# Equations - July 2007, Stage 4&5

## Problems

### Matchless

##### Stage: 4 Challenge Level:

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

### Gutter

##### Stage: 4 Challenge Level:

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

### Perpendicular Lines

##### Stage: 4 Challenge Level:

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

### Reasonable Algebra

##### Stage: 4 Challenge Level:

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

### Equation Attack

##### Stage: 5 Challenge Level:

The equation a^x + b^x = 1 can be solved algebraically in special cases but in general it can only be solved by numerical methods.

### Production Equation

##### Stage: 5 Challenge Level:

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

### Resistance

##### Stage: 5 Challenge Level:

Find the equation from which to calculate the resistance of an infinite network of resistances.