A rectangular box has a length of 1 in, width of 2 in, and height of 3 in. Find the cosine of the angle between the diagonal of the box and the diagonal of its base.

what is the volume of the parallelepiped by <-5; 2; 1>, <1; -1; -3> and
<-1; -1; -4>?

(a) 9 
(b) 10 
(c) 11 
(d) 12 
(e) None of the above

The set R^2 with addition and scalar multiplication defined by

(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)

c(x1, y1) = (cx1, y1)

is not a vector space. Determine which axiom fails and find a counterexample that shows that it fails.

Give the domains and ranges of the following functions. A) ln x, B) cos X, C) 10^x D) f(x)= 1/ x^2-12x-45 E) g(x)= sqrt(82-x).

Determine the following : a) what is the angle formed by an arc length of 3 radii?

b) what is the angle formed by an arc length of 1 radii?

c) what is the angle formed by an arc length of pi/2 radii?

d) what is the angle formed by an arc length of 4pi/3 radii?

Use the Midpoint Rule for triple integral to estimate the value of the integral. Divide B into eight sub-boxes of equal size. (Round your answer to three decimal places.)

B =

(x, y, z) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 2, 0 ≤ z ≤ 1

The derivative of ln|x| is an odd function

is it true/false

In a class of 130 shs students,41 offers agricultural science ,62 offers business,and 31 offers visual arts .16 of these students offer both agricultural science and business only and 14 offers both business and visual arts only.Assuming none of the students offers all the three subjects ,illustrate the information on a venn diagram

find the area of the shaded region.

1. y = x² − 4x,  y = 4x

2. y = 2x²,  y = 7x − 5x²

1) Equations for two lines L1 and L2 are given. Find the angle between L1 and L2.

L1: ? = 7 + 2?, ? = 8 − 4?, ? = −9 + ?

L2: ? = −8 − ?, ? = 3 − 3?, ? = 4 + 3?

2) Find polar form of complex number z :

?) ? = 4√3 − 4?

?) ? = 2√3 − 2i

Physiologically based pharmacokinetic (PBPK) models have been shown to be useful in predicting drug distribution throughout the body.

You will be examining the distribution of a model drug, Elheats. Elheats is broken down in the liver into Cirekram. Cirekram has many detrimental effects in the body, and therefore, it is desirable to track and document both compounds after ingestion of the drug.

ELHEATS DISTRIBUTION:

STOMACH:

Elheats is ingested orally, and you can approximate the starting compartment as the stomach itself-no Elheats is absorbed or metabolized between the mouth and the stomach.

The initial dose of interest is 25g/kg, which is consumed with 100mL of fluid. Assume the standard adult mass of 69.4kg, and neglect any other stomach contents.

Elheats dissolves into the fluid in the stomach very quickly (assume immediately), and thestomach can be considered to be well-mixed. The rate of stomach emptying is dependent

upon the volume of the stomach contents, with a rate constant ks= 0.045 min-1(this a first-order rate constant).

Hint: Since the concentration of Elheats in the stomach is constant, and it is only the rate at which it leaves that varies, try to end up with an equation to describe the

stomach contents volume at any time (dVs/dt).

GI TRACT:

The GI Tract is approximately 2.40L in total water volume. Elheats leaves the stomach and enters the GI tract. The GI tract also receives input from the blood (the CENTRAL

COMPARTMENT), which carries both Elheats and Cirekram. The volumetric flow rate of blood is 900mL/min. From the GI tract, it enters the first compartment of the liver (see

below). The volumetric flow rate of material leaving the GI tract to the liver is 900mL/min.

LIVER:

The liver is approximately 1.08L in total water volume. Elheats is metabolized intoCirekram in the liver, so unlike the remainder of the compartments, the liver cannot beconsidered to be well-mixed. To deal with this, consider the liver as 5 separate compartments. Material enters the liver (L1) from the GI tract (as above), where it is mixed with blood from the central compartment (entering the liver at 450mL/min).

One reaction occurs to metabolize Elheats into Cirekram. (For Deliverable 1, consider thisreaction to be a consumption/generation term that is dependent upon the concentration of Elheats and the concentration of Cirekram. Use r1(units of mass/(vol*time)).)

Another reaction occurs to metabolize Cirekram into an innocuous compound, whose distribution we are not interested in at this point. (For Deliverable 1, consider this reaction to be a consumption term that is dependent upon the concentration of Cirekram. Use r2 (units of mass/(vol*time)).)

The L1liver compartment empties at a volumetric flow rate of 1350mL/min into the L2 liver compartment, where the same reactions occur. The L2 liver compartment empties into the L3 liver compartment, at the same volumetric flow rate, and with the same reactions. This continues for the L4 compartment. For the L5 compartment, the output is directly to the bloodstream (the central compartment). Note that by separating the liver into 5 compartments, we are able to capture the spatial nature of the reaction rates and the dependence on concentration more accurately. Each liver compartment itself can be considered to be well-mixed without requiring the physiologically inaccurate assumption that the entire liver is well-mixed.

CENTRAL COMPARTMENT (BLOODSTREAM)

The central compartment is a catch-all for every other part of the human body (excluding the stomach, GI Tract, liver, and muscle), which we lump together with the carrying bloodstream to call the central compartment. The central compartment as defined here has a total water volume of 11.56L. The central compartment receives blood carrying Elheats and Cirekram from the L5 compartment of the liver (at a volumetric flow rate of 1350mL/min) and from the muscles (950mL/min), and delivers blood to the muscles (950mL/min), liver (450mL/min), and GI tract (900mL/min). No reactions occur in the central compartment.

MUSCLES

The muscles compartment has a total water volume of 25.76L. Although you are not required to do so here, you could decouple any entity from the central compartment in the

same way that the muscles are decoupled. The muscles receive Elheats and Cirekram from the central compartment at a volumetric flow rate of 950mL/min, and return the blood to

the central compartment after mixing well in the muscles at a rate of 950mL/min. There are no reactions that occur in the muscles.

ANSWER THE QUESTIONS BELOW

Problem 1A: From the information given above, create two schematics that show the

distribution paths of Elheats and Cirekram (one schematic each-they should be similar,but not identical). Your schematics may take any form, but you may wish to review Figure1 of Zhuang and Lu for an example. You may, however, find it easier to simplify their representation and draw a simple flow diagram that shows the path of distribution.

Problem 1B:

From the information given above, generate a list of ordinary differential equations that you would use to develop a PBPK model of Elheats and Cirekram. Submit the list of equations as well as all of your work to develop those equations. You may use either numerical values (with units) or parameters (e.g. Vc) in these equations. If you choose to use parameters, make sure to define the parameters somewhere easily accessible in your submission.

Hint: Label each compartment with a number or letter, and label each mass flow stream.Then use a system boundary around each compartment to calculate the concentration of Elheats or Cirekram (one equation for each) in that compartment. Recall that concentration is mass/volume, and volume is constant for each non-stomach compartment.

Find the maximum volume of a box inscribed in the tetrahedron bounded by the coordinate planes and the plane

x + 1/9y + 1/4z = 1

(A) Find the number c which satisfies the conclusion of the Mean Value Theorem for the function f(x) = x³ -2x on the interval [-2,2]

(B) Then use Calculus to find the critical numbers and intervals of increase and decrease for f(x) =(x²+x)^1/3