Concepts of Calculus - I need help?

Question

I am not so good w/ word problems. I appreciate any help.

Thanks a lot.

1) S(x) = -x3 + 6x2 + 288x + 4000, 4 ≤ x ≤ 20 is an approximation to the number of salmon swimming upstream to spawn, where x represents the water temperature in degrees Celsius. Find the temperature that produces the maximum number of salmon.

2) Assume that the temperature T of a person during a certain illness is given by T(t) = -0.1t2 + 1.3t + 98.6, 0 ≤ t ≤ 12 where T = the temperature (°F) at time t, in days. Find the maximum value of the temperature and when it occurs. Round your answer to the nearest tenth, if necessary.

3) The total-revenue and total-cost functions for producing x clocks are R(x) = 520x - 0.01x2 and C(x) = 120x + 100,000, where 0 ≤ x ≤ 25,000. What is the maximum annual profit?

Answer 1

1) S(x) = - x³ + 6x² + 288x + 4000, 4 ≤ x ≤ 20

S'(x) = - 3x² + 12x + 288 = 0 for max or min.

x² - 4x - 96 = 0

(x + 8)(x - 12) = 0

x = - 8, 12

but - 8 is outside the domain, so

x = 12°C

S''(12) = - 6(12) + 12 = - 60

so S(12) is a maximum

2) T(t) = - 0.1t^2 + 1.3t + 98.6, 0 ≤ t ≤ 12

T(t) = - 0.2t + 1.3 =  0 for max or min.

t = 0.65, which is a max since T''(t) < 0

T(0.65) = - 0.1(0.65)^2 + 1.3(0.65) + 98.6

T(0.65) = 99.40275 ≈ 99.4 °F

3)

R(x) = 520x - 0.01x^2

C(x) = 120x + 100,000

0 ≤ x ≤ 25,000

P(x) = R(x) - C(x)

P(x) = - 0.01x^2 + 400x - 100,000

P'(x) = - 0.02x + 400

x = 400/0.02 = 20,000

P(x) = - 0.01(20,000)^2 + 400(20,000) - 100,000

P(x) = 3,900,000