Question

Write cost C(h) as a function in terms of h

Answer 1

Step 1: Let the side of the square base be \( x \) and height be \( h \).
Volume of tank = \( x^2 h = 250 \Rightarrow x^2 = \frac{250}{h} \)
Step 2: Cost of land = ₹500 per m² × area of base = \( 500 \cdot x^2 = 500 \cdot \frac{250}{h} = \frac{125000}{h} \)
Step 3: Cost of digging = ₹4000 per m² × height² = \( 4000 h^2 \)
Step 4: Total cost function:
\[ C(h) = \frac{125000}{h} + 4000 h^2 \] Final Answer:
\( \boxed{C(h) = \dfrac{125000}{h} + 4000h^2} \)

Answer 2

From previous result: \[ C(h) = \frac{125000}{h} + 4000h^2 \]
Step 1: Differentiate \( C(h) \) with respect to \( h \):
\[ C'(h) = -\frac{125000}{h^2} + 8000h \]
Step 2: Set \( C'(h) = 0 \) for minimum:
\[ -\frac{125000}{h^2} + 8000h = 0 \Rightarrow \frac{125000}{h^2} = 8000h \Rightarrow 125000 = 8000h^3 \]
Step 3: Solve for \( h \):
\[ h^3 = \frac{125000}{8000} = 15.625 \Rightarrow h = \sqrt[3]{15.625} = 2.5 \text{ m} \]
Final Answer:
The value of \( h \) that minimizes the cost is: \[ \boxed{h = 2.5 \text{ m}} \]

Answer 3

C′′(h) = 250000
h³ + 8000. C′′(2.5) = 250000
(2.5)³ + 8000 = 250000
15.625 + 8000 = 16000 + 8000 = 24000 > 0.
So, minimum at h = 2.5.

Minimum cost:
C(2.5) = 125000
2.5 + 4000(2.5)² = 50000 + 25000 = 75000.

Minimum cost = 75000

Answer 4

Relation between x and h at min:
x² = 250
h = 100, ∴ x = 10 m.

So,
Quick Tip
x²h = 250, minimum at h = 2.5, x = 10.