Class 12 Applications Of Derivatives · Case Study4 Markscase_based_application
Class 12 Maths · Applications Of Derivatives · Case Study #31
A rainwater tank has a square base of side x metres, depth h metres and fixed volume 250 m³. Land costs ₹5,000 per square metre of base, and the total digging cost is ₹40,000h². Take x,h>0 and C to be the sum of these two costs.
Part (1) 1 Marks
Express C as a function of x.
Answer: C(x)=5000x²+2500000000/x⁴
- The volume constraint x²h=250 gives h=250/x².
- C=5000x²+40000(250/x²)²=5000x²+2500000000/x⁴.
Part (2) 1 Marks
Find dC/dx.
Answer: 10000x−10000000000/x⁵
- Differentiate each term using the power rule.
- C′(x)=10000x−10000000000/x⁵.
Part (3) 2 Marks
Find x that minimizes the cost.
Answer: 10 m
- C′=0 gives 10000x⁶=10000000000, hence x⁶=10⁶ and x=10.
- For positive x, C′ has the sign of x⁶−10⁶: negative before 10 and positive afterwards.
- Thus C has its global minimum at x=10 m.
Part (4) 2 Marks
Alternatively, determine whether C is increasing on all x>0, and give its monotonicity intervals.
Answer: Decreasing on (0,10), increasing on (10,∞).
- C′(x)=10000(x⁶−10⁶)/x⁵.
- For x>0 the denominator is positive, so C′<0 below 10 and C′>0 above 10.
- C is not increasing on all positive x: it decreases on (0,10) and increases on (10,∞).