Answer:
14.8 ml of the diphosphonate kit to the syringe to correct the dose to 20 mCi
Step-by-step explanation:
To solve this problem, we can use a formula for the sum of a geometric sequence:
S = a(1 - r^n) / (1 - r)
where S is the sum of the geometric sequence, a is the first term, r is the common ratio, and n is the number of terms.
In this case, the first term is $10, the common ratio is 1.1 (since Patrick receives 10% more each week), and there are 52 terms. So we have:
a = $10
r = 1.1
n = 52
Plugging these values into the formula, we get:
S = 10(1 - 1.1^52) / (1 - 1.1)
S = 10(1 - 5.68 x 10^-6) / (-0.1)
S = 10(0.99999432) / (-0.1)
S = -$999.94
So over the 52 weeks, Patrick received a total of -$999.94. Note that the negative sign indicates that Patrick actually paid money, rather than receiving it, since the increasing payments were not enough to offset the initial payment of $10.
A diphosphonate kit contian 180 mCi of Tc99m in 30 ml when it is prepared at 8am. Immediately, a 20 mCi dose is withdrawn for a bone scan. if the patient arrives late at 9:30 and half the volume is accidentally discharged, how much volume from the kit must now be added to the syringe to correct the dose to 20 mCi? (no other doses have been withdrawn meanwhile, and the decay factor for 1.5 hrs is 0.841)
To solve this problem, we need to use the fact that the decay factor for 1.5 hours is 0.841. This means that if we wait 1.5 hours, the amount of Tc99m will have decreased by a factor of 0.841.
When the kit is prepared at 8am, it contains 180 mCi of Tc99m in 30 ml. If a 20 mCi dose is immediately withdrawn for a bone scan, the remaining amount of Tc99m in the kit is:
180 mCi - 20 mCi = 160 mCi
If we wait 1.5 hours, the amount of Tc99m in the kit will be:
160 mCi x 0.841 = 134.56 mCi
However, half of the volume is accidentally discharged, which means that the remaining volume is 15 ml. To correct the dose to 20 mCi, we need to calculate how much volume from the kit must be added to the syringe. Let's call this volume "x".
We can set up a proportion to solve for x:
(134.56 mCi / 30 ml) = (20 mCi / (15 ml + x))
Cross-multiplying, we get:
134.56 mCi (15 ml + x) = 20 mCi (30 ml)
Expanding and simplifying, we get:
2018.4 mCiml = 20 mCiml + 134.56 mCi*x
Subtracting 20 mCi*ml from both sides and dividing by 134.56 mCi, we get:
x = (2018.4 mCiml - 20 mCiml) / (134.56 mCi)
x ≈ 14.8 ml
Therefore, we need to add approximately 14.8 ml of the diphosphonate kit to the syringe to correct the dose to 20 mCi.