Write in words how to read each of the following out loud.a. {x∈R'| 0<x<1}b. {x∈R| x<=0 or x=>1}c. {n∈Z| n is a factor of 6}d. {n∈Z*| n is a factor of 6}

An adventurous archaeologist crosses between two rock cliffs by slowly going hand-over-hand along a rope stretched between the cliffs. He stops to rest at the middle of the rope (Fig. 1) The rope will break if the tension in it exceeds $4.50\cdot {10}^4$N and our hero's mass is 86.5 kg.

 Figure 1 a) If the angle $\theta$ is ${13.0}^{\circ }$, find the tension iin the rope.

b) What is the smallest value the angle $\theta$ can have if the rope is not to break?

For tests using a ballistocardiograph, a patient lies on a horizontal platform that is supported on jets of air. Because of the air jets, the friction impeding the horizontal motion of the platform is negligible. Each time the heart beats, blood is pushed out of the heart in a direction that is nearly parallel to the platform. Since momentum must be conserved, the body and the platform recoil, and this recoil can be detected to provide information about the heart. For each beat, suppose that 0.050 kg of blood is pushed out of the heart with a velocity of +0.30 m/s and that the mass of the patient and the platform is 85 kg. Assuming that the patient does not slip with respect to the platform, and that the patient and the platform start from rest, determine the recoil velocity.

An elevator (mass 4850 kg) is to be designed so that themaximum acceleration is 0.0600g. What is the maximum and the minimum forces the motor should exert on the supportingcable?
I already know how to find the maximum force (just multiplyingthe maximum acceleration by the mass), but I've got no clue as tohow to find the minimum acceleration when no minimum accelerationhas been found.

A rifle with a weight of 30 N fires a 5.0 g bullet with a speed of 300m/s. (a) Find the recoil speed of the rifle. (b) If a700 N man holds the rilfe firmly against his shoulder, find there coil speed of the man and his rifle.

Which transition in the hydrogen atom produces emitted light with the longest wavelength?
a. ${\displaystyle n=4\to n=3,}$
b. ${\displaystyle n=2\to n=1}$
c. ${\displaystyle n=3\to n=2}$

Prove that there is a positive integer that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive?

Determine whether or not the following transformation T : V \rightarrow W is a linear transformation. If T is not a linear transformation, provide a counter example. If it is, then: (i) find the nullspace N(T) and nullity of T, (ii) find the range R(T) and rank of T , (iii) determine if T is one-to-one, (iv) determine if T is onto.
${\displaystyle T:M_{n\times n}\left(R\right)\to RdefinedbyT\left(A\right)=det\left(A\right).}$