Question

A spray can is used to paint a wall. The thickness of the paint on the wall is t. The distance of the spray can from the wall is d. t is inversely proportional to the square of d. $t=0.2$ when $d=8$. Find t when $d=10$.

Answer 1

Whenever we are given a proportionality relation between two quantities, it becomes easy to evaluate one unknown quantity via equating the proportionality over two sets of values by taking their ratio in accordance with the proportionality. Use this to arrive at the appropriate solution for the thickness of the paint, given that it is inversely proportional to the square of the distance.
Formula Used:
$\dfrac{t_1}{t_2} = \dfrac{d_2^2}{d_1^2}$

Complete answer:
We have a spray can that is used to paint a wall. The spray can is held at a distance d from the wall where it lays paint of thickness t. The correlation between the thickness of the paint sprayed and the distance of the can from the wall is given as:
$t\propto \dfrac{1}{d^2}$
Now, we are given that at a distance of $d_1=8$, the thickness of the paint sprayed is $t_1=0.2$. We are required to find the thickness of the paint $t_2$, when sprayed from a distance of $d_2=10$.
Whenever we have a correlation between two quantities defined by a proportionality, given a set of values corresponding to those quantities, we can scale the relation to determine one of the quantities for a different value of the other. This is done by taking the ratio of the two sets of values in accordance with the correlation to eliminate the proportionality in order to equate them and solve for the unknown arithmetically.
Given that $t\propto \dfrac{1}{d^2} \Rightarrow \dfrac{t_1}{t_2} = \dfrac{\left(\dfrac{1}{d_1^2}\right)}{\left(\dfrac{1}{d_2^2}\right)}$
$\Rightarrow \dfrac{t_1}{t_2} = \dfrac{d_2^2}{d_1^2}$
Plugging in the given values, we get:
$\dfrac{0.2}{t_2} = \dfrac{8^2}{10^2} \Rightarrow t_2 = \dfrac{64 \times 0.2}{100} = 0.128$
Therefore, when the spray can is at a distance of 10 units from the wall, the thickness of the sprayed paint is 0.128 units.

Note:
From the above question we see how we can implement ratios and proportions in order to establish a mathematical comparison between two quantities. Remember that proportion is essentially an equation which defines two given ratios as equivalent to each other. This is what we used to solve our problem above. We treated the ratio of the thickness and the ratio of the squares of distance to be equivalent to each other when treated inversely. Thus, the ratio is used to compare the size of two quantities with the same unit, whereas the proportion is used to express a relation between two ratios.