[Problem Solving 1 - 25 points]

A 1.00-gram sample contains two radioactive isotopes A and B. The half-life of isotope A is 2 hrs. After 1 hr, the amounts of A and B in grams left in the sample are equal. After 3 hrs, the ratio of the amounts of A and B left in the sample is 5:3.

a) What is the half-life of isotope B?

b) What are the amounts of isotopes A and B in the initial sample?

[Problem Solving 2 - 25 points]

A hemispherical tank of radius 8 ft. is initially filled with water. At the bottom of the tank, there is a hole 2 in. in radius through which water drains under the influence of gravity. Use a coefficient of 0.6 to account for friction and sudden contraction in the hole. How long will it take the tank to drain completely?

[Problem Solving 3 - 25 points]

The growth of malignant tumors can be modeled using the Gompertz model.

where r > 0 is the intrinsic growth rate and K > 0 is a constant.

a) Find the general solution to the Gompertz model and state the significance of the constant K.

b) A tumor with 2,500 cells is initially growing at a rate of 500 cells/day. Assuming the maximum size of the tumor is 100,000 cells, how large will the tumor be after 50 days?