The weight of an object above Earth’s surface varies inversely with the square of the distance from Earth’s centre. If a body weighs 50 pounds when it is 3960 miles from the Earth’s centre, what would it weigh if it was 3970 miles from Earth’s centre?
In this question, we have two variables. The weight of the object and the distance from Earth’s centre. We’re told that these two variables vary inversely. This is the same as an inverse variation or an inverse proportion. However, we’re told that the weight of the object is inversely proportional to the square of the distance. So we would say the weight of the object is inversely proportional to the distance squared. If we use 𝑤 to represent the weight of the object and 𝑑 for the distance, then we can write the statement of proportionality as 𝑤 is directly proportional to one over 𝑑 squared.
We could recall that for two variables 𝑥 and 𝑦, where 𝑥 is inversely proportional to 𝑦, and if we have 𝑦 is directly proportional to one over 𝑥, then we can write that 𝑦 equals 𝑘 over 𝑥 where 𝑘 is the constant of variation or proportion. So for our statement of proportionality, we can then write that 𝑤 equals 𝑘 over 𝑑 squared, where 𝑘 is the constant of proportion. We can now work at the value of our constant 𝑘 by using values that we’re given for the weight and the distance. We’re told that the weight is 50 pounds when it is 3960 miles from Earth’s centre.
So we can substitute 𝑤 equals 50 and 𝑑 equals 3960 into our equation, giving us 50 equals 𝑘 over 3960 squared. Using our calculator then, we can evaluate this as 50 equals 𝑘 over 15681600. To find 𝑘 by itself, we multiply both sides of our equation by 15681600. So 𝑘 is equal to 784080000. And now that we have found the value of 𝑘, we can then substitute it into our equation of proportionality. Therefore, our complete equation of proportionality will be 𝑤 equals 784080000 over 𝑑 square. And we can now use this equation to help us find the unknown weight when the object is 3970 miles from Earth’s centre.
So given our equation, we can substitute 𝑑 equals 3970 into it, giving us 784080000 over 3970 squared. Using a calculator, we can evaluate our denominator as 15760900. So 𝑤 can be calculated as 49.748428 and so on pounds. We now need to round this to a more sensible answer. So rounding to two decimal places, we check our third decimal place digit to see if it is five or more. In this case, our value of eight means that we will round the answer up to 49.75 pounds. This means that when the object is 3970 miles from Earth’s centre, it will weigh 49.75 pounds.