Governor's School Final 2003 Part 2
Part 2A
1. The function y = 2 t^2 - 3 t + 12 is quadratic. Evaluate the function at t = 0, 3, 6 and 9 and show that for this function the rate at which the rate changes is constant.
2. The function y = 3 t^2 + 4 t + 2 is quadratic. Evaluate the function at t = 0, 3, 6 and 9 and show that for this function the rate at which the rate changes is constant.
3. The function y = 4 t^2 - 2 t + 8 is quadratic. Evaluate the function at t = 0, 3, 6 and 9 and show that for this function the rate at which the rate changes is constant.
4. The function y = 5 t^2 - 2 t + 22 is quadratic. Evaluate the function at t = 0, 3, 6 and 9 and show that for this function the rate at which the rate changes is constant.
Part 2B
1. The function y = 2^x is exponential. Evaluate the function on the interval from x = 3 to x = 3.1 and again on the interval from x = 6 to x = 6.1 and show that the ratio of the average value of y to the rate at which y changes with respect to x is about the same on both intervals.
2. The function y = 3^x is exponential. Evaluate the function on the interval from x = 5 to x = 5.2 and again on the interval from x = 9 to x = 9.2 and show that the ratio of the average value of y to the rate at which y changes with respect to x is about the same on both intervals.
3. The function y = 4^x is exponential. Evaluate the function on the interval from x = 3.9 to x = 4 and again on the interval from x = 6.9 to x = 7 and show that the ratio of the average value of y to the rate at which y changes with respect to x is about the same on both intervals.
4. The function y = 5^x is exponential. Evaluate the function on the interval from x = 6 to x = 6.1 and again on the interval from x = 10 to x = 10.1 and show that the ratio of the average value of y to the rate at which y changes with respect to x is about the same on both intervals.
Part 2C
1. The frequency of oscillations of a mass on a certain spring, in cycles per minute, is given by f = 100 / sqrt(m), where m is the mass of the spring in grams. At what average rate is the period of oscillation, in seconds, changing with respect to mass between masses of 50 grams and 70 grams? Based on this rate how much would you expect the period to change if you added 5 grams to a 57 gram mass?
2. The period of oscillation of a mass on a certain spring, in seconds, is given by T = .03 * sqrt(m), where m is the mass of the spring in grams. At what average rate is the frequency of oscillation, in cycles / minute, changing with respect to mass between masses of 30 grams and 60 grams? Based on this rate how much would you expect the frequency to change if you added 2 grams to a 41 gram mass?
Part 2D
1. The cross-sectional area of a container at the point where water depth in cm is y is A = 100 sqrt(y), where A is in cm^2. How many cm^3 of water must be lost to reduce the depth from 8 cm to 7.7 cm? If this water flows out through a tube whose cross-sectional area is .4 cm^2 then how much tube is needed to contain it? If this water flows out in 3 seconds then how fast is the water flowing through this tube?
2. The cross-sectional area of a container at the point where water depth in cm is y is A = 50 * 1.1^y, where A is in cm^2. How many cm^3 of water must be lost to reduce the depth from 4.2 cm to 4.0 cm? If this water flows out through a tube whose cross-sectional area is .7 cm^2 then how much tube is needed to contain it? If this water flows out in 5 seconds then how fast is the water flowing through this tube?
3. The cross-sectional area of a container at the point where water depth in cm is y is A = 1000 / sqrt(y), where A is in cm^2. How many cm^3 of water must be lost to reduce the depth from 40 cm to 39 cm? If this water flows out through a tube whose cross-sectional area is 2 cm^2 then how much tube is needed to contain it? If this water flows out in 10 seconds then how fast is the water flowing through this tube?
4. The cross-sectional area of a container at the point where water depth in cm is y is A = 500 / y, where A is in cm^2. How many cm^3 of water must be lost to reduce the depth from 5 cm to 4.9 cm? If this water flows out through a tube whose cross-sectional area is .1 cm^2 then how much tube is needed to contain it? If this water flows out in .5 seconds then how fast is the water flowing through this tube?
Part 2E
1. The cross-sectional area of a container at the point where water depth in cm is y is A = 100 sqrt(y), where A is in cm^2. The exit velocity of water through a long tube in the bottom of this cylinder, in cm/sec, is sqrt(1960 y). If the tube has cross-sectional area .4 cm^2 then what is your best estimate of how long it will take for depth to change from 10 cm to 9 cm?
2. The cross-sectional area of a container at the point where water depth in cm is y is A = 50 * 1.1^y, where A is in cm^2. The exit velocity of water through a long tube in the bottom of this cylinder, in cm/sec, is sqrt(1960 y). If the tube has cross-sectional area .5 cm^2 then what is your best estimate of how much the water level will change in 10 seconds, starting from depth 6 cm? From the new level how much will the water level change in the next 10 seconds?
3. The cross-sectional area of a container at the point where water depth in cm is y is A = 1000 / sqrt(y), where A is in cm^2. The exit velocity of water through a long tube in the bottom of this cylinder, in cm/sec, is sqrt(1960 y). If the tube has cross-sectional area .3 cm^2 then what is your best estimate of how long it will take for depth to change from 10 cm to 9 cm?
4. The cross-sectional area of a container at the point where water depth in cm is y is A = 500 / y, where A is in cm^2. The exit velocity of water through a long tube in the bottom of this cylinder, in cm/sec, is sqrt(1960 y). If the tube has cross-sectional area .8 cm^2 then what is your best estimate of how much the water level will change in 10 seconds, starting from depth 6 cm? From the new level how much will the water level change in the next 10 seconds?
Part 2F
1. A cell wall is transports a certain nutrient by diffusion, with concentration being 100 units just outside the cell and 10 just units in the interior of the cell. The outside concentration doesn't change. At the outside wall the concentration is 80 units and at the inside wall the concentration is 20 units. The concentration in the middle is 40 units. The transition rate from outside to just inside is 50% while the transition rate from the inside wall to the interior of the cell is 40%. The transition rate within the cell is 30%. None of the nutrient goes 'backwards' but the interior of the cell utilizes 5% of the nutrient.
Sketch a trapezoidal graph representing this configuration, including the concentrations outside and interior to the cell. Include slopes and rates of slope change.
Use information from your graph to determine the concentrations after one transition. How much of the nutrient is transferred to the interior of the cell?
Will the amount transferred to the interior increase or decrease in the next transition?
Give your best estimate of what will happen to the concentrations over a long period of time.