5. Suppose a cup of cocoa cooled from 90° C to 60° C after 10 minutes in a room whose temperature was 20° C . Use Newton’s law of cooling [T − Ts = (T0 − Ts )⋅ e−k⋅t ] to answer the following questions: a) How much longer would it take the cocoa to cool to 35° C ? b) Instead of being left to stand in a room, the cup with initial temperature 90° C is placed in a freezer whose temperature is −15° C . How long will it take the cocoa to cool from 90° C to 35° C ?

a) t = 1051.6 sec = 17.5 min

b) t = 795.5 sec = 13.25 min

Explanation:

First of all we use the initial data to find out constant 'K'.

T - Ts = (T₀ - Ts) e^(-kt)

Here, we have:

T = Final Temperature = 60° C

Ts = Surrounding Temperature = 20° C

T₀ = Initial Temperature = 90° C

t = time = 10 min = 600 sec

k = constant = ?

Therefore,

60° C - 20° C = (90° C - 20° C).e^(-k600)

40° C/70° C = e^(-k600)

ln (0.57142) = -600k

k = 9.327 x 10⁻⁴ sec⁻¹

a)

Now, for this case we have:

T = Final Temperature = 35° C

Ts = Surrounding Temperature = 20° C

T₀ = Initial Temperature = 60° C

t = time = ?

k = constant = 9.327 x 10⁻⁴ sec⁻¹

Therefore,

35° C - 20° C = (60° C - 20° C).e^(-9.327 x 10⁻⁴ sec⁻¹ x t)

15° C/40° C = e^(-9.327 x 10⁻⁴ sec⁻¹ x t)

ln (15/40) = - 9.327 x 10⁻⁴ sec⁻¹ x t

t = 1051.6 sec = 17.5 min

b)

Now, for this case we have:

T = Final Temperature = 35° C

Ts = Surrounding Temperature = -15° C

T₀ = Initial Temperature = 90° C

t = time = ?

k = constant = 9.327 x 10⁻⁴ sec⁻¹

Therefore,

35° C + 15° C = (90° C + 15° C).e^(-9.327 x 10⁻⁴ sec⁻¹ x t)

50° C/105° C = e^(-9.327 x 10⁻⁴ sec⁻¹ x t)

ln (50/105) = - 9.327 x 10⁻⁴ sec⁻¹ x t

t = 795.5 sec = 13.25 min

elly's long hair:

a. if the salon only takes cash, then this isn't the problem. it doesn't matter how much she has in her checking account.   she can't get a haircut until she has the cash.

b. if she has overdraft protection, then there's no problem.   she can go ahead and write a check that her account can't cover, then sometime later, pay the bank some outrageous usury for loaning her a few bucks for a few days, but with beautiful hair.

c. if elly has some principles, and doesn't want to write a bad check, then now she has a problem.   her desire for the new hairstyle is in conflict with her honesty and integrity.

d.   if the bank will not process a check she writes, then even if she had a million $$ in her account, she couldn't use any of it.   the amount she has in her account is not the problem.   and if i may suggest, she might as well close her checking account, and put the money where it'll work for her.   it's not doing her a bit of good in a checking account, since she can't write any checks.

justin can't pay the full balance:

a. he can't even handle the credit limit he has now.   under no circumstances should he ask for a higher limit.

b.   he can ask for the grace period to be extended.   but that's no solution, the bank will almost certainly refuse anyway, and there are much better options.   he knew the rules of his credit card when he started using it.

c.   he definitely should pay the interest on the balance, plus as much of the balance as he can cover at the same time.

d.   he definitely should pay the minimum otherwise they could shut down his card.   he should also pay as much more as he possibly can.   he should also remember the awkward situation he's in, stop buying stuff on credit that he can't afford, and avoid adding any more debt.

i'd say that justin's best best option isn't on this list of choices.   what he has to do:   pay as much as he can at least the minimum payment plus as much more as he can and then avoid adding more debt.