Your swimming pool containing 84,000 gal of water has been contaminated by 6 kg of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of 250 gal/min.

(a) What is the initial value problem for the filtering process? Let q be the amount of dye in grams in the pool at any time t. dq dt = − 250q/84000​ Correct: Your answer is correct. g/min, q(0) = Correct: Your answer is correct. grams

(b) Solve the problem in part (a).

(d) Find the time T at which the concentration of the dye first reaches the value of 0.03 g/gal. (Round your answer to two decimal places.) T = hr

(e) Find the flow rate r that is sufficient to achieve the concentration 0.03 g/gal within 4 h. (Round your answer to the nearest integer.) r = gal/min

Suppose f : [a, b] −→ R is continuous on [a, b] and that f attains its minimum value at ξ ∈ (a, b). If f ' (ξ) exists, prove that f ' (ξ) = 0.

5. The Rams are playing the Aggies in the last conference game of the season. The Rams are trailing the Aggies 21 to 14 with 7 seconds left in the game, when they score a touchdown. Still trailing 21 to 20, the Rams can either go for two points and win or go for one point to send the game into overtime. The conference championship will be determined by the outcome of this game. If the Rams win, they will go to the Candy Ball with a payoff of $6.4 million; if they lose they go to the Crocodile Bowl with a payoff of $1.2 million. If the Rams go for two points there is 30% chance they will be successful and 70% chance they will fail and lose. If they go for one point there is a 0.97 probability of success and 0.03 probability of failure. If they tie they will play overtime, in which the Rams have a 40% of winning because of fatigue.

1. Draw a decision tree for this problem.
2. What should the Rams do to achieve the highest expected monetary value (EMV) and what is the EMV?
3. What would the Rams’ probability of winning the game in overtime have to be to make the Rams indifferent between going for one point or two points?

Using the factorization of wave operator, derive the D’Alembert formula.

show (b1+b2+...+bn)/n >= (b1b2...bn)^(1/n)

Hint: do induction over k when n = 2k . Then for 2k−1 < n < 2k append to b1, b2, . . . , bn, 2 k − n equal numbers all equal to the arithmetic mean of the first n.

QUESTION 4 [20 MARKS] T A company manufactures XPRINGLE and YPRINGLE products from Gold and Silver. One XPRINGLE requires 5 grams of Gold and 4 grams of Silver while one YPRINGLE needs 4 grams of Gold and 3 grams of silver. The company has only 400 grams of Gold while it can use at least 120 grams of Silver. The production process must produce at least 10 of XPRINGLE. It should also produce at least 10 of YPRINGLE but cannot exceed 80 of this type. The profit for one XPRINGLE is P100 and for one YPRINGLE is P150. Let ? represent the number of XPRINGLE and ? represent the YPRINGLE. Required:

a. Formulate the linear programme for this production process clearly stating all the
constraints and the objective function.
b. Show the linear programme graphically.
c. Use the corner points to recommend the production mix that will maximize profit.
(4 minutes)
d. If the production process is allocated 990 min with the XPRINGLE taking 11 minutes
YPRINGLE taking 9 minutes, would the result obtained in (d)) above change? Explain
fully.

Apply single linkage clustering to these schools until only one option remains. What conclusions can you make from this analysis?

A manufacturer of sports equipment has developed a new synthetic fishing line that he claims has a mean breaking strength of 8 kilograms. If a random sample of 20 lines is tested and found to have a sample mean breaking strength of 7.8 kilograms with a sample variance of 0.25. By using hypothesis testing, does this suggest at a 0:01 level of significant that the mean breaking strength is not 8 kilograms? Assume the population of the breaking strength to be normal.

. Use the Taylor expansion of the function f(z) = 1 1+z [8] 4 centred at the origin z = 0, together with the extended Cauchy Integral Formula to evaluate the contour integrals I C dz/ z^ k (z^ 4 + 1), k = 0, 1, . . . , where C is any positively oriented simple contour going around the origin that is interior to the circle of radius 1 centred at z = 0.