Amit M Agarwal Solutions for Chapter: Monotonicity, Maxima and Minima, Exercise 5: Proficiency in 'Monotonicity, Maxima and Minima' Exercise 2

A closed rectangular box with a square base is to be made to hold $1000$ cubic feet. The cost of the material per sq foot for the bottom is $15$ paise, for the top $25 \mathrm{p}\mathrm{a}\mathrm{i}\mathrm{s}\mathrm{e}$ and for the sides $20 \mathrm{p}\mathrm{a}\mathrm{i}\mathrm{s}\mathrm{e}$. The labour charge for making the box is $\text{₹}3/-$.Find the dimensions of the box when the cost is minimum.

A running track of $440$ yards is to be laid out encircling a football field the shape of which is rectangle with a semi-circle at each end. If the area of the rectangular position is to be maximum find the lengths of its sides.

Show that a right circular conical tent of volume $V$ will require the least amount of canvas to make it if it's height is $\sqrt{2}$ times the radius of its base.

Two corridors of width $a$ and $b$ are at right angles. Show that the length of the longest pipe that can be passed round the corner horizontally is ${(a^{2/3}+b^{2/3})}^{3/2}.$

A point $P$ is given on the circumference of a circle of radius $r$. Chord $QR$ is parallel to the tangent at $P$. Determine the maximum possible area of the $∆PQR$.

Two cars are travelling along two roads which cross each other at right angles at $A$. One car is travelling towards $A$, at a speed of $21$ miles per hour while the other is travelling towards it at a speed of $28$ miles per hour. If initially, their distance from $A$ are, $1500$ feet and $2100$ feet respectively, prove that the least distance between them is $60$ feet.

Two roads $OA$ & $OB$ intersect at an angle of ${60}^{\mathrm{o}}$. A car driver approaches $O$ from $A$, where $AO=800 \mathrm{m}$ at a uniform speed of $20 \mathrm{m}/\mathrm{s}$. Simultaneously, a runner starts running from $O$ towards $B$ at a uniform speed of $5 \mathrm{m}/\mathrm{s}$. Find the time when the car and the runner are closest.

A swimmer $S$ is in the sea at the closest distance $d \mathrm{k}\mathrm{m}$ from $A$ on a straight shore. The house of the swimmer is on this shore at a distance $L \mathrm{k}\mathrm{m}$ from $A$. He can swim at a speed of $u \mathrm{k}\mathrm{m} {\mathrm{h}}^{-1}$ and walk at a speed of $v \mathrm{k}\mathrm{m}$ ${\mathrm{h}}^{-1}\mathrm{}(v>u)$. At what point on the shore should he land so that he reaches his house in the shortest possible time.