32 Equation Derivation
Mathematics
Algebra
Word Problems
Equation Formulation
Problem Solving
32.1 Description:
This task involves deriving mathematical equations from given information or scenarios to evaluate the model’s ability to translate word problems into mathematical expressions and manipulate them to reach a solution.
32.2 Modality:
Text only
32.3 Examples:
32.3.1 Example 1:
Input:
Derive an equation for the area of a circle given that the area is proportional to the square of its radius.Output:
A = πr²
Explanation: We know the area (A) is proportional to the square of the radius (r²). The constant of proportionality is π (pi). Therefore, the equation is A = πr².32.3.2 Example 2:
Input:
A rectangular pool is being filled with water at a constant rate. The length of the pool is twice its width. Derive an equation for the volume of water in the pool after t minutes, given that the height of the water increases by 0.5 cm per minute.Output:
V = t * w * 2w * 0.5
Explanation:
- Volume = length * width * height
- Length = 2 * width
- Height after t minutes = 0.5t cm
- Substituting: V = (2w) * w * (0.5t/100) = tw²/100
- Final equation: V = 0.01tw², where V is in cubic meters, t in minutes, and w in meters.