3. Discuss the pros and cons of using tangent line approximations in the two situations you explored above. If you were in the position of advising someone on these issues, what would you need to consider in order to decide you should use the linear approximation? What are the potential harms or benefits of doing so? Use graphs of the functions and their approximations to help with your discussion. deal of emitido es or gatured (o) Hinate the air pres 4)(3290 4. A streamgage is a structure installed beside a river which contains instruments to measure the height of the river. Due to a broken streamgage, you must manually measure the height of a river each hour of a rainstorm. The cross sections of the river are equilateral right triangles. The water started at a height of 30in. After 1 hour, the height had increased by 6in. (a) Find a formula relating the area of the water in the cross section to the height of the water. Use the diagram below to help you. boy diw.esueel froide bas asidemedtem (b) besesen part A and Cor same. In part Air AC (b) What was the area of the water in a cross section when the height was 30in? What was the area of the water in a cross section when the height was 36in? Subtract these two answers to get the total increase in area in the first hour. (c) Assuming the area continues to increase at the same rate, at what rate is the height of the water changing when the height is 50 inches? (d) Suppose that after an hour, you realize you are coming down with a bad cold and decide to leave. You record that the river continued to increase 6 inches per hour. Explain the mathematical and ethical issues with your decision to extrapolate the data.

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