CALCULUS

Assignment 6 problems Suppose that sand is stored in a large bag, but at time t = f] the bag develops a leak. The amount of sand 5' {measured in kilograms} in the bag at time t {measured in minutes] can be modeled by the following function: 3 so) = K(1::) . where K is a positive constant and ,u is a positive constant called the coarseness constant of the sand. The domain of S is [[11 T] where T is the time when the bag becomes empty. 1. {a} (1 mark} How much sand was in the bag when it started leaking? Your answer may involve K andfor u. You should specify units. {b} (1 mark} How fast is the sand leaking from the bag at time 1. minutes? Your answer may involve K andfor it. You should specify units. {c} (2 marks} For this part only, suppose Jr; 2 B. At time 1' = 2 minutes. is the rate at which sand is leaking From the bag speeding up or slowing down? Why? Support your answer with some computations and a written explanation. 2. (a) (1 mark} How long does it take for all of the sand to leak out? Your answer may involve K and/or p. You should specify units. {b} (1 mark} Use your answers to Qllb} and Q2fa} to check that r13 Iim = D I >T- (It (Remember that T is the time when the bag becomes empty} Does this limit being zero make physical sense? Write a sentence explaining why or why not. 3. (2 marks} 3 is an invertible function. You do not need to prove that. For this question, suppose that H = 80 and p = 8. Find [5 1}f [10} using any method. [You do not need to provide units here unless you want to.) 4. Laila, a MATHS student, is investigating how different types of sand affect the rate of the leak. Laila sets t = l. and thinks of this as a function of III only (with t being constant at 1): slot} = If (1 if [1. _ r131 [a]: (1 mark} Find E. {b} (1 mark) From her calculations, Laila concludes that for larger values ofgt1 more sand has leaked out of the bag at time t = 1. Is this correct? Explain