Question

I am really struggling with a bunch of problems. Any help with any of the following would be greatly appreciated. Thanks

1. ln e^(ln x) - ln(x - 3) = ln 8

2.3e^(6x - 2) = 18

The word problems, especially, are killing me.

3. In the formula A = Iekt, A is the amount of radioactive material remaining from an initial amount I at a given time t, and k is a negative constant determined by the nature of the material. An artifact is discovered at a certain site. If it has 68% of the carbon-14 it originally contained, what is the approximate age of the artifact? (carbon-14 decays at the rate of 0.0125% annually.) (Round to the nearest year.)

4. Coyotes are one of the few species of North American animals with an expanding range. The future population of coyotes in a region of Mississippi can be modeled by the equation P=52 + 12 ln(11t + 1) , where t is time in years. Use the equation to determine when the population will reach 150. (Round to the nearest tenth of a year.)

5. Coffee is best enjoyed at a temperature of 121° F. A restaurant owner wants to discover the temperature T at which he should serve his coffee so that it will have cooled to this ideal temperature in 6 minutes. He discovers that a cup of coffee served at 198° F cools to 185° F in one minute when his restaurant is at 70° F. If he maintains the restaurant temperature at 70° F, at what temperature should he serve the coffee to meet his goal?

6. The decay of 307 mg of an isotope is given by A(t) = 307e^-0.016t, where t is time in years. Find the amount left after 54 years.

Answer 1

1.
ln e^(ln x) - ln(x - 3) = ln 8 ----- apply the log rules
(lnx)(lne) - ln(x-3) = ln 8 ----- recall that ln e = 1
lnx - ln(x-3) = ln 8
ln (x/(x-3)) = ln 8 ---- recall : ln (A/B) - ln A - ln B
"anti-ln" it
x/(x-3) = 8
cross-multiply
8x - 24 = x
7x = 24
x = 24/7

2. Divide both sides by 3
take ln of both sides,

3. Your formula should be A = l e^(kt)
solve :
.68 = e^(-.0125t) for t , (I got about 30.86 years)

4. set
150 = 52 + 12ln(11t+1)
12ln(11t+1) = 98
ln(11t + 1) = 8.16667
11t+1 = 3521.5
11t = 3520.5
t = appr. 320

5. This is based on Newton's Formula of Cooling, which you must have been given.
Just sub in the values and solve

6. replace t with 54 and solve for A(54)
Just button-pushing.