Consider the table of values for 𝑥 and 𝑦. Given that 𝑦 is inversely proportional to the square of 𝑥, calculate the value of 𝑚.
If two quantities are inversely proportional to one another, then this means that as one increases, the other decreases at the same rate. So if one doubles, the other will halve, and so on. We need to read the question carefully.
We’re told that 𝑦 is inversely proportional to the square of 𝑥. So that means 𝑥 squared. A common mistake would just be to think that 𝑦 was inversely proportional to 𝑥 or to misread square as square root.
We can express this relationship using the proportionality symbol. And if 𝑦 is inversely proportional to 𝑥 squared, it actually means that 𝑦 is directly proportional to the reciprocal of 𝑥 squared. That’s one over 𝑥 squared. It also means that 𝑦 is equal to some multiple of one over 𝑥 squared.
So we can rewrite this relationship as 𝑦 equals 𝑘 over 𝑥 squared, where 𝑘 is a constant, the constant of proportionality which we need to find. To do so, we can use the pair of values in our table. We know that when 𝑦 is equal to one, 𝑥 is equal to 10. So we can substitute one for 𝑦 and 10 for 𝑥 into our proportional relationship. And it gives that one is equal to 𝑘 over 10 squared. 10 squared is equal to 100. So we have that one equals 𝑘 over 100.
To solve this equation for 𝑘 then, we need to multiply both sides by 100. And we find that 𝑘 is equal to 100. We now know the exact relationship between 𝑦 and 𝑥 squared. It’s that 𝑦 is equal to 100 over 𝑥 squared.
The value we’re looking to calculate, 𝑚, is paired with a 𝑦-value of 25 in the table. So we’ll now substitute 25 for 𝑦 and 𝑚 for 𝑥 to give an equation we can solve to find 𝑚. This gives 25 equals 100 over 𝑚 squared.
The first step in solving this equation is to multiply both sides by 𝑚 squared, as this will eliminate the denominator of 𝑚 squared on the right-hand side. When we multiply the left-hand side by 𝑚 squared, we get 25𝑚 squared. So our equation becomes 25𝑚 squared equals 100.
The next step is to divide both sides of the equation by 25, giving 𝑚 squared equals four. To find the value of 𝑚 then, we need to take the square root of each side of the equation. But be careful here. Nowhere in the question does it say that 𝑚 has to be positive. So when we’re taking the square root, we need both the positive and negative square roots of four, as there are two possible values of 𝑚.
The square root of four is just two. So we have that 𝑚 is equal to plus or minus two. You must make sure that you write down plus or minus or write 𝑚 equals two or negative two, as both are valid solutions for 𝑚.