Pharmacy exam 2018-12-17

Degrees: Pharmacy and Biotechnology
Date: Dic 17, 2018

Question 1

A organism metabolizes alcohol at a rate half of the present amount per minute. If initially there is no alcohol and we start to introduce alcohol in the organism at a constant rate of 2 ml/min, how much alcohol will there be in the organism after 5 minutes?

Solution

Let

a

be the alcohol in the organism and

t

the time.
Differential equation:

a^{'} = 2 - a / 2

.
Solution:

a (t) = 4 - 4 e^{- t / 2}

a (5) = 3.6717

ml.

Question 2

The amount $y$ of bacteria of type $B$ (in thousands) in a culture is related to the amount $x$ of bacteria of type $A$ (also in thousands) according to the function $y = f (x)$ . Knowing that the equation $x^{2} y^{3} - 6 x^{3} y^{2} + 2 x y = 1$ is satisfied in this culture and that $f (1 / 2) = 2$ , study if $f$ could have a local maximum at $x = 1 / 2$ .

Solution

Implicit derivative:

y^{'} = \frac{- 2 x y^{3} + 18 x^{2} y^{2} + 2 y}{3 x^{2} y^{2} - 12 x^{3} y + 2 x}

y^{'} (1 / 2) = 6 \neq 0

, so

f

has no local maximum at

x = 1 / 2

Question 3

A capsule has pyramidal shape with base a rectangle of sides $a = 3$ cm, $b = 4$ cm, and height $h = 6$ cm.

How must change the dimensions of the capsule to increase the volumen the most? What would be the rate of change of the volume if we changed the dimensions in such a way?
If we start to change the dimensions of the capsule such that the largest side of the rectangle decreases half of the increase of the smaller side, and the height increases the double of the increase of the smaller side, what will the rate of change of the volume be?

Remark: The volume of a pyramid is $1 / 3$ of the base area times the height.

Solution

$\nabla V (3, 4, 6) = (8, 6, 4)$ and the volume will increase $| \nabla V (3, 4, 6) | = 10.7703$ cm $^{3}$ /s if we change the dimensions of the capsule following this direction.
Directional derivative of $V$ in $(3, 4, 6)$ along the vector $u = (1, - 1 / 2, 2)$ : $V_{u}^{'} (3, 4, 6) = 5.6737$ cm $^{3}$ /s.

Question 4

The yield of a crop $y$ depends of the concentrations of nitrogen $n$ and phosphor $p$ according to the function $y (n, p) = n p e^{- (n + p)} .$ Compute the amount of $n$ and $p$ that maximizes the yield of the crop.

Solution

n = 1

and

p = 1

Exam

Last updated on Sep 15, 2020