2) A geometry textbook gives the equation of a parabola as y =x2, where x and y are measured in centimetres. How can this equation be dimensionally correct?
y = x^2 = x * x cm * cm or cm^2
So it is not correct.
It is probably a typo or they just mean that lengths are in centimeters, leaving you to figure out that area is therefore in square centimeters.
To understand how the equation y = x^2 can be dimensionally correct in the context of a geometry textbook, let's break it down.
In this equation, both x and y represent measurements in centimeters, which means they have the dimension of length.
In the equation y = x^2, the variable x is squared, which means its dimension is also squared. So, the dimension of x^2 is (length)^2 or length squared.
Now, we need to check if the dimension of y is also length squared. Since y is the dependent variable, its dimension should match the dimension of x^2.
Since x^2 has the dimension of length squared, it is correct for y to have the same dimension, which makes the equation dimensionally correct.
To summarize, the equation y = x^2 is dimensionally correct because both x and y have the dimension of length (centimeters), and squaring x maintains the dimension of length squared.