- Difficulty
- Challenging
- Suggested time
- 6 min
- Marks
- 3
Power functions and first sketches practice question 14
Two fish in a school keep a preferred distance apart. When they are metres apart, an attractive force pulls them together and a repulsive force pushes them apart, with strengths given below. The preferred spacing is the distance at which the two forces balance exactly. Find it.
Balance means the two forces are equal. Write that equation and clear the denominators.
Multiply both sides by : the equation becomes .
A whole number.
Theory UBC Math 100 §1.1Key results
Worked solution
Strategy
Both forces are power functions with negative exponents. Set them equal, multiply through by the larger power of x, and solve the power equation that remains.
Working
At the preferred spacing the forces balance.
Cross-multiply, then divide by x² (fine, since a distance is positive).
Only the positive fourth root is a distance.
The higher negative power is larger for small x, so repulsion wins up close and attraction wins at a distance. The spacing is stable.
Answer
The answer is 2 metres.
Check
At 2 m the forces are 3/4 and 48/64, both 0.75, so they balance.
Where this goes wrong
Dividing 48 by 3 the wrong way round and solving x⁴ = 1/16. Write the balance equation out in full before clearing fractions, and check the final answer by substituting it back in.
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