one injured person, so the insurance will cover up to 50,000 USD. The total amount paid is 56,000 USD, so the insurance will cover:

56,000 - 50,000 6,000 USD

The property damage is 5,800 USD, which is covered fully:

5,800 USD

The deductible for the injured person is 250 USD, so the total payment to the injured person is:

6,000 - 250 6,250 USD

The insurance will pay:

50,000 - (5,800 250) 55,550 USD

Step 3: Consider Legal Liabilities

The driver of the Furnishing Office violated traffic rules, so the insurance company may file a lawsuit against the driver, the Furnishing Office, or both. The payment amounts may change until there is a legal agreement.

Conclusion

Through these examples, we have demonstrated how to approach complex word problems in mathematics and physics, as well as the more practical task of insurance calculations and legal liabilities. By breaking each problem into smaller, manageable steps, you can solve even the most challenging situations effectively.