1) V is the volume of a cube with edge length x and the cube expands as time passes. a) write an equation with amounts V and x . . . dV dx b t -- ) WI'l e an equation w1th rates (it and dt . dx . dV f :4 d _ : ' c)i x an [ dim 2 find dt 2) A circle with radius r has area A, and the circle expands as time passes. a) write an equation with amounts A and r . . . dA dr b write an e nation w th rat d ) q 1 es dt an dt c) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 2 mfs, how fast is the area of the spill increasing when the radius is 30 m? . dx . dy =x3 + 2x = = 3) y iix 2and ldt 1m 5 ,then find (it 4) A plane flying horizontally at an altitude of 1 mi and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station. (a) What quantities are given in the problem? Which are amounts and which are rates? (b) What is the unknown? (c) Draw a picture of the situation for any time t. ((1) Write an equation that relates the amounts. (e) Finish solving the problem. 5) The total personal income is rising in the Richmond-Petersburg, Virginia, metropolitan area. In 1999, the population of this area was 961,400, and the population was increasing at roughly 9200 people per year. The average annual income was $30,593 per capita, and this average was increasing at about $1400 per year (a little above the national average of about $1225 yearly). Estimate the rate at which total personal income was rising in the Richmond-Petersburg area in 1999. [hintc you will use the product rule] 6) Abaseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s (a) At what rate is his distance from second base decreasing when he is halfway to first base? (b) At what rate is his distance from third base increasing at the same moment? 7) A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the ground decreasing when 200 ft of string have been let out