A newly formed colony of ants are about to build an anthill. The number of ants in the colony is 𝑎. It takes 𝑡 hours to complete the anthill. 𝑡 is inversely proportional to 𝑎 squared. When 𝑎 is equal to 450, 𝑡 is equal to 36. Some new ants join the colony before they build the anthill. Work out 𝑡 when 𝑎 is equal to 540.
As 𝑡 is inversely proportional to 𝑎 squared, we can say that 𝑡 is proportional to one over or one divided by 𝑎 squared. When dealing with any proportion questions, we can replace the proportion symbol with equals 𝑘, where 𝑘 is the constant of proportionality. Our equation can be rewritten as 𝑡 is equal to 𝑘 divided by 𝑎 squared.
Our next step is to substitute in the values of 𝑎 and 𝑡 that we know to help us calculate 𝑘. We know that 𝑎 is 450 when 𝑡 is equal to 36. This means that 36 is equal to 𝑘 divided by 450 squared. We can multiply both sides of this equation by 450 squared. This means that 𝑘 is equal to 36 multiplied by 450 squared. Typing this into the calculator gives us a value of 𝑘 equal to 7290000. Our equation linking the time 𝑡 and the number of ants 𝑎 can therefore be rewritten as 𝑡 equals 7290000 divided by 𝑎 squared.
We were asked to calculate the value of 𝑡 when 𝑎 equals 540. Substituting 𝑎 equals 540 into the equation gives us 𝑡 is equal to 7290000 divided by 540 squared. Once again, we can type this straight into our calculator, giving us a value of 𝑡 equal to 25.
As the time was measured in hours, we can say that it would take 540 ants 25 hours to build the anthill.