Variations: Practice Questions

If $$y$$ varies directly as $$x$$, and $$y=15$$ when $$x=5$$, find $$y$$ when $$x=8$$.

The variable Z varies jointly as the variable X and the square root of the variable Y. If X is increased by 20% and Y is decreased by 36%, find the percentage change in Z.

The cost $$C$$ of producing a certain item partially varies as the number of items produced $$N$$ and partially as the square of the number of items produced $$N^2$$. When $$N=100$$, $$C=5000$$. When $$N=200$$, $$C=12000$$. Find the cost $$C$$ when $$N=150$$.

The cost \(C\) of manufacturing a certain product partly varies directly as the number of units produced \(n\), and partly varies inversely as the working hours \(H\).

When \(n=1\) unit is produced in \(H=2\) hours, the cost is \(C=10\) thousand dollars. When \(n=2\) units are produced in \(H=4\) hours, the cost is \(C=13\) thousand dollars.

Find the cost \(C\) (in thousand dollars) if \(n=3\) units are produced in \(H=2\) hours.

The quantity R varies partly directly as the square of x and partly inversely as y. It is given that:

• R = 1 when x = 1 and y = 1
• R = 11 when x = 2 and y = 2

Find the positive value(s) of x when R = 23 and y = 0.5.

If the quantity \(R\) varies inversely as the square root of \(T\). If \(R = 6\) when \(T = 4\), find the value of \(R\) when \(T = 9\).

The total cost \(C\) of printing flyers is partly constant and partly varies directly as the number of flyers printed \(N\). If it costs HKD \(500\) for \(100\) flyers and HKD \(900\) for \(300\) flyers, find the cost of printing \(500\) flyers.

It is given that \(C\) is the sum of two parts, where one part is a constant and the other part varies directly as the square of \(n\). If \(C = 2300\) when \(n = 10\) and \(C = 3200\) when \(n = 20\), find the positive value of \(n\) when \(C = 12800\).

The resistance $$R$$ of a wire varies directly as its length $$L$$ and inversely as the square of its radius $$r$$. When $$L=10$$ cm and $$r=0.5$$ cm, the resistance $$R=40$$ ohms.

(a) Express $$R$$ in terms of $$L$$, $$r$$, and a constant $$k$$.

(b) Find the constant $$k$$.

(c) If the length of the wire is 15 cm and its radius is 0.25 cm, find its resistance.

The rate of fuel consumption, R (in litres per hour), of a specialized fishing trawler is modeled by a partial variation relationship. It is observed that R partly varies directly as the square of the speed, v (in km/h), and partly varies inversely as the speed, v.

(a) Express R in terms of v and constants a and b.

It is found that:

• When the speed is 10 km/h, the rate of fuel consumption is 20 litres per hour.
• When the speed is 20 km/h, the rate of fuel consumption is 45 litres per hour.

Find the values of the constants a and b.

(b) If the trawler needs to travel a fixed distance D (in km), express the total fuel consumption F (in litres) required for the journey in terms of v and D.

(c) If the total distance of the journey is 300 km, find the speed v (in km/h) that minimizes the total fuel consumption F. Give your answer in simplest radical form and correct to 3 significant figures.