Matemagica

This is a pretty basic one, it could even be done without calculus, but let’s use this as an opportunity to get used to solving these optimization problems. First, how could we write this algebraically? The difference between two numbers is a subtraction. Therefore, we could write it as: $$ x-y $$ But, we know our second number is actually the square or the first! This means \( y = x^2 \). Because of that, we can write \( x - y \) as this instead: ...

Let’s start by drawing the rectangle and naming one side \(x\) and the orher side \(y\): We want to maximize the area of this rectangle, so let’s write the area formula, which for a rectangle is: $$ A = xy $$ Since the area is what is changing, we can change both of these values at will as the area depends on both of them, but we need to express the area as a single variable function to be able to apply what we know about extreme values. ...