Application of Derivatives 4 Question 50
53. A window of perimeter (including the base of the arch) is in the form of a rectangle surmounted by a semi-circle. The semi-circular portion is fitted with coloured glass while the rectangular part is fitted with clear glass. The clear glass transmits three times as much light per square meter as the coloured glass does. What is the ratio for the sides of the rectangle so that the window transmits the maximum light? (1991, 4M)
Show Answer
Answer:
Correct Answer: 53. (2)
Solution:
- Let $2 b$ be the diameter of the circular portion and $a$ be the lengths of the other sides of the rectangle.
Total perimeter $=2 a+4 b+\pi b=K$
Now, let the light transmission rate (per square metre) of the coloured glass be $L$ and $Q$ be the total amount of transmitted light.
Then, $\quad Q=2 a b(3 L)+\frac{1}{2} \pi b^{2}(L)$
$\Rightarrow \quad Q=\frac{L}{2}\left(\pi b^{2}+12 a b\right)$
$\Rightarrow \quad Q=\frac{L}{2}\left[\pi b^{2}+6 b(K-4 b-\pi b)\right] \quad$ [from Eq. (i)]
$\Rightarrow \quad Q=\frac{L}{2}\left(6 K b-24 b^{2}-5 \pi b^{2}\right)$
On differentiating w.r.t. $b$, we get
$$ \frac{d Q}{d b}=\frac{L}{2}(6 K-48 b-10 \pi b) $$
For maximum, put $\frac{d Q}{d b}=0$
$$ \Rightarrow \quad b=\frac{6 K}{48+10 \pi} $$
Now, $\quad \frac{d^{2} Q}{d b^{2}}=\frac{L}{2}(-48-10 \pi)<0$
Thus, $Q$ is maximum and from Eqs. (i) and (ii), we get $(48+10 \pi) b=6{2 a+4 b+\pi b}$
$$ \therefore \quad \text { Ratio }=\frac{2 b}{a}=\frac{6}{6+\pi}=6: 6+\pi $$
