The decay of 200 particles of a particular radioactive substance is given by y=200(0.93)^x, where x is the number of days and y is the number of particles remaining. It costs the laboratory $1.50 per day to store each particle. On which day will the cost to store the particles be $135.
a) day 4
b) day 11
c) day 9
d) day 5
y = 135/1.50 = 90 particles
90 = 200 * .93^x
0.45 = .93^x
log 0.45 = x log 0.93
-.347 = x * -.0315
x = 11
Lo and behold, 11 is an answer choice.
To find out on which day the cost to store the particles will be $135, we need to determine the number of particles on that day and multiply it by the cost per day.
Given: y = 200(0.93)^x
To calculate the number of particles remaining on a particular day, we substitute the value of x into the equation. Let's calculate the value of x for each option and check which one gives a cost of $135.
a) day 4:
Substituting x = 4 into the equation:
y = 200(0.93)^4
y ≈ 200(0.716)
y ≈ 143.2
The cost on day 4 would be:
Cost = y * cost per particle = 143.2 * 1.50 = $214.80
b) day 11:
Substituting x = 11 into the equation:
y = 200(0.93)^11
y ≈ 200(0.736)
y ≈ 147.2
The cost on day 11 would be:
Cost = y * cost per particle = 147.2 * 1.50 = $220.80
c) day 9:
Substituting x = 9 into the equation:
y = 200(0.93)^9
y ≈ 200(0.755)
y ≈ 151
The cost on day 9 would be:
Cost = y * cost per particle = 151 * 1.50 = $226.50
d) day 5:
Substituting x = 5 into the equation:
y = 200(0.93)^5
y ≈ 200(0.814)
y ≈ 162.8
The cost on day 5 would be:
Cost = y * cost per particle = 162.8 * 1.50 = $244.20
Therefore, the correct answer is option c) day 9, where the cost to store the particles will be $135.
To find the day when the cost to store the particles is $135, we need to calculate the cost per day as the product of the number of particles remaining and the storage cost per particle.
Let's start by substituting the given equation for y into the cost per day equation:
Cost per day = number of particles remaining * storage cost per particle
Cost per day = (200 * (0.93)^x) * $1.50
Now, we can set up an equation to find the value of x (the number of days) when the cost per day is $135:
$135 = (200 * (0.93)^x) * $1.50
To simplify the equation, divide both sides by $1.50:
$135 / $1.50 = 200 * (0.93)^x
Now divide both sides by 200:
($135 / $1.50) / 200 = (0.93)^x
Simplify the left side:
$135 / ($1.50 * 200) = (0.93)^x
Now, we can solve for x by taking the logarithm of both sides:
log($135 / ($1.50 * 200)) = log((0.93)^x)
Using logarithm properties, we can bring down the exponent x:
log($135 / ($1.50 * 200)) = x * log(0.93)
Now, divide both sides by log(0.93):
log($135 / ($1.50 * 200)) / log(0.93) = x
Using a calculator, compute the left side to find the value of x.
x ≈ 11.05
Since x represents the number of days, we can round it to the nearest whole number:
x ≈ 11
Therefore, the cost to store the particles will be $135 on day 11.
The correct answer is option b) day 11.