Step 1 :First, we need to find the temperature difference between the base of the mountain and the desired elevation. The temperature at the base is \(25^\circ\mathrm{C}\) and the desired temperature is \(17^\circ\mathrm{C}\). So, the temperature difference is \(25^\circ\mathrm{C} - 17^\circ\mathrm{C} = 8^\circ\mathrm{C}\).
Step 2 :Next, we know that the temperature drops by \(4^\circ\mathrm{C}\) for every \(1000\mathrm{m}\) of elevation. So, we can set up a proportion to find the elevation: \(\frac{8^\circ\mathrm{C}}{4^\circ\mathrm{C}/1000\mathrm{m}}\).
Step 3 :Now, we can solve the proportion: \(\frac{8^\circ\mathrm{C}}{4^\circ\mathrm{C}/1000\mathrm{m}} = \frac{8^\circ\mathrm{C}}{1^\circ\mathrm{C}} * 1000\mathrm{m} = 8 * 1000\mathrm{m} = 8000\mathrm{m}\).
Step 4 :So, the elevation at which the air temperature will be \(17^\circ\mathrm{C}\) is \(\boxed{8000\mathrm{m}}\).