15_Practice Quiz in Mathematics_College Algebra_Trigonometry_Probability and Statistics_Calculus with Answer and Solution

Refresher - MATHEMATICS Quiz 15
PROBLEM 1:


A rectangle has its length equal to (5x = 2) millimeters, find its width if it
has an area of (15 x2 – 29x – 14) square millimeter.


Solution:
A = LW
15 x 2 - 29x - 14 = (5x + 2)W
15 x 2 - 29x - 14
W=
5x + 2
(3x - 7)(5x + 2)
W=
(5x + 2)
W = 3x - 7

, Refresher - MATHEMATICS Quiz 15
PROBLEM 2:

A physicist finds that an unknown radioactive substance registers 2000 counts per
minute on a Geiger counter. Ten days later the substance registers 1500 counts
per minute. Using calculus, it can be shown that after t days the amount of
radioactive material and hence the number of counts per minute N(t) is directly
proprotional to ect for some constant c. Determine the half life of this substance.

Solution:
N(t) = K ect
When t = 0 N(0) = 2000
2000 = K ec(0)
2000 = K e0
K = 2000
N(t) = 2000 ect
N(10) = 1500
1500 = 2000ec(10)
¾ = e10c
10 c ln e = ln ¾
1
c = 10 ln (¾)
Half life of the substance N(t) = 1000
N(t) = 2000ect
1000 = 2000ect
½ = ect
ln ½ = ct ln e
ct = ln (½)
1 ⎛ 1⎞
t= ln ⎜ ⎟
1 ⎛ 3⎞ ⎝ 2⎠
ln
10 ⎜⎝ 4 ⎟⎠
t = 24 days