Q1. Express y as a function of x. The constant C is a positive number. ln y = ln
ID: 3028762 • Letter: Q
Question
Q1. Express y as a function of x. The constant C is a positive number.
ln y = ln 4x + ln C
a. y = 4Cx
b. y = 4x + C
c. y = (4x)C
d. y = x + 4C
Q2. The half-life of silicon-32 is 710 years. If 100 grams is present now, how much will be present in 600 years? (Round your answer to three decimal places.)
a. 0
b. 0.286
c. 94.311
d. 55.668
Q3. Change the exponential expression to an equivalent expression involving a logarithm.
ex = 25
a. log 25 x = e
b. log x e = 25
c. ln x = 25
d. ln 25 = x
Q4. A fossilized leaf contains 12% of its normal amount of carbon 14. How old is the fossil (to the nearest year)? Use 5600 years as the half-life of carbon 14.
a. 1031
b. 17,099
c. 20,040
d. 36,108
Explanation / Answer
Q1. ln y = ln 4x + ln C = ln (4Cx) so that y = 4Cx ( as log a + lig b = log ab)
The answer a. is correct.
Q2. An exponential decay can be described by the following formula:
N(t) = N0 (1/2) t/t1/2 , where N0 is the initial quantity of the substance that will decay, N(t) is the quantity that still remains and has not yet decayed after a time t, and t12 is the half-life of the decaying matter. Here, t1/2 is 710 years, N0= 100gms and t is 600 years. Then N(t) = 100(1/2)600/710 = 100( 0.5)60/71 = 100* 0.556683635 = 55.66836352 gms = 55.668 gms ( on rounding off to 3 decimal places).The answer d is correct.
Q3. ex =25 so that, on taking natural logarithms of both sides, x ln e = ln 25 or, x = ln 25
( as log ab) = b log a and ln e = 1) The answer d is correct.
Q4. We will use the same formula as in Q 2. Here, N(t)/N0 = 12 % = 0.12, and t1/2 = 5600 years. Then N(t)/N0 = (1/2)t/5600 or, 0.12 = (0.5)t/5600 . On taking logarithms of both sides, we have log(0.12) = t/5600(log 0.5) or, -0.920818754 =( t/5600)* ( - 0.301029995) or, t = 5600*( 0.920818754/ 0.301029995) = 5600*3.058893696 = 17129.8047 say, 17130 years ( on rounding off to the nearest whole number).The option b is nearest to the correct answer.
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