Question

(a) Find formulas for the distance between (0,0) and \(\left(a, a^{2}\right)\) along the line between these points and along the parabola \(y=x^{2}\) (b) Use the formulas from part (a) to find the distances for \(a=1\) and \(a=10\) (c) Make a conjecture about the difference between the two distances as \(a\) increases.

Short answer

The distance along the line is \(\sqrt{a^2+a^4}\) and along the parabola is \(\int_{0}^{a}\sqrt{1+4x^2}dx\). Also, for a=1, the line distance is \(\sqrt{2}\) and for a=10 is \(\sqrt{1010}\). The parabola distance needs to be calculated using the integral expression above. The conjecture regarding the difference of the distances needs numerical evaluations of the integrals.

Step-by-step solution

Derive distance formula along the line

A line can be described by \(y = mx+b\). In this case, we are speaking about a line that crosses the origin and the point \((a, a^2)\), which means that the slope \(m=a\). Therefore, the line equation is \(y=ax\). The distance \(D_l\) between two points in a line can be calculated by the formula \(D_l=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\), where \((x_1, y_1)\) is the origin and \((x_2, y_2)\) is \((a, a^2)\), so plug in these values and get \(D_l=\sqrt{a^2+(a^2)^2}=\sqrt{a^2+a^4}\)

Derive distance formula along the parabola

The equation of the parabola is \(y = x^2\). The arc length \(D_p\) (the distance on the parabola from the origin to the point \((a, a^2)\)) can be calculated by integrating over the interval from 0 to \(a\), with the formula \(\sqrt{1+(y')^2}\) dx, being \(y'\) the derivative of \(y = x^2\), which is \(2x\). Carry out the integration and get \(D_p=\int_{0}^{a} \sqrt{1+4x^2}dx\)

Evaluate distances for a=1 and a=10

Evaluate the distances by plugging in the values of \(a\) into the formulas derived earlier. For \(a=1\), get \(D_l=\sqrt{2}\) and \(D_p\) is calculated by evaluating the integral, with the boundaries being from 0 to 1. For \(a=10\), find \(D_l=\sqrt{1010}\) and again, evaluate the definite integral from 0 to 10 to find \(D_p\).

Make a conjecture about the difference in distances

Observe these two sequences, for the line and the parabola distances respectively, as \(a\) increases. It could be conjectured that the difference between these two distances may be approaching a fixed number or increasing at a consistent rate, depending on calculations. The conjecture depends on the particular numerical evaluation of the integral for \(D_p\) and will most likely involve a comparison or ratio between \(D_l\) and \(D_p\).