CAlculus in Context #3 Math 113 Chapter 3 How does our understanding of derivatives give us insight into air pressure estimates used by climbers? Higher altitudes can present challenges to mountain climbers and hikers. One of those challenges is dealing with air pressure changes. A rule of thumb is that air pressure (using inches of mercury) drops about 1 inch for every 1000-foot increase in height above sea level. 1. Given that the air pressure at sea level is 30 inches of mercury, write a formula for air pressure as a function of height above sea level for this rule of thumb. Call this function the Climber's Estimate and use the function notation C(h). 2. The summit of Babad Do'ag (Mount Lemmon) north of Tucson is roughly 9200 feet above sea level. Use the Climber's Estimate function to estimate the air pressure at the summit of Babad Do'ag. Show your work appropriately. 3. At roughly 14500 feet above sea level, the summit of Tumanguya (Mount Whitney) in California is the highest point in the lower 48 states of the US. Use the Climber's Estimate function to estimate the air pressure at the summit of Tumanguya. Show your work appropriately. 4. At what elevation does the Climber's Estimate predict that the air pressure would be zero? Show your work appropriately. 5. Find the derivative of the Hiker's Estimate function. Explain why the value of the derivative makes sense within the framework of this rule of thumb. by Page 17

Question 7. (AQA June 2005 Intermediate Paper 1 NO Calculator) The diagram shows a cylinder, The diameter of the cylinder is 10 cm. The height of the cylinder is 10 cm. (a) Calculate the volume of the cylinder. Give your answer in terms of T. 10 cm TIX5 x10 = 50 T -10 cm T0OS ado [3 marks] Twenty of the cylinders are packed in a box of height 10cm. The diagram shows how the cylinders are arranged inside the box. The shaded area is the space between the cylinders. (b) Work out the volume inside the box that is not filled by the cylinders. Give your answer in terms of n. Not drawn accurately [4 marks]

Questions 2-5. Consider the data in the table. Time Value Absolute Change Relative Change Percent Increase |0 13.60 XX XX XX 1 14.85 2 16.10 3 17.35 |4 18.60 19.85 Complete the table by computing the absolute and relative change.

Questions 6-9. Consider the data in the table. Absolute Relative Percent |Change Change Increase Time Value 13.60 17.00 2 21.25 3 26.56 |4 33.20 41.50 Which model would better represent this data, linear or exponential? Why? An exponential model would be better because the relative change is constant. A linear model would be better because the relative change is constant. A linear model would be better because the absolute change varies. An exponential model would be better because the absolute change is constant.