H S Hall and S R Knight Solutions for Chapter: Variation, Exercise 1: Examples. III .

When a body falls from rest, its distance from the starting point varies as the square of the time it has been falling. If a body falls through $122.6m$ in $5$ seconds, how far does it fall in $10$ seconds? Also, how far does it fall in the $10\mathrm{th}$ second?

Given that the volume of the sphere varies as the cube of it's radius and where the radius is $3.5 \mathrm{cm},$ the volume is $179.7$ cubic $\mathrm{cm}$. Find the volume when the radius is $1.75 \mathrm{cm}.$

The weight of the circular disc varies as the square of the radius when the thickness remains the same, it also varies as the thickness when the radius remains the same. Two discs have their thicknesses in the ratio of $9 : 8,$ find the ratio of their radii if the weight of the first is twice that of the second.

At a certain regatta, the number of races on each day varied jointly as the number of days from the beginning and end of the regatta up to and including the day in question. On three successive days there were respectively $6, 5,$ and $3$ races. Which days were these, and how long did the regatta last?

The price of a diamond varies as the square of its weight. Three rings of equal weight, each composed of a diamond set in gold, have values Rs. $a, \mathrm{Rs}. b$ and $\mathrm{Rs}. c,$ the diamonds in them weighing $3, 4, 5$ carats respectively. Show that the value of a diamond of one carat is $Rs \left(\frac{a+c}{2}-b\right).$ The cost of workmanship being the same for each ring.

Two persons are awarded pensions in proportion to the square root of the number of years they have served. One has served $9$ years longer than the other and receives a pension greater by $\mathrm{Rs}. 250,$ if the length of the service of the first had exceeded that of the second by $4\frac{1}{4}$ years, their pensions would have been in the proportion of $9 : 8.$ How long had they served and what were their respective pensions?

The attraction of a planet on its satellites varies directly as the mass $\left(\mathrm{M}\right)$ of the planet, and universe as the square of the distance $\left(\mathrm{D}\right),$ also the square of a satellite's time of revolution varies directly as the distance and inversely as the force of attraction. If $m_1, d_1, t_1$ and $m_2, d_2, t_2$ are simultaneous values of $M, D, T$, respectively, prove that

$\frac{m_1{t}_1^2}{m_2{t}_2^2}=\frac{d_1^3}{d_2^3}.$

Hence, find the time of revolution of that moon of Jupiter whose distance is to the distance of our Moon as $35; 31,$ having given that the masses of Jupiter is $343$ times that of the earth and that the Moon's period is $27.$ $2$ days.

The consumption of coal by a locomotive varies as the square of the velocity, when the speed is $16 \mathrm{km}$ an hour the consumption of coal per hour is $2$ tonnes, if the price of coal be $\mathrm{Rs}. 10$ per tonne and the other expenses of the engine be $\mathrm{Rs}. 11.25$ an hour, find the least cost of a journey of $100 \mathrm{Km}.$