Vitamin A is stored primarily in our plasma and our liver. suppose that 40% of the vitamin A in the plasma is filtered out by the kidneys each day and that 30% of the vitamin A in the plasma is absorbed into the liver each day. Also assume that 1% of the vitamin A in the liver is absorbed back into the plasma each day.

Question

Vitamin A is stored primarily in our plasma and our liver. suppose that 40% of the vitamin A in the plasma is filtered out by the kidneys each day and that 30% of the vitamin A in the plasma is absorbed into the liver each day. Also assume that 1% of the vitamin A in the liver is absorbed back into the plasma each day.

(a) how much vitamin A should you ingest each day so that the equilibrium amount of vitamin A in the plasma is 4 mg? Assume all of the vitamin A ingested goes into the plasma.
(b) what is the equilibrium amount of vitamin A in your liver given the answer to part(a)

Answer 1

Draw a figure

whole body
intake=loss from body

liver
loss to plasma=gain from plasma

plasma=loss to liver+loss to body-gain from liver+intake
so you have three equations.
whole body:
intake=.4(4mg)=1.6mg/day
liver
.1L=.3(4mg) or L=12mg
Plasma
.3*4mg+.4*4mg-.1L+intake=0
1.2mg+1.6mg-1.2mg+1.6mg=0
checks. The system is in equilibrium.

Answer 2

To solve this problem, we need to consider the equilibrium amount of vitamin A in both the plasma and the liver. Let's break it down step by step:

(a) To find the amount of vitamin A you should ingest each day for an equilibrium amount of 4 mg in the plasma, we need to consider the filtration and absorption rates.

Let's say the amount of vitamin A ingested each day is x mg.

The filtration rate removes 40% of vitamin A in the plasma each day, so we have 0.4 * x mg of vitamin A filtered out.

The absorption rate into the liver is 30%, so 0.3 * x mg of vitamin A is absorbed into the liver.

Now let's calculate the total change in vitamin A in the plasma:

Change in vitamin A in the plasma = vitamin A ingested - filtered out + absorbed from the liver

Change in vitamin A in the plasma = x mg - 0.4 * x mg + 0.3 * x mg

To reach equilibrium, the change in vitamin A in the plasma must be zero, meaning the amount ingested equals the amounts filtered out and absorbed:

x - 0.4 * x + 0.3 * x = 0

Simplifying the equation:

0.9 * x = 0

This equation tells us that the equilibrium amount of vitamin A in the plasma is zero, meaning there is no vitamin A remaining in the plasma.

However, we want to achieve an equilibrium amount of 4 mg in the plasma. Therefore, we need to adjust our equation:

0.9 * x = 4 mg

Solving for x:

x = 4 mg / 0.9

x ≈ 4.44 mg

Therefore, you should ingest approximately 4.44 mg of vitamin A each day to achieve an equilibrium amount of 4 mg in the plasma.

(b) Now let's determine the equilibrium amount of vitamin A in the liver given the answer to part (a).

We know that 1% of the vitamin A in the liver is absorbed back into the plasma each day. So, the equilibrium amount in the liver can be calculated using the following equation:

Equilibrium amount in the liver = absorbed from the plasma / absorption rate

Equilibrium amount in the liver = 0.01 * (0.4 * x) / 0.01

Substituting the value of x from part (a):

Equilibrium amount in the liver = 0.01 * (0.4 * 4.44 mg) / 0.01

Equilibrium amount in the liver ≈ 0.4 * 4.44 mg

Equilibrium amount in the liver ≈ 1.776 mg

Therefore, the equilibrium amount of vitamin A in your liver is approximately 1.776 mg given the ingestion rate from part (a).