急! F5 About equations 1exq05

41.
The n be the number of members of the club.

[2000/(n - 4)] - (2000/n) = 25
[2000n - 2000(n - 4)] / [n(n - 4)] = 25
8000/(n² - 4n) = 25
25n² - 100n = 8000
n² - 4n - 320 = 0
(n - 20)(n + 16) = 0
n = 20 or n = -16 (rejected)

Hence, there are 20 members of the club.


2.
Let t hours be the time taken for the longer route.
Then the time taken for the shorter route = (t + 0.5) hours

(24/t) - (20/(t + 0.5) = 4
[24(t + 0.5) - 20t] / [t(t + 0.5)] = 4
(4t + 12)/(t² + 0.5t) = 4
4t² + 2t = 4t + 12
2t² + t = 2t + 6
2t² - t - 6 = 0
(2t + 3)(t - 2) = 0
t = -3/2(rejected) or t = 2

Hence, the cyclist takes 2 hours for the longer route.


3.
Let u = 2^-0.02x

When the machine cannot produce paint, P = 0
11000(2^-0.02x) - 8000(2^-0.04x) - 3000 = 0
8(2^-0.04x) - 11(2^-0.02x) + 3 = 0
8(2^-0.02x)^2 - 11(2^-0.02x) + 3 = 0
8u^2 - 11u + 3 = 0
(8u - 3)(u - 1) = 0
u = 3/8 or u = 1
2^-0.02x = 3/8 or 2^-0.02x = 1
log(2^-0.02x) = log(3/8) or log(2^-0.02x) = log(1)
(-0.02x)log(2) = log(3/8) or (-0.02x)log(2) = 0
x = [1/(-0.02)] [log(3/8)/log(2)] or x = 0 (rejected)
x » 70.75

Hence, the machine produce paint at most 70 years.

41)
"a" assumes total member
"x" assumes payment per member

a*x = 2000 eqn 1
(x+25)*(a-4) = 2000 eqn 2
2 eqns 2 unknows, you find out yourself

42)
"x" assume speed for longer route in km/hr
30 min. = .5 hr

[20/(x-4)] - [24/(x)] = .5
solve it by yourself

43)
hint: use log