Your Turn 3-An arch is the shape of a rectangular hyperbola. If it is $300 \mathrm{~m}$ wide at its base and has a maximum height of $100 \mathrm{~m}$, how high is the arch $30 \mathrm{~m}$ from the end?

Step 1 ：Given a rectangular hyperbola with width 300m and maximum height 100m, we need to find the height 30m from the end.

Step 2 ：A rectangular hyperbola can be represented by the equation: \(xy = k\), where x and y are the coordinates of a point on the hyperbola, and k is a constant.

Step 3 ：Using the given width and height, we can find the value of k: \(x = 150\) (half of the width), \(y = 100\) (maximum height), so \(k = 150 \times 100 = 15000\).

Step 4 ：Now that we have the value of k, we can find the height of the arch 30m from the end. Since the width of the arch is 300m, the x-coordinate of the point we're interested in is 30m from the end, which is \(x = 270\).

Step 5 ：Calculate the height (y-coordinate) using \(x = 270\) and \(k = 15000\): \(y = \frac{k}{x} = \frac{15000}{270} \approx 55.56\).

Step 6 ：\(\boxed{\text{The height of the arch 30m from the end is approximately 55.56m}}\)