SOLUTION: unsolved 2020-08-22 10:07:30 It is believed that two quantities, z and d are Connected by the relationship of the form z=kd^n where k and n are provided that d doesn't exceed some

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: unsolved 2020-08-22 10:07:30 It is believed that two quantities, z and d are Connected by the relationship of the form z=kd^n where k and n are provided that d doesn't exceed some       Log On


   

Question 1163697: unsolved 2020-08-22 10:07:30 It is believed that two quantities, z and d are Connected by the relationship of the form z=kd^n where k and n are provided that d doesn't exceed some fixed (but unknown)values D.An experiment produced the following data
D 750 810 870 930 990 1050 1110 1170
Z 2.1 2.6 3.2 4.0 4.8 5.6 5.9 6.1
a)Plot the values of log10Z against log10 d.Use these points to suggest a
value for D.
b)It is known tht for d < D,n is a whole number.Use your graph to find the value of n.Show also that k=5×10^-9
c) Use your value of n and the estimate k=5×10^-9 to find the value of d for which z=3.0.
Answer by KMST(5403) About Me  (Show Source):
You can put this solution on YOUR website!
If z=k%2Ad%5En , then log%2810%2Cz%29=log%2810%2Ck%29%2Bn%2Alog%2810%2Cd%29
The graph of log%2810%2Cz%29 against log%2810%2Cd%29 would be expected to be a straight line with slope n up to a certain value log%2810%2Cd%29=log%2810%2CD%29 .
We need to calculate and tabulate log%2810%2Cz%29 and log%2810%2Cd%29


a)Then we plot log%2810%2Cz%29 against log%2810%2Cd%29
The points in green, up to log%2810%2Cd%29=3.021 , corresponding to d=1050 fit well enough on a line. That supports suggesting highlight%28D=1050%29 .

b) We could estimate n as the slope between points %22%28%22log%2810%2Cd%29%22%2C%22log%2810%2Cz%29%22%29%22 %22%28%222.875%22%2C%220.3222%22%29%22 and %22%28%223.021%22%2C%220.7482%22%29%22
That slope is %280.7482-0.3222%29%2F%283.021-2.875%29=0.4260%2F0.146=2.92%28%22rounded%22%29
Since n is supposed to be a whole number, it must be highlight%28n=3%29.

c) For k=5%C3%9710%5E-9 log%2810%2Ck%29=log%2810%2C5%2A10%5E-9%29=-8.30103 (rounded)
Substituting values for z=3.0 , n=3 k=5%C3%9710%5E-9, with log%2810%2Ck%29=-8.30103%0D%0A and log%2810%2C3%29=0.47712 into log%2810%2Cz%29=log%2810%2Ck%29%2Bn%2Alog%2810%2Cd%29 we get
0.47712=-8.30103%2B3log%2810%2Cd%29
0.47712%2B8.30103=3log%2810%2Cd%29
8.77815=3log%2810%2Cd%29
8.77815%2F3=log%2810%2Cd%29
2.92615=log%2810%2Cd%29
d=10%5E2.92615 to get highlight%28d=843%29 as the value of d for which z=3.0.